Annals of Mathematics, 150 (1999), 77–150

Classification of irreducible holonomies of torsion-free affine connections

By Sergei Merkulov and Lorenz Schwachhöfer

Contents

1. Introduction

2. Preliminary facts and results 2.1. Holonomy groups 2.2. Berger’s criteria 2.3. Real Berger algebras 2.4. Conformal representations 2.5. Review of representation theory 2.6. Homogeneous manifolds and vector bundles 2.7. The tangent bundle of a compact homogeneous manifold

3. Spencer cohomology and complex homogeneous manifolds 3.1. Spencer cohomology 3.2. Twistor formulae for Spencer cohomology groups

4. Holonomy groups associated with compact Hermitean-symmetric manifolds 5. Classification of Segre holonomies

6. Proof of the Main Theorem 6.1. Adjoint representations 6.2. Classification of complex Berger algebras 6.3. Skies of complex Berger algebras 6.4. Proof of the Main Theorem

7. Existence and supersymmetry of exotic holonomies 7.1. Existence theorem 7.2. Sypersymmetry

8. Classification of complex homoegeneous-rational manifolds X and ample line bundles L → X such that H^{1}(X, T X ⊗ L^{∗}) ≠ 0

1. Introduction

One of the most informative characteristics of an affine connection on a smooth connected manifold M is its holonomy group H. At any particular point p ∈ M one may represent H as the set of all linear automorphisms of the associated tangent space T_{p}M which are induced by parallel translation along p-based loops. If M is simply connected, then this set is known to be a Lie subgroup of End(T_{p}M). The notion of the holonomy group was introduced in the 1920s by Élie Cartan who used it to classify all Riemannian locally symmetric spaces [19]. This concept is widely used in many branches of mathematics and theoretical physics. One of the fundamental questions in differential geometry is the irreducible holonomy problem:

Which irreducible Lie subgroups G ⊂ End(V), where V is a real or complex finite dimensional vector space, can occur as the holonomy group of a torsion-free affine connection?

The requirement of torsion-freeness is imposed because of a result of Hano and Ozeki [23] which says that any (closed) Lie subgroup G ⊂ End(V) can occur as the holonomy of some affine connection (with torsion, in general).

If a connection is (locally) symmetric, then its holonomy group, if irreducible, equals essentially the (local) isotropy group in the transvection group. Hence, the holonomy problem for these connections is equivalent to the classification problem of symmetric spaces which was solved long ago [19], [9]. Therefore, throughout this paper we shall use the term holonomy group for an irreducible Lie subgroup G ⊂ End(V) which can occur as the holonomy of a torsion-free affine connection which is not locally symmetric.

The next milestone toward the solution of the holonomy problem was set in 1955 by Berger [8]. He gave necessary conditions which a holonomy group must satisfy, and then classified the groups satisfying these conditions. His classification was given in two parts.

The first part (“metric part”) consists of all holonomy groups which leave some symmetric bilinear form invariant. The connections with one of these holonomies are always Levi-Civita connections of a (pseudo-)Riemannian metric on M and have been studied extensively in the intervening years. In fact, it is by now well-known which of the entries in this list do actually occur as holonomy groups and how the holonomy relates to both the geometry and the topology of the underlying manifold. We summarize the possible metric holonomies in Table 1 which represents a culmination of efforts of many people (see e.g. [4], [10], [14], [16], [36] and the references cited therein).

The second part (“non-metric part”) was stated to contain all remaining holonomy groups, up to a finite number of missing terms. It is given in Table 2,

Table 1: Complete list of metric holonomies

Group G  Representation space  Group G Representation space
SO(p, q)  ℝ^{p+q}, p + q ⩾ 3  G_{2}  ℝ^{7}
SO(n, ℂ)  ℂ^{n} ≃ ℝ^{2n}, n ⩾ 3  G′_{2}  ℝ^{7}
U(p, q)  ℂ^{p+q} ≃ ℝ^{2(p+q)}, p + q ⩾ 2  Gℂ_{2}  ℝ^{14}
SU(p, q)  ℂ^{p+q} ≃ ℝ^{2(p+q)}, \{ \begin{matrix}p + q ⩾ 2 \\ (p, q) ≠ (1, 1) \end{matrix}  Spin(7)  ℝ^{8}
Sp(p, q) · Sp(1)  H^{p+q} ≃ ℝ^{4(p+q)}, p + q ⩾ 2  Spin(4, 3)  ℝ^{8}
Sp(p, q)  H^{p+q} ≃ ℝ^{4(p+q)}, p + q ⩾ 2  Spin(7, ℂ)  ℝ^{16}
Sp(n, ℝ) · SL(2, ℝ)  ℝ^{2} ⊗ ℝ^{2n} ≃ ℝ^{4n}, n ⩾ 2
Sp(n, ℂ) · SL(2, ℂ)  ℂ^{2} ⊗ ℂ^{2n} ≃ ℝ^{8n}, n ⩾ 2

modulo some slight modifications due to Bryant [16]. The holonomy groups which are missing from this list are called exotic holonomies. That the list of exotic holonomies is nonempty was established by Bryant [15]; several other (but still finitely many) exotic holonomies followed [16], [22], [21]. However, in [20], Chi et al. discovered an infinite family of exotic holonomies, thereby showing the incompleteness of the nonmetric part of Berger’s list.

In Table 3 we list all known exotic holonomies, including some new examples; that is, we have the following result.

Theorem A. (i) The representations of the groups

Sp(3, ℝ), Sp(3, ℂ), SL(6, ℝ), SU(1, 5), SU(3, 3), SL(6, ℂ), Spin(2, 10), Spin(6, 6), Spin(12, ℂ)

listed in Table 3 occur as holonomies of torsion-free affine connections.

(ii) Any torsion-free affine connection whose holonomy is one of those listed in (i) is analytic.

(iii) The moduli space of torsion-free affine connections whose holonomy is one of those listed in (i) is finite dimensional.

These properties are due to the fact that all connections with these holonomies are constructed from certain quadratic deformations of linear Poisson structures; this method was developed for the construction of the infinite family of exotic holonomies [20].

Table 2: Berger’s list of non-metric holonomies, modified by Bryant [16]

Group G  Representation V  Restrictions
T_{ℝ} · SL(n, ℝ)  \begin{matrix}ℝ & n \\ ⊙^{2}ℝ^{n} ≃ ℝ^{n(n+1)/2} \\ Λ^{2}ℝ^{n} ≃ ℝ^{n(n−1)/2} \end{matrix}  \begin{matrix}n ⩾ 2 \\ n ⩾ 3 \\ n ⩾ 5 \end{matrix}

T_{ℂ} · SL(n, ℂ)  \begin{matrix}ℂ^{n} ≃ ℝ^{2n} \\ ⊙^{2}ℂ^{n} ≃ ℝ^{n(n+1)} \\ Λ^{2}ℂ^{n} ≃ ℝ^{n(n−1)} \end{matrix}  \begin{matrix}n ⩾ 1 \\ n ⩾ 3 \\ n ⩾ 5 \end{matrix}
Tℝ · SL(n, ℂ)  {A ∈ M_{n}(ℂ) : A = A^{∗}} ≃ ℝ^{n^{2}}  n ⩾ 3
T_{ℝ} · SL(n, ℍ)  \begin{matrix}ℍ^{n} ≃ ℝ^{4n} \\ \{A ∈ M_{n}(ℍ) : A = −A^{∗}\} ≃ ℝ^{n(2n+1)} \\ \{A ∈ M_{n}(ℍ) : A = A^{∗}\} ≃ ℝ^{n(2n−1)} \end{matrix}  \begin{matrix}n ⩾ 1 \\ n ⩾ 2 \\ n ⩾ 3 \end{matrix}

\begin{matrix}Sp(n, ℝ) \\ Sp(n, ℂ) \end{matrix}  \begin{matrix}ℝ & 2n \\ ℂ^{2n} ≃ ℝ^{4n} \end{matrix}  \begin{matrix}n ⩾ 2 \\ n ⩾ 2 \end{matrix}
\begin{matrix}ℝ^{∗} · SO(p, q) \\ T_{ℂ} · SO(n, ℂ), T_{ℂ} ≠ 0 \end{matrix}  \begin{matrix}ℝ & p+q \\ ℂ^{n} ≃ ℝ^{2n} \end{matrix}  \begin{matrix}p + q ⩾ 3 \\ n ⩾ 3 \end{matrix}
Tℝ · SL(m, ℝ) · SL(n, ℝ)  ℝ^{m} ⊗ ℝ^{n} ≃ ℝ^{mn}  m > n ⩾ 2 or m ⩾ n > 2
T_{ℂ} · SL(m, ℂ) · SL(n, ℂ)  ℂ^{m} ⊗ ℂ^{n} ≃ ℝ^{2mn}  m > n ⩾ 2 or m ⩾ n > 2
Tℝ · SL(m, ℍ) · SL(n, ℍ)  ℍ^{m} ⊗ ℍ^{n} ≃ ℝ^{4mn}  m > n ⩾ 1 or m ⩾ n > 1
Notation: T_{𝔽} denotes any connected Lie subgroup of 𝔽^{∗},

M_{n}(𝔽) denotes the algebra of n × n matrices with entries in 𝔽.

Our main result is to show that these are the last entries missing from Berger’s list, thus completing the holonomy classification.

Main Theorem. If G ⊂ End(V) is an irreducible Lie subgroup which occurs as the holonomy of a torsion-free affine connection which is not locally symmetric, then G is one of the entries in Tables 1–3.

Moreover, due to the efforts of a number of people over the last 40 years (see the fundamental papers of Bryant [13]–[16], the most recent works on exotic holonomies [20], [21], the books [10], [36] and the references cited therein), all entries of Tables 1–3 are known to occur as holonomies, except for the 4-dimensional representations of H_{λ} · SU(2) and H_{λ} · SL(2, ℝ) which were discovered as candidates for holonomy groups by Bryant [16].

There is no apparent pattern in Tables 1–3. A certain pattern does emerge, however, when one switches from the usual representation of holonomy as a pair (Lie group G, representation space V) to the Borel-Weil picture (homogeneous manifold X, line bundle L → X) which is defined as follows. To any irreducible non-abelian real matrix subgroup G ⊂ End(V), one first associates, by complexifying G or complexifying both G and V, an irreducible complex matrix subgroup G_{ℂ} ⊂ End(V_{ℂ}). Then the projectivized G_{ℂ}-orbit X ⊂ ℙ(V_{ℂ}) of a maximal weight vector is well defined and is called the sky of G; define L = O(1) |_{X}. Clearly, for any initial G ⊂ End(V) the associated sky X is a compact complex homogeneous-rational manifold. Then the main theorem easily implies the following:

Corollary. Let G ⊂ End(V) be an irreducible (real or complex) Lie subgroup. If G occurs as the holonomy group of a torsion-free connection which is not locally symmetric, then its sky is biholomorphic to a compact hermiteansymmetric manifold.

Remarkably, the implication Main Theorem ⟹ Corollary can be reversed. This is a central idea of our approach to the solution of the holonomy problem. In a sense, this is a holomorphic analogue of Simons’ [37] approach to the classification of compact irreducible holonomies of Levi-Civita connections.

Using the above corollary as a working hypothesis we first were able to reproduce all the known holonomy groups in Tables 1–3 and second find new examples of Theorem A. This part of our work is explained in Sections 2-4 of the paper. Then using a combination of standard representation-theoretic methods (applicable in the picture (G, V)) with the methods of complex analysis and the Bott-Borel-Weil theorem (which are applicable in picture (X, L)) we showed that the working hypothesis gives rise to all possible holonomies, thereby completing the Berger classification (and giving, in particular, a new proof of the entire original Berger list). This second part of our work is explained in Sections 5 and 6 of the paper.

While working on the solution of the holonomy problem, we obtained the following classification result which is of interest on its own.

Theorem B. Let X be a compact complex homogeneous-rational manifold and L an ample line bundle on X. Then

(i) H^{0}(X, T X ⊗ L^{∗}) = \{ \begin{matrix}ℂ & for (X, L) = (ℂℙ_{1}, O(2)), \\ ℂ^{n} & for ,(X, L) = (ℂℙ_{n}, O(1)), n ⩾ 1 \\ 0 & otherwise. \end{matrix}

(ii) H^{1}(X, T X ⊗ L^{∗}) = 0 unless (X, L) is one of the entries in Table 4.

82 SERGEI MERKULOV AND LORENZ SCHWACHHÖFER

Table 3: List of exotic holonomies

group G representation V restrictions/remarks

T_{ℝ} · Spin(5, 5) T_{ℝ} · Spin(1, 9) T_{ℂ} · Spin(10, ℂ)

\begin{matrix}ℝ & 16 \\ ℝ & 16 \\ ℂ^{16} ≃ ℝ^{32} \end{matrix}

\begin{matrix}T_{ℝ} · E_{6}^{1} \\ T_{ℝ} · E_{6}^{4} \\ T_{ℂ} · Eℂ_{6} \end{matrix}  \begin{matrix}ℝ & 27 \\ ℝ & 27 \\ ℂ^{27} ≃ ℝ^{54} \end{matrix}

T_{ℝ} · SL(2, ℝ)

SL(2, ℂ) ℂ^{∗} · SL(2, ℂ) ℝ^{∗} · Sp(2, ℝ) ℂ^{∗} · Sp(2, ℂ)

ℝ^{∗} · SO(2) · SL(2, ℝ) ℂ^{∗} · SU(2) H_{λ} · SU(2) H_{λ} · SU(1, 1)

\begin{matrix}⊙^{3}ℝ^{2} ≃ ℝ^{4} \\ ⊙^{3}ℂ^{2} ≃ ℝ^{8} \\ ⊙^{3}ℂ^{2} ≃ ℝ^{8} \\ ℝ & 4 \\ ℂ^{4} ≃ ℝ^{8} \\ ℝ^{2} ⊗ ℝ^{2} ≃ ℝ^{4} \\ ℂ^{2} ≃ ℝ^{4} \\ ℂ^{2} ≃ ℝ^{4} \\ ℂ^{2} ≃ ℝ^{4} \end{matrix}

existence unknown existence unknown

SL(2, ℝ) · SO(p, q)  ℝ^{2} ⊗ ℝ^{p+q} ≃ ℝ^{2(p+q)}  p + q ⩾ 3
Sp(1) · SO(n, ℍ)  ℍ^{n} ≃ ℝ^{4n}  n ⩾ 2
SL(2, ℂ) · SO(n, ℂ)  ℂ^{2} ⊗ ℂ^{n} ≃ ℝ^{4n}  n ⩾ 3
E_{7}^{5}  ℝ^{56}
E_{7}^{7}  ℝ^{56}
Eℂ_{7}  ℝ^{112} ≃ ℂ^{56}
Sp(3, ℝ)  ℝ^{14} ⊂ Λ^{3}ℝ^{6}
Sp(3, ℂ)  ℝ^{28} ≃ ℂ^{14} ⊂ Λ^{3}ℂ^{6}
SL(6, ℝ)  ℝ^{20} ≃ Λ^{3}ℝ^{6}
SU(1, 5)  ℝ^{20}
SU(3, 3)  ℝ^{20}
SL(6, ℂ)  ℝ^{40} ≃ Λ^{3}ℂ^{6}
Spin(2, 10)  ℝ^{32}
Spin(6, 6)  ℝ^{32}
Spin(12, ℂ)  ℂ^{32} ≃ ℝ^{64}
Notation: T_{𝔽} denotes any connected Lie subgroup of 𝔽^{∗},
H_{λ} = {e^{(2πi+λ)t} | t ∈ ℝ} ⊆ ℂ^{∗}, λ > 0.

Table 4: The list of all (X, L) with H^{1}(X, T X ⊗ L^{∗}) ≠ 0
Aut^{0}X  (X, L)  H^{1}(X, T X ⊗ L^{∗})
× k (k ⩾ 4)  ⊙^{k−4}ℂ^{2}
A_{l} (l ⩾ 1)  × • 3 0  ℂ
• × • 0 2 0  ℂ
… × • • • × 1 0 0 0 1  ℂ
B_{l} (l ⩾ 3)  • • 0 0 × > 2  ℂ
… • ×  • • > • 2 0 0 0 0  ℂ
C_{l} (l ⩾ 2)  • × < 0 2  ℂ
… • × • • < • 0 1 0 0 0  ℂ

D_{l} (l ⩾ 4)  [figure: …; ×; •; •; •; •; •; •; 2; 0; 0; 0; 0; 0; 0]  ℂ

F_{4}  • • • × > 0 0 0 1  ℂ
A_{1} × A_{1}  × k ⨂ × 2 (k ⩾ 2)  ⊙^{k−2}ℂ^{2}
A_{1} × A_{l} (l ⩾ 1) × k ⨂  … × • • • 1 0 0 0 (k ⩾ 2) ⊙^{k−2}ℂ^{2} ⊗ ℂ^{l+1}

Notation: Aut^{0}X denotes the universal covering of the component, the identity of the group of all automorphisms of X

This theorem is a “nonvanishing” counterpart to the vanishing theorem of Kobayashi and Ochiai [27] which says that if X is a compact complex rational manifold and L → X a line bundle such that det(T X) ⊗ L^{∗} is ample, then H^{i}(X, T X ⊗ L^{∗}) = 0 for all i ⩾ 2.

Another by-product result of our work on the holonomy problem unveils a striking supersymmetry property of all complex symplectic representations and their real forms which occur as holonomies of torsion-free affine connections. With any symplectic Lie algebra 𝔤 ⊂ 𝔰𝔭(V), dim 𝔤 = m, dim V = 2n, one may associate naturally an (m|2n)-dimensional supermanifold M𝔤 ≃ 𝔤 ⊕ ΠV, Π being the parity change functor [29], which comes equipped with a SUSY (supersymmetry)-structure induced by the canonical projection ⊙^{2}V^{∗} → 𝔤.

The structure sheaf O_{M} has supersymmetry automorphisms which mix purely even and purely odd sectors; in physics literature such automorphisms are often called SUSY-translations.

Theorem C. Given an irreducible symplectic representation 𝔤 ⊂ 𝔰𝔭(V), the structure sheaf of the associated supermanifold M𝔤 contains 𝔤-invariant and SUSY-invariant polynomials of order two in odd coordinates and of order at least two in even coordinates if and only if 𝔤 occurs as the holonomy of a torsion-free affine connection which is not locally symmetric, i.e. 𝔤 is an entry of the following table:

Group G  Representation space  Group G  Representation space
Sp(n, ℝ)  ℝ^{2n}  E_{7}^{5}  ℝ^{56}
Sp(n, ℂ)  ℂ^{2n}  E_{7}^{7}  ℝ^{56}
SL(2, ℝ)  ℝ^{4} ≃ ⊙^{3}ℝ^{2}  Eℂ_{7}  ℂ^{56}
SL(2, ℂ)  ℂ^{4} ≃ ⊙^{3}ℂ^{2}  Spin(2, 10)  ℝ^{32}
SL(2, ℝ) · SO(p, q)  ℝ^{2(p+q)}, p + q ⩾ 3  Spin(6, 6)  ℝ^{32}
SL(2, ℂ) · SO(n, ℂ)  ℂ^{2n}, n ⩾ 3  Spin(12, ℂ)  ℂ^{32}
Sp(1) · SO(n, ℍ)  ℍ^{n} ≃ ℝ^{4n}, n ⩾ 2  Sp(3, ℝ)  ℝ^{14} ⊂ Λ^{3}ℝ^{6}
SL(6, ℝ)  ℝ^{20} ≃ Λ^{3}ℝ^{6}  Sp(3, ℂ)  ℂ^{14} ⊂ Λ^{3}ℂ^{6}
SU(1, 5)  ℝ^{20}
SU(3, 3)  ℝ^{20}
SL(6, ℂ)  ℂ^{20} ≃ Λ^{3}ℂ^{6}

A few words about our notation. A short exact sequence of sheaves

0⟶A_{1}⟶C⟶A_{2}⟶0

is written sometimes shortly as C = A_{1} + A_{2}. One may check that A_{1} + (A_{2} + A_{3}) = (A_{1} + A_{2}) + A_{3} so that the notation ∑_{i=1}^{n} A_{i} is non-ambigously defined (and should not be confused with ⊕_{i=1}^{n}A_{i}).

We also identify vector bundles with the associated locally free sheaves.

2. Preliminary facts and results

2.1. Holonomy groups. Let M be a smooth connected and simply connected n-manifold and ∇ an affine connection on M, i.e. a linear connection in the tangent bundle T M. Fix a point x ∈ M and let

L_{x} = {γ : [0, 1] → M | γ(0) = γ(1) = x}

be the set of piecewise smooth loops based at x. For γ ∈ L_{x}, denote by P_{γ} : T_{x}M⟶T_{x}M the linear automorphism induced by the ∇-parallel translations along γ.

The (restricted) holonomy of ∇ at x ∈ M is defined as a subset H_{x} := {P_{γ} | γ ∈ L_{x}} ⊂ GL(T_{x}M). Its basic properties are: (i) H_{x} is a connected Lie subgroup of GL(T_{x}M); (ii) if one fixes an isomorphism i : T_{x}M ≃ V, where V is any fixed vector space with dim V = dim M (typically, V = ℝ^{n}), then the conjugacy class of i(H_{x}) ⊂ GL(V) does not depend on the choice of x ∈ M (see, e.g., [10]). The holonomy group of ∇ is defined as any linear subgroup G ⊂ GL(V) in the conjugacy class of i(H_{x}) for some x ∈ M. The Lie algebra 𝔤 ⊂ 𝔤𝔩(V) of G is called the holonomy algebra of ∇.

2.2. Berger’s criteria. Let V be a vector space and 𝔤 ⊂ 𝔤𝔩(V) a Lie subalgebra. We shall now describe a necessary condition for 𝔤 to be the holonomy algebra of a torsion-free connection which is due to Berger [8]. We define the space of formal curvature maps

K(𝔤) := {R ∈ Λ^{2}V^{∗} ⊗𝔤 | R(x, y)z +R(y, z)x+R(z, x)y = 0 for all x, y, z ∈ V },

and the space of formal curvature derivatives

K^{1}(𝔤) := {ϕ ∈ V^{∗} ⊗ K(𝔤) | ϕ(x)(y, z) + ϕ(y)(z, x)
+ϕ(z)(x, y) = 0 for all x, y, z ∈ V }.

We also let \underline{𝔤} := \{R(x, y) | R ∈ K(𝔤), x, y ∈ V \} ⊂ 𝔤.

The equation defining the elements of K(𝔤) (K^{1}(𝔤) respectively) is called the first (second) Bianchi identity. The significance of these spaces is that for a torsion-free connection ∇ on M, the curvature R and its covariant derivative ∇R at a point x ∈ M lie in K(𝔥𝔬𝔩_{x}) and K^{1}(𝔥𝔬𝔩_{x}), respectively, where 𝔥𝔬𝔩_{x} ⊂ 𝔤𝔩(T_{x}M) is the holonomy algebra of ∇ at x.

Note that K(𝔤) and K^{1}(𝔤) are defined by the exact sequences

(1)  0 → K(𝔤) ⟶ Λ^{2}V^{∗} ⊗ 𝔤 ⟶ Λ^{3}V^{∗} ⊗ V

and

(2)  0 → K^{1}(𝔤) ⟶ V^{∗} ⊗ K(𝔤) ⟶ Λ^{3}V^{∗} ⊗ V,

where in each case, the last map is given by the composition of the natural inclusion and the skew-symmetrization map, i.e. Λ^{2}V^{∗}⊗𝔤 ↪ Λ^{2}V^{∗}⊗V^{∗}⊗V → Λ^{3}V^{∗} ⊗ V in the first and V^{∗} ⊗ K(𝔤) ↪ V^{∗} ⊗ Λ^{2}V^{∗} ⊗ 𝔤 → Λ^{3}V^{∗} ⊗ 𝔤 in the second case.

Definition 2.1. A Lie subalgebra 𝔤 ⊂ 𝔤𝔩(V) is called a Berger algebra if

1. \underline{𝔤} = 𝔤, and 2. K^{1}(𝔤) ≠ 0.

A Lie subgroup G ⊂ GL(V) is called a Berger group if its Lie algebra 𝔤 ⊂ 𝔤𝔩(V) is a Berger algebra.

In the literature, the two criteria in this definition are usually referred to as Berger’s first and second criterion. The following simple observation of Berger which is an immediate consequence of the Ambrose-Singer Holonomy Theorem [5] explains the relevance of this definition.

Proposition 2.2 [8]. Let G ⊂ GL(V) be an irreducible Lie subgroup which occurs as the holonomy group of a torsion-free affine connection on some manifold M which is not locally symmetric. Then G must be a Berger group, and its Lie algebra 𝔤 ⊂ 𝔤𝔩(V) must be a Berger algebra.

We shall often utilize the following simple result.

Lemma 2.3. If 𝔤 ⊂ 𝔤𝔩(V) is a Berger algebra, then K(𝔤) is a nontrivial 𝔤-module.

Proof. Without loss of generality, we may assume that dim V > 2. Suppose K(𝔤) is a trivial 𝔤-module. Then K^{1}(𝔤) ⊂ V^{∗} ⊗ K(𝔤) is a submodule and thus, since V is irreducible, we have K^{1}(𝔤) = V^{∗} ⊗ W for some subspace W ⊂ K(𝔤). Suppose there is a 0 ≠ R ∈ W. Pick independent elements x, y, z ∈ V such that R(x, y) ≠ 0, and define ϕ : V → W such that ϕ(x) = ϕ(y) = 0 and ϕ(z) = R. Then it follows that ϕ ∉ K^{1}(𝔤) which is a contradiction. Therefore, W = 0, i.e. K^{1}(𝔤) = 0, and thus 𝔤 is not Berger.

2.3. Real Berger algebras. In this subsection we shall use the following notation: If W is a complex vector space, then the Lie algebras of real and complex endomorphisms of W are denoted by End_{ℝ}(W) and End_{ℂ}(W), respectively.

Let V be a finite dimensional real vector space, and let 𝔥 ⊂ End_{ℝ}(V) be a real Lie subalgebra. We denote their complexifications by V_{ℂ} := V ⊗_{ℝ} ℂ and 𝔥_{ℂ} := 𝔥 ⊗_{ℝ} ℂ. Then obviously, 𝔥_{ℂ} ⊂ End_{ℂ}(V_{ℂ}), and by complexifying the exact sequences (1) and (2), we obtain

K(𝔥_{ℂ}) = K(𝔥) ⊗_{ℝ} ℂ and K^{1}(𝔥_{ℂ}) = K^{1}(𝔥) ⊗_{ℝ} ℂ.

In particular, 𝔥 ⊂ End_{ℝ}(V) is a (symmetric, respectively, nonsymmetric) Berger algebra if and only if 𝔥_{ℂ} ⊂ End_{ℂ}(V_{ℂ}) is.

Let us now assume that 𝔥 ⊂ End_{ℝ}(V) is irreducible. Then there are two cases to be distinguished.

First, suppose that 𝔥 is of real type, i.e. there is no complex structure on V which commutes with the elements of 𝔥. This happens if and only if 𝔥_{ℂ} ⊂ End_{ℂ}(V_{ℂ}) is also irreducible.

Second, suppose that 𝔥 is not of real type, i.e. there is a complex structure J on V which commutes with the elements of 𝔥. That is, 𝔥 ⊂ End_{ℂ}(V) with respect to this complex structure J. In this case, V_{ℂ} = W ⊕ \overline{W} decomposes into two irreducible 𝔥_{ℂ}-submodules of equal dimension given by

W = \{x + iJx | x ∈ V \} and \overline{W} = \{x − iJx | x ∈ V \}.

Let 𝔥_{1} := {A ∈ 𝔥 | JA ∈ 𝔥}. Then 𝔥_{1} < 𝔥, and J induces a complex Lie algebra structure on 𝔥_{1}; (𝔥_{1})_{ℂ} can be written as the direct sum of complex Lie algebras (𝔥_{1})_{ℂ} = 𝔥_{1}^{+} ⊕ 𝔥_{1}^{−} with

𝔥_{1}^{+} = {A + iJA | A ∈ 𝔥_{1}} and 𝔥_{1}^{−} = {A − iJA | A ∈ 𝔥_{1}}.

Let R ∈ K(𝔥_{ℂ}). Then for u, v ∈ W and \overline{w} ∈ \overline{W} the first Bianchi identity implies that R(u, v)\overline{w} = 0. Since this is true for all \overline{w} ∈ \overline{W}, it follows that R(u, v) ∈ 𝔥_{1}^{+}. Likewise, for \overline{u}, \overline{v} ∈ \overline{W}, we have R(\overline{u}, \overline{v}) ∈ 𝔥_{1}^{−}.

Next, for any R ∈ K(𝔥_{ℂ}) the first Bianchi identity also implies that R(\overline{u}, v)w = R(\overline{u}, w)v for all \overline{u} ∈ \overline{W}, v, w ∈ W. Thus, we have a map

\overline{W} ⟶ (𝔥_{ℂ}|_{W})^{(1)}, \overline{u} ⟼ R(\overline{u}, ).

If (𝔥_{ℂ}|_{W})^{(1)} = 0 then this implies that R(W, \overline{W}) = 0, and hence by the above, R(V_{ℂ}, V_{ℂ}) ⊂ 𝔥_{1}^{+} ⊕ 𝔥_{1}^{−} = (𝔥_{1})_{ℂ} for all R ∈ K(𝔥_{ℂ}); that is, \underline{𝔥}_{ℂ} ⊂ (𝔥_{1})_{ℂ}. Hence 𝔥_{ℂ} is not Berger unless 𝔥_{1} = 𝔥; i.e. 𝔥 is a complex Lie algebra which acts irreducibly on the complex vector space V.

We define a map ı : 𝔥_{ℂ} → End_{ℂ}(V) by

(3)  ı(A + iB) := A + JB.

In fact, it is easy to see that ı(𝔥_{ℂ}) ⊂ End_{ℂ}(V) is congruent to (𝔥_{ℂ})|_{W} ⊂ End_{ℂ}(W), and hence (𝔥_{ℂ}|_{W})^{(1)} = 0 if and only if (ı(𝔥_{ℂ}))^{(1)} = 0. Thus, we obtain the following.

Proposition 2.4. Let V be a finite dimensional real vector space, and let 𝔥 ⊂ End_{ℝ}(V) be an irreducible real subalgebra with complexification 𝔥_{ℂ} ⊂ End_{ℂ}(V_{ℂ}).

1. If 𝔥 is of real type, i.e. if there is no complex structure on V which commutes with the elements of 𝔥, then 𝔥 is a Berger algebra if and only if 𝔥_{ℂ} ⊂ End_{ℂ}(V_{ℂ}) is an irreducible Berger algebra.

2. If 𝔥 is not of real type, i.e. if there is a complex structure J on V which commutes with the elements of 𝔥, and if the subalgebra ı(𝔥_{ℂ}) ⊂ End_{ℂ}(V) given by (3) satisfies (ı(𝔥_{ℂ}))^{(1)} = 0, then 𝔥 is a Berger algebra if and only if J𝔥 = 𝔥 and 𝔥 ⊂ End_{ℂ}(V) is a complex irreducible Lie subalgebra.

Thus, in order to classify all Berger algebras we need to classify all irreducible complex Berger subalgebras 𝔥_{ℂ} ⊂ End_{ℂ}(V_{ℂ}), add all their real forms of real type, and finally, to investigate the real algebras 𝔥 ⊂ End_{ℂ}(V) for which ı(𝔥_{ℂ}) ⊂ End_{ℂ}(V) is one of the entries of Table 5. The latter task has been completed by Bryant [16]; hence, we shall mainly concern ourselves with the investigation of complex Berger algebras.

2.4. Conformal representations. In this subsection, we shall prove the following result.

Proposition 2.5. Let 𝔤 ⊂ 𝔰𝔬(V) be a proper irreducible Lie subalgebra on some finite dimensional euclidean vector space V with dim V ⩾ 5. If Hom𝔤(⊙^{2}Ad(𝔤), Λ^{2}V) = 0, i.e. if the 𝔤-modules ⊙^{2}Ad(𝔤) and Λ^{2}V have no irreducible summand in common, then K(𝔤 ⊕ ℂ Id) = K(𝔤). In particular, 𝔤 ⊕ ℂ Id is not a Berger algebra.

Proof. We identify 𝔰𝔬(V) and Λ^{2}V by the equation (x ∧ y) · z := (x, z)y − (y, z)x. It is well known that K(𝔰𝔬(V) ⊕ ℂ Id) = K(𝔰𝔬(V)) ⊕ K^{c}(V), where the subspace K^{c}(V) is isomorphic to 𝔰𝔬(V) via

R_{A} := (A, ) Id + ad(A) for all A ∈ 𝔰𝔬(V).

Here, ( , ) denotes a suitable multiple of the Killing form on 𝔤. Moreover, it is known that K(𝔰𝔬(V)) ⊂ ⊙^{2}(Ad(𝔰𝔬(V))), and therefore, K(𝔤) ⊂ ⊙^{2}(Ad(𝔤)).

By hypothesis, K(𝔤) and K^{c}(V) ≅ Λ^{2}V have no irreducible summand in common, and hence

K(𝔤 ⊕ ℂ Id) = K(𝔤) ⊕ [K^{c}(V) ∩ K(𝔤 ⊕ ℂ Id)].

Let

𝔥 := {A ∈ 𝔰𝔬(V) | R_{A} ∈ K^{c}(V)∩K(𝔤⊕ℂ Id)} = {A ∈ 𝔰𝔬(V) | [A, 𝔰𝔬(V)] ⊂ 𝔤}.

It is then straightforward to show that 𝔥 ⩽ 𝔰𝔬(V). But dim(V) ⩾ 5, thus 𝔰𝔬(V) is simple, and obviously, 𝔥 ≠ 𝔰𝔬(V) if 𝔤 is a proper subalgebra. Thus, 𝔥 = 0, and this completes the proof.

Corollary 2.6. Let 𝔤 ⊂ 𝔤𝔩(V) be one of the following subalgebras:

𝔤_{2}^{ℂ} ⊂ 𝔤𝔩(ℂ^{7}), 𝔰𝔭𝔦𝔫(7, ℂ) ⊂ 𝔤𝔩(ℂ^{8}), 𝔰𝔭𝔦𝔫(9, ℂ) ⊂ 𝔤𝔩(ℂ^{16}).

Then 𝔤 ⊕ ℂ Id ⊂ 𝔤𝔩(V) is not a Berger algebra.

Proof. It is well-known that all these representations are metric. The decomposition of the 𝔤-modules ⊙^{2}(Ad(𝔤)) and Λ^{2}V is straightforward and yields that Proposition 2.5 applies in each case.

2.5. Review of representation theory [7], [24]. Let 𝔤 be a semisimple complex Lie algebra and G the associated simply connected Lie group. Fix a maximally Abelian self-normalizing subalgebra 𝔱 ⊂ 𝔤 (any two such subalgebras, called Cartan subalgebras, are conjugate under the adjoint action of G). If ρ : 𝔤 → 𝔤𝔩(V) is a representation of 𝔤 in a complex vector space V, then with any ω ∈ 𝔱^{∗} ≡ Hom_{ℂ}(𝔱, ℂ) one may associate the weight space of V by V_{ω} = {v ∈ V : ρ(h)v = ω(h)v for all h ∈ 𝔱}. An element ω ∈ 𝔱^{∗} is called a weight of V if V_{ω} ≠ 0.

In the particular case when V = 𝔤 and ρ : 𝔤 → 𝔤𝔩(𝔤) is the adjoint representation of 𝔤 on itself, the nonzero weights of 𝔤 are called the roots of 𝔤. Thus

𝔤 = 𝔱 ⊕ ∑_{α∈Φ} 𝔤_{α}

where Φ is the set of all roots of 𝔤 and all sums are direct. A subset Δ = {α_{1}, …, α_{r}} ⊂ Φ with the property that every ω ∈ Φ may be expressed as a linear combination ω = ∑_{i=1}^{r} a_{i}α_{i} with all a_{i} being nonnegative or all nonpositive integers is called a system of simple roots of 𝔤. Such a Δ exists, and any two such sets are conjugate under the adjoint action of G. If we fix Δ, then Φ = Φ^{+} ∪ Φ^{−}, where Φ^{+} = {ω ∈ Φ : ω = ∑_{i=1}^{r} a_{i}α_{i} with a_{i} ⩾ 0} is the set of positive roots and Φ^{−} = {ω ∈ Φ : ω = ∑_{i=1}^{r} a_{i}α_{i} with a_{i} ⩽ 0} is the set of negative roots (both with respect to Δ).

For any root α ∈ Φ^{+} there is a unique element H_{α} in [𝔤_{α}, 𝔤_{−α}] ⊂ 𝔱 such that α(H_{α}) = 2. If Δ = {α_{1}, …, α_{r}} is the set of simple roots, then the associated set {H_{α_{1}}, …, H_{α_{r}}} forms a basis of 𝔤. Its dual basis {ω_{α_{1}}, …, ω_{α_{r}}} of 𝔱^{∗} is called the set of fundamental weights. One may use it to define the following three important subsets of 𝔱^{∗}: the set of (integral) weights Λ = {λ ∈ 𝔱^{∗} : ω = ∑_{i=1}^{r} λ_{i}ω_{i} with λ_{i} ∈ ℤ}; the set of dominant weights Λ^{+} = {λ ∈ Λ : λ = ∑_{i=1}^{r} λ_{i}ω_{i} with λ_{i} ⩾ 0}; and the set of strongly dominant weights Λ^{++} = {λ ∈ Λ^{+} : λ = ∑_{i=1}^{r} λ_{i}ω_{i} with λ_{i} > 0}. Note that λ_{i} = λ(H_{α_{i}}). The minimal integral element ω_{1}+ω_{2}+…+ω_{r} in Λ^{++} is denoted by η. Any integral weight λ of 𝔤 can be graphically represented by inscribing the integer λ_{i} over the i^{th} node of the Dynkin diagram for 𝔤. For example, the fundamental weight

ω_{1} of 𝔰𝔩(3, ℂ) is [figure: 1; 0] .

Let λ ∈ Λ be an integral weight. It is called singular if λ(H_{α}) = 0 for some α ∈ Φ^{+}, and regular otherwise. The index of λ is defined to be the number of positive roots α for which λ(H_{α}) < 0 holds; it is denoted by ind(λ).

If ρ : 𝔤 → 𝔤𝔩(V) is an irreducible representation of 𝔤, then there exists a unique weight ω(V) ∈ Λ^{+} of V, called the highest weight of V (relative to 𝔱 and Δ) such that dim V_{ω} = 1 and ρ(𝔤_{α})V_{ω} = 0 for all α ∈ Δ. This establishes a one-to-one correspondence, V ⇔ ω(V), between finite-dimensional irreducible 𝔤-modules and dominant weights, and allows us to use the graphical description of ω(V) to represent ρ : 𝔤 → 𝔤𝔩(V). For example, the standard representation

of 𝔰𝔩(3, C) in ℂ^{3} is denoted by [figure: 1; 0] .

If 𝔤 is simple, then the adjoint representation ρ : 𝔤 → 𝔤𝔩(𝔤) is irreducible. The associated highest weight of V = 𝔤 is a root μ ∈ Λ^{+} which is called the maximal root of 𝔤. The following is the list of all maximal roots [7], [24]:

(4)

[figure: 2; >; 1; 0; 0; 0]
[figure: …; 1; 0; 0; 0; 1; 0; 0; 0; 0; 0; 1]

[figure: …; >; 0; 1; 0; 0; 0] (⩾ 3 nodes)  [figure: 1; 0; 0; 0; 0; 0; 0]

[figure: …; <; 2; 0; 0; 0; 0] (⩾ 2 nodes)  [figure: 0; 0; 0; 0; 0; 0; 1; 0]

[figure: …; 0; 1; 0; 0; 0; 0; 0] (⩾ 4 nodes)  [figure: >; 1; 0] .

For any simple root α_{i} ∈ Δ, denote by σ_{i} the reflection in the hyperplane perpendicular to α_{i}. The Weyl group W of 𝔤 is the group generated by all the simple reflections σ_{i}. If 𝔤 is simple then W acts irreducibly on 𝔱, and moreover, there is a W -invariant inner product ( , ) on 𝔱 (the so-called Killing form). Given a weight λ ∈ Λ and a root α of 𝔤, the Cartan number, which is known to be an integer, is given by

⟨λ, α⟩ = \frac{2(λ, α)}{(α, α)} ∈ ℤ.

Note that ⟨ , ⟩ is linear in the first entry only. Its significance is the following. If λ occurs as the weight of an irreducible representation of 𝔤 and ⟨λ, α⟩ > 0 (⟨λ, α⟩ < 0, respectively) then λ − kα (λ + kα, respectively) is also a weight of that representation for k = 1, … | ⟨λ, α⟩ |. For the adjoint representation, it is well-known that | ⟨α, β⟩ | ⩽ 3, and | ⟨α, β⟩ | = 3 occurs if and only if 𝔤 contains 𝔤_{2} as a direct summand.

The action of the simple reflection σ_{i} on a weight λ ∈ Λ can be described by the following rule [7]: to compute σ_{i}(λ), let c = λ(H_{α_{i}}) be the coefficient of the node associated to α_{i}; add c to the adjacent coefficients, with multiplicity if there is a multiple edge directed towards the adjacent node, and then replace c by −c. For example

[figure: >; a; b; c; d; ⟶^{σ_{2}}; >; a+b; -b; c+2b; d]
[figure: >; a; b; c; d] ⟶^{σ_{3}}  [figure: >; a; b+c; -c; d+c] .

For any w ∈ W, there exists a minimal integer l(w) such that w can be expressed as a composition of l(w) simple reflections. This integer is called the length of w.

2.6. Homogeneous manifolds and vector bundles. A maximal solvable subalgebra of a semisimple Lie algebra 𝔤 is called a Borel subalgebra. A subalgebra 𝔭 ⊂ 𝔤 is called parabolic if it contains a Borel subalgebra. Every Borel subalgebra is G-conjugate to the standard one,

𝔟 := 𝔱 ⊕ 𝔫

where 𝔫 := ∑_{α∈Φ^{+}} 𝔤_{α}. There is a standard form 𝔭 for any parabolic subalgebra as well. Let Δ𝔭 be a subset of Δ and let Φ𝔭^{+} = span {Δ𝔭} ∩ Φ^{+}. Then

𝔭 = 𝔱 + 𝔫 + ∑_{α∈Φ𝔭^{+}} 𝔤_{−α}

is the standard parabolic subalgebra of 𝔤. A useful notation for a standard parabolic 𝔭 ⊂ 𝔤 (and for the associated subgroup P ⊂ G) is to cross all nodes in the Dynkin diagram for 𝔤 which correspond to simple roots of 𝔤 in Δ \ Δ𝔭.

It is well-known that any compact complex homogeneous-rational manifold X is isomorphic to the quotient space G/P, where G is a simply connected Lie group and P ⊂ G is a parabolic subgroup. It is then very useful to denote X by the same Dynkin diagram as 𝔭, the Lie algebra of P. For example, the odd dimensional quadric Q_{2n−1} is denoted by [figure: …; ×; >] .

The number of crossed nodes in the Dynkin diagram for X is called the rank of X and is denoted by rankX. This number is independent of the representation of X as a quotient G/P.

A vector bundle E → X = G/P is called G-homogeneous if there is a holomorphic representation ρ : P → GL(V) such that E = G ×_{ρ} V; i.e., E is the quotient G × V/P, where every p ∈ P acts on G × V as follows

G × V ⟶ G × V
(g, v) ⟶ (g · p, ρ(p^{−1})v).

If ρ : P → GL(V) is irreducible, then E is said to be irreducible as well.

The finite-dimensional irreducible representations of P are in one-to-one correspondence with integral weights λ ∈ Λ whose Dynkin diagram has nonnegative coefficients over the uncrossed nodes for 𝔭. A useful notation for an irreducible homogeneous vector bundle E → X is to combine the Dynkin diagram for the associated integral weight λ with the Dynkin diagram for 𝔭 into

one picture. For example, if X = × is the projective plane ℂℙ_{2}, then

O(−1) = × -1 0 and T X =× 1 1 .

The cohomology ring H^{∗}(X, E) of an irreducible homogeneous vector bundle E → X with integral weight λ ∈ Λ can be computed, according to Bott [11], as follows:

(i) If λ + η is singular, then H^{∗}(X, E) = 0;

(ii) If λ+η is regular and if ind(λ+η) = p, then there is a unique element σ_{λ} (of length p) in the Weyl group of Φ such that σ_{λ}(λ + η) ∈ Λ^{++}. Then H^{∗}(X, E) = H^{p}(X, E) and H^{p}(X, E) is an irreducible 𝔤-module whose highest weight is σ_{λ}(λ + η) − η.

2.7. The tangent bundle of a compact homogeneous manifold. Let X = G/P be a compact complex homogeneous-rational manifold. The associated number |Δ\Δ𝔭| of crossed nodes does not depend on the representation of X as the quotient G/P of a semisimple complex Lie group G by a parabolic subgroup P; i.e., if G′ is another semisimple complex Lie group acting transitively on X, X = G′/P′, then |Δ \ Δ𝔭| = |Δ′ \ Δ𝔭′^{′}|; this number is called the rank of X.

The tangent bundle T X is the homogeneous vector bundle on X associated with the natural representation of 𝔭 on 𝔤/𝔭. Since 𝔭 is never semisimple, T X is not, in general, decomposable into a direct sum of irreducible homogeneous vector bundles. Nevertheless, there is a simple recipe to exhibit the weight structure of T X which works a follows. First define the map

π : Φ^{+} \ Φ𝔭^{+}⟶ℤ^{rankX}

by taking coefficients of positive roots from Φ^{+} \ Φ𝔭^{+} over crossed nodes. Let Π = Imπ. Define the height of a positive root α ∈ Φ^{+} to be the sum of all coefficients of α in the basis Δ. Define the section of π

\begin{aligned}α : & Π & ⟶ & Φ^{+} \ Φ𝔭^{+} \\ & n & ⟶ & α(n) \end{aligned}

by the condition that α(n) is the root in the fibre π^{−1}(n) which has the maximal height. By considering root strings, one easily obtains the following decomposition [7]:

(5)  T X = ∑_{i⩾1} [ ⨁_{|n|=i} J(α(n))] .

where J(λ) denotes the irreducible homogeneous bundle on X with the highest weight λ and the meaning of ∑ is as explained at the end of the introduction.

If 𝔮 ⊂ 𝔭 are parabolic subalgebras of a semisimple Lie algebra 𝔤, then there is a natural fibration

X = G/Q⟶X_{0} = G/P

whose fibre X_{v} can be identified by the following rule [7]: delete from the Dynkin diagram for 𝔮 all crossed nodes (and incident) edges shared with 𝔭 and then delete all connected components with no crossed through nodes.

For example, the fibration × × ⟶ [figure: ×] has the fibre ℙ^{2} = × . If ν : X → X_{0} is the fibration of the above type, then

(6)  T X = T_{v} + ν^{∗}(T X_{0})

where T_{v} is the homogeneous vector bundle on X whose restriction to the fibre X_{v} of ν is isomorphic to T X_{v}. Any ample line bundle L → X can be uniquely factored as L_{v} ⊗ ν^{∗}(L_{0}), where L_{0} is an ample line bundle on X_{0} and the line bundle L_{v} → X restricted to the fibers of ν is an ample line bundle on X_{v}.

Then

(7)  ⊙^{2}T X ⊗ L^{∗2} = ⊙^{2}T_{v} ⊗ L^{∗2} + (T_{v} ⊗ L_{v}^{∗2}) ⊗ ν^{∗}(T X_{0} ⊗ L_{0}^{∗2})
+ L_{v}^{∗2} ⊗ ν^{∗}(⊙^{2}T X_{0} ⊗ L_{0}^{∗2})

and

(8) ⊙^{3}T X ⊗ L^{∗2} = ⊙^{3}T_{v} ⊗ L^{∗2} + (⊙^{2}T_{v} ⊗ L_{v}^{∗2}) ⊗ ν^{∗}(T X_{0} ⊗ L_{0}^{∗2})
+ (T_{v} ⊗ L_{v}^{∗2}) ⊗ ν^{∗}(⊙^{2}T X_{0} ⊗ L_{0}^{∗2})
+ L_{v}^{∗2} ⊗ ν^{∗}(⊙^{3}T X_{0} ⊗ L_{0}^{∗2}).

If rankX ⩾ 2, then there is always a fibration X → X_{0} of the above type whose fibre X_{v} is either a projective space ℙ_{n} or a Grassmanian G(k; ℂ^{n+1}), k ⩾ 2. We shall find very useful the following simple statement.

Lemma 2.7. (i) If X_{v} = ℙ_{n} with n ⩾ 2, then
(9) H^{1}(X, ⊙^{2}T X ⊗ L^{∗2}) ⊂ H^{1}(X, ⊙^{2}T_{v} ⊗ L^{∗2}),
(10) H^{1}(X, ⊙^{3}T X ⊗ L^{∗2}) ⊂ H^{1} (X, ⊙^{3}T_{v} ⊗ L^{∗2})

⊕ H^{1} (X_{0}, ν_{∗}^{0}(⊙^{2}T_{v} ⊗ L_{v}^{∗2}) ⊗ T X_{0} ⊗ L_{0}^{∗2}) .

In particular, if L_{v}|_{X_{v}} = O(k) with k ⩾ 2, then

H^{1}(X, ⊙^{2}T X ⊗ L^{∗2}) = H^{1}(X, ⊙^{3}T X ⊗ L^{∗2}) = 0.
(ii) If X_{v} = G(k, ℂ^{n+1}) with (k, n) ≠ (3, 6) or (4, 8), then
(11)  H^{1}(X, ⊙^{2}T X ⊗ L^{∗2}) = H^{1}(X, ⊙^{3}T X ⊗ L^{∗2}) = 0.

Proof. The statements easily follow from (7), (8), the Leray spectral sequence and the fact that for X_{v} = ℙ_{n} with n ⩾ 2 one has, for any ample line bundle L_{v} → X_{v},

H^{1}(X_{v}, ⊙^{2}T X_{v} ⊗ L_{v}^{∗2}) = H^{i}(X_{v}, L_{v}^{∗2}) = H^{i}(X_{v}, T X_{v} ⊗ L_{v}^{∗2}) = 0, i = 0, 1,

while for (X_{v}, L_{v}) = (ℙ_{n}, O(k)), n, k ⩾ 2 and for X_{v} = G(k, ℂ^{n+1}) with (k, n) ≠ (3, 6) or (4, 8) one has, in addition to the above, the vanishing of the following cohomology groups:

H^{0}(X_{v}, ⊙^{2}T X_{v} ⊗ L_{v}^{∗2}) = H^{i}(X_{v}, ⊙^{3}T X_{v} ⊗ L_{v}^{∗2}) = 0, i = 0, 1.

3. Spencer cohomology and complex homogeneous manifolds

3.1. Spencer cohomology. Let V be a vector space and 𝔤 a Lie subalgebra of 𝔤𝔩(V) := V ⊗ V^{∗}. Define recursively the 𝔤-modules

𝔤^{(−1)} = V
𝔤^{(0)} = 𝔤
𝔤^{(k)} = [𝔤^{(k−1)} ⊗ V^{∗}] ∩ [V ⊗ ⊙^{k+1}V^{∗}] , k = 1, 2, …,

and define the map

∂ : 𝔤^{(k)} ⊗ Λ^{l−1}V^{∗}⟶𝔤^{(k−1)} ⊗ Λ^{l}V^{∗}

as the antisymmetrisation over the last l indices. Here and elsewhere the symbols ⊙^{k} and Λ^{k} stand for k^{th} order symmetric and antisymmetric powers, respectively.

Since ∂^{2} = 0, there is a complex

𝔤^{(k)} ⊗ Λ^{l−1}V^{∗} ⟶^{∂} 𝔤^{(k−1)} ⊗ Λ^{l}V^{∗} ⟶^{∂} 𝔤^{(k−2)} ⊗ Λ^{l+1}V^{∗}

whose cohomology at the center term is denoted by H^{k,l}(𝔤) and is called the (k, l) Spencer cohomology group. In particular,

(12)  H^{k,1}(𝔤) = 0,
H^{k,2}(𝔤) = \frac{Ker : 𝔤^{(k−1)} ⊗ Λ^{2}V^{∗} →^{∂} 𝔤^{(k−2)} ⊗ Λ^{3}V^{∗}}{Image : 𝔤^{(k)} ⊗ V^{∗} →^{∂} 𝔤^{(k−1)} ⊗ Λ^{2}V^{∗}} .

The 𝔤-module H^{k,2}(𝔤) has the following geometric meaning: if G is a matrix Lie group whose Lie algebra is 𝔤 and G → M is a G-structure on a manifold M which is infinitesimally flat to k^{th} order, then the obstruction for G to be infinitesimally flat to (k + 1)^{th} order is given by a section of the associated vector bundle G ×_{G} H^{k,2}(𝔤).

The 𝔤-module 𝔤^{(1)} has a clear geometric interpretation as well. If a Gstructure G → M is infinitesimally flat to 1st order (which is equivalent to saying that G admits a torsion-free affine connection), then the set of all torsionfree affine connections in G is an affine space modelled on the vector space H^{0}(M, G ×_{G} 𝔤^{(1)}). In particular, if G ⊂ GL(V) is such that 𝔤^{(1)} = 0, then any G-structure admits at most one torsion-free affine connection. If K(𝔤) denotes the 𝔤-module of formal curvature tensors of 𝔤 defined in subsection 2.2. then

H^{1,2}(𝔤) = \frac{K(𝔤)}{∂(𝔤^{(1)} ⊗ V^{∗})};

i.e., the cohomology group H^{1,2}(𝔤) represents the part of K(𝔤) which is invariant under 𝔤^{(1)}-valued shifts in a formal torsion-free affine connection with holonomy in 𝔤. For example, if (G, V) = (CO(n, ℝ), ℝ^{n}), then 𝔤^{(1)} = V^{∗} and H^{1,2}(𝔤) is the vector space of formal Weyl tensors. Note that K(𝔤) fits into the exact sequence

𝔤^{(1)} ⊗ V^{∗} ⟶ K(𝔤) ⟶ H^{1,2}(𝔤) ⟶ 0.

In particular, if both 𝔤^{(1)} and H^{1,2}(𝔤) vanish, then K(𝔤) = 0 and hence, 𝔤 is not Berger.

If 𝔤^{(1)} = 0, then H^{1,2}(𝔤) is exactly K(𝔤), the 𝔤-module which plays a key role in the theory of torsion-free affine connections with holonomy in 𝔤. The case 𝔤^{(1)} = 0 is generic — there are very few irreducibly acting Lie subgroups 𝔤 ⊂ 𝔤𝔩(V) which have 𝔤^{(1)} ≠ 0. For future reference we list in Table 5 all complex irreducible Lie subgroups G ⊂ GL(V) with 𝔤^{(1)} ≠ 0 which is due to Cartan [17] and Kobayashi and Nagano [26]. We emphasize that our classification result also yields the classification of those irreducibly acting subalgebras 𝔤 ⊂ 𝔤𝔩(V) with H^{1,2}(𝔤) ≠ 0.

Table 5: The list of all irreducible complex matrix Lie groups G with 𝔤^{(1)}≠0

group G  representation V  𝔤^{(1)}
SL(n, ℂ)  V = ℂ^{n}, n ⩾ 2  (V ⊗ ⊙^{2}V^{∗})_{0}
GL(n, ℂ)  V = ℂ^{n}, n ⩾ 1  V ⊗ ⊙^{2}V^{∗}
GL(n, ℂ)  V ≃ ⊙^{2}ℂ^{n}, n ⩾ 2  V^{∗}
GL(n, ℂ)  V ≃ Λ^{2}ℂ^{n}, n ⩾ 5  V^{∗}
GL(m, ℂ) · GL(n, ℂ) V ≃ ℂ^{m} ⊗ ℂ^{n}, m, n ⩾ 2  V^{∗}
Sp(n, ℂ)  V = ℂ^{2n}, n ⩾ 2  ⊙^{3}V^{∗}
ℂ^{∗} · Sp(n, ℂ)  V = ℂ^{2n}, n ⩾ 2  ⊙^{3}V^{∗}
CO(n, ℂ)  V = ℂ^{n}, n ⩾ 5  V^{∗}
ℂ^{∗} · Spin(10, ℂ)  V = ℂ^{16}  V^{∗}
ℂ^{∗} · E_{6}^{ℂ}  V = ℂ^{27}  V^{∗}

Unless otherwise explicitly stated, ℤ_{ℂ} denotes in what follows either a trivial group or the multiplicative group ℂ^{∗} and z_{ℂ} denotes the Lie algebra of ℤ_{ℂ}.

3.2. Twistor formulae for Spencer cohomology groups. Let V be a finite dimensional complex vector space and G ⊂ GL(V) an irreducible representation of a reductive complex Lie group in V. Then G also acts irreducibly in V^{∗} via the dual representation. Let X̃ be the G-orbit of a highest weight vector in V^{∗} \ 0. Then the quotient X := X̃/ℂ^{∗} is a compact complex homogeneousrational manifold called the sky of 𝔤. There is a commutative diagram

\begin{matrix}X̃ & ↪ & V^{∗} \ 0 \\ ↓ & & ↓ \\ X & ↪ & ℙ(V^{∗}) \end{matrix} .

In fact, X = G_{s}/P, where G_{s} is the semisimple part of G and P is the parabolic subgroup of G_{s} leaving a highest weight vector in V^{∗} invariant up to a scale. Let L be the restriction of the hyperplane section bundle O(1) on ℙ(V^{∗}) to the submanifold X. Clearly, L is an ample homogeneous line bundle on X.

In summary, there is a natural map

(G, V)⟶(X, L)

which associates with an irreducibly acting reductive Lie group G ⊂ GL(V) a pair (X, L) consisting of a compact complex homogeneous-rational manifold X and an ample line bundle L on X. We call (X, L) the Borel-Weil data associated with (G, V).

Can this map be reversed? According to Borel-Weil, the representation space V can be reconstructed very easily:

V = H^{0}(X, L).

What about G? According to Onishchik, with a few (but notable) exceptions, G can be reconstructed as well.

Fact 3.1 [2]. Assume that G is simple. The Lie algebra of G is isomorphic to H^{0}(X, T X) unless one of the following holds:

(i) G is the representation of Sp(n, ℂ) in ℂ^{2n} in which case H^{0}(X, T X) ≃ 𝔰𝔩(2n, ℂ);

(ii) G is the representation of G_{2} in ℂ^{7} in which case H^{0}(X, T X) ≃ 𝔰𝔬(7, ℂ);

(iii) G is the fundamental spinor representation of Spin(2n + 1, ℂ) in which case H^{0}(X, T X) ≃ 𝔰𝔭𝔦𝔫(2n + 2, ℂ).

Another proof of this fact is given in [39]. Therefore, if G ⊂ GL(V) is semisimple then, with few exceptions, G can be reconstructed from (X, L). However, it is often undesirable to restrict oneself to semisimple groups only (especially in the context of the holonomy classification problem). There is a natural central extension of the Lie algebra H^{0}(X, T X):

Fact 3.2. For any (X, L), 𝔤 := H^{0}(X, L ⊗ (J^{1}L)^{∗}) is a reductive Lie algebra canonically represented in H^{0}(X, L). This fact is easy to explain —

H^{0}(X, L ⊗ (J^{1}L)^{∗}) is exactly the Lie algebra of the Lie group G of all global biholomorphisms of the line bundle L which commute with the projection L → X.

In summary, with a given irreducible representation G ⊂ GL(V) there is canonically associated a pair (X, L) consisting of a compact complex homogeneous-rational manifold X and a very ample line bundle on X such that much of the original information about G can be restored from (X, L). For our purposes the crucial observation is that the 𝔤-modules 𝔤^{(k)} and H^{k,2}(𝔤) also admit a simple description in terms of (X, L).

Theorem 3.3. For a compact complex homogeneous manifold X and a very ample line bundle L on X, there is an isomorphism

𝔤^{(k)} = H^{0} (X, L ⊗ ⊙^{k+1}N^{∗}) , k = 0, 1, 2, …,

and an exact sequence of 𝔤-modules,

0⟶H^{k,2}(𝔤)⟶H^{1} (X, L ⊗ ⊙^{k+2}N^{∗})
⟶H^{1} (X, L ⊗ ⊙^{k+1}N^{∗}) ⊗ V^{∗}, k = 1, 2, …,

where 𝔤 := H^{0}(X, L ⊗ N^{∗}), N := J^{1}L, and H^{k,2}(𝔤) are the Spencer cohomology groups associated with the canonical representation of 𝔤 in the vector space V := H^{0}(X, L).

Proof. Since L is very ample, there is a natural “evaluation” epimorphism

V ⊗ O_{X} → J^{1}L → 0

whose dualization gives rise to the canonical monomorphism 0 → N^{∗} → V^{∗} ⊗ O_{X}. Then one may construct the following sequences of locally free sheaves,

(13)  0 ⟶ L ⊗ ⊙^{k+1}N^{∗}⟶L ⊗ ⊙^{k}N^{∗} ⊗ V^{∗}⟶L ⊗ ⊙^{k−1}N^{∗} ⊗ Λ^{2}V^{∗}

and

0 ⟶ L ⊗ ⊙^{k+2}N^{∗}⟶L ⊗ ⊙^{k+1}N^{∗} ⊗ V^{∗}⟶L ⊗ ⊙^{k}(N^{∗}) ⊗ Λ^{2}V^{∗} (14)

⟶ L ⊗ ⊙^{k−1}N^{∗} ⊗ Λ^{3}V^{∗},

and verify that both are exact. (Hint: For any vector space W one has W ⊗ Λ^{2}W mod Λ^{3}W ≃ W ⊗ ⊙^{2}W mod ⊙^{3}W.)

Then computing H^{0}(X, …) of (13) and using the inductive definition of 𝔤^{(k)} one easily obtains the first statement of the theorem.

The second statement follows from (14) and the definition (12) of H^{k,2}(𝔤). Indeed, define E_{k} by the exact sequence

0⟶L ⊗ ⊙^{k+2}N^{∗}⟶L ⊗ ⊙^{k+1}N^{∗} ⊗ V^{∗}⟶E_{k}⟶0.

The associated long exact sequence implies the following exact sequence of vector spaces,

0 ⟶ H^{0}(X, E_{k})/∂[𝔤^{(k)} ⊗ V^{∗}] ⟶ H^{1} (X, L ⊗ ⊙^{k+2}N^{∗})
⟶ H^{1} (X, L ⊗ ⊙^{k+1}N^{∗}) ⊗ V^{∗}.

On the other hand, the exact sequence

0⟶E_{k}⟶L ⊗ ⊙^{k}N^{∗} ⊗ Λ^{2}V^{∗}⟶L ⊗ ⊙^{k−1}N^{∗} ⊗ Λ^{3}V^{∗}

implies

H^{0}(X, E_{k}) = ker : 𝔤^{(k−1)} ⊗ Λ^{2}V^{∗} ⟶^{∂} 𝔤^{(k−2)} ⊗ Λ^{3}V^{∗},

which in turn implies

H^{k,2}(𝔤) = H^{0}(X, E_{k})/∂[𝔤^{(k)} ⊗ V^{∗}].

This completes the proof of the second part of the theorem.

Corollary 3.4. If a complex irreducible matrix group G ⊂ GL(V) is a Berger group, then the associated Borel-Weil data (X, L) satisfy at least one of the following inequalities:

(15)  dim H^{0} (X, L ⊗ ⊙^{2}N^{∗}) > 0,
(16)  dim H̃^{1} (X, L ⊗ ⊙^{3}N^{∗}) > 1,

where N = J^{1}L, and where

H̃^{1} (X, L ⊗ ⊙^{3}N^{∗}) = Ker : H^{1}(X, L ⊗ ⊙^{3}N^{∗}
⟶ H^{1} (X, L ⊗ ⊙^{2}N^{∗}) ⊗ (H^{0}(X, L))^{∗}.

The short exact sequence N^{∗} = L^{∗} + T X ⊗ L^{∗} implies

(17) L ⊗ ⊙^{3}N^{∗} = ⊙^{2}N^{∗} + ⊙^{3}T X ⊗ L^{∗2}
(18)  = L^{∗2} + T X ⊗ L^{∗2} + ⊙^{2}T X ⊗ L^{∗2} + ⊙^{3}T X ⊗ L^{∗2}.

Lemma 3.5. If dim X > 1 and X is not a quadric, then

(19) H^{1}(X, L ⊗ ⊙^{3}N^{∗}) ⊂ H^{1}(X, ⊙^{2}T X ⊗ L^{∗2}) ⊕ H^{1}(X, ⊙^{3}T X ⊗ L^{∗2}),
(20) H^{1}(X, L ⊗ ⊙^{2}N^{∗}) ⊂ H^{1}(X, T X ⊗ L^{∗}) ⊕ H^{1}(X, ⊙^{2}T X ⊗ L^{∗}).

Proof. If dim X > 1, then H^{1}(X, L^{∗2}) = 0 by the Kodaira vanishing theorem. If X is not a quadric, then, by Theorem B, for any ample line bundle L on X one has H^{1}(X, T X ⊗ L^{∗2}) = 0. Then long exact sequences of the short exact sequences (18) immediately imply (19). The second estimation can be checked analogously.

In conclusion, every irreducibly acting holonomy group 𝔤 ⊂ End(V) has an associated compact homogeneous-rational manifold X, the sky of 𝔤. The class of all such manifolds contains a very special subclass — the family of compact hermitean-symmetric manifolds. Therefore, there is a natural question: Which holonomy groups have the members of this subfamily as their skies? The answer to this question is given in the next section and provides us with a surprising insight into the structure of the completed Berger list.

4. Holonomy groups associated with

compact Hermitean-symmetric manifolds

The list of irreducible compact hermitean symmetric manifolds is very short:

(i) Projective spaces ℙ_{n} = … × • • • (n nodes, n ⩾ 1). Note also that there is another representation for odd-dimensional projective spaces; cf. Fact 3.1.

ℙ_{2n+1} =  … • ×  • • < • .

(ii) Grassmanians G(k, ℂ^{n+1}) = … • • • × • … • • (n nodes, k^{th} node is crossed, 2 ⩽ k ⩽ (n + 1)/2, n ⩾ 3).

(iii) Quadrics Q_{2n−1} = [figure: …; ×; >] (n nodes, n ⩾ 2) and

Q_{2n−2} = [figure: …; ×; •; •; •; •] (n nodes, n ⩾ 4).

Note that Q_{4} = G(2, ℂ^{4}), and × > is biholomorphic to Q_{5}; cf. Fact 3.1.

(iv) Manifolds Y_{n} = [figure: …; •; •; •; •; •; •; ×] (n nodes in both cases, n ⩾ 5).

Note that they are biholomorphic to … • • • • > × (n − 1 nodes); cf. Fact 3.1.

(v) [figure: …; <; ×] (n nodes, n ⩾ 3),

(vi) [figure: ×],

(vii) [figure: ×] .

Theorem 4.1. Let 𝔤 ⊂ 𝔤𝔩(V) be an irreducible Berger algebra. Then its sky is biholomorphic to an irreducible compact complex Hermitean manifold if and only if 𝔤 is an entry of Table 6.

Table 6

No.  group G  representation V restrictions
1  ℤ_{ℂ} · SL(n, ℂ)  ℂ^{n}  n ⩾ 1
2  GL(n, ℂ)  ⊙^{2}ℂ^{n}  n ⩾ 3
3  GL(n, ℂ)  Λ^{2}ℂ^{n}  n ⩾ 3
4  ℤ_{ℂ} · SL(2, ℂ)  ⊙^{3}ℂ^{2}
5  SL(6, ℂ)  Λ^{3}ℂ^{6}
6  ℤ_{ℂ} · SO(n, ℂ)  ℂ^{n}  n ⩾ 3
7  ℤ_{ℂ} · Sp(n, ℂ)  ℂ^{2n}  n ⩾ 2
8  Sp(3, ℂ)  ℂ^{14} ⊂ Λ^{3}ℂ^{6}
9  G_{2}^{ℂ}  ℂ^{7}
10  ℤ_{ℂ} · E_{6}^{ℂ}  ℂ^{27}
11  E_{7}^{ℂ}  ℂ^{56}
12  Spin(7, ℂ)  ℂ^{8}
13 ℤ_{ℂ} · Spin(10, ℂ)  ℂ^{16}
14  Spin(12, ℂ)  ℝ^{64} ≃ ℂ^{32}

Notation: ℤ_{ℂ} denotes either the trivial group or ℂ^{∗}.

Proof. All the entries in Table 6 are known to be Berger algebras (for the new exotic entries, numbers 5, 8, 11 and 12, this will be shown in detail in §7) and to have hermitean symmetric skies. So Theorem 4.1 will be proved if we show that there are no other irreducible matrix Lie algebras with these properties.

Case (i). Let L be an ample line bundle on X = ℙ_{n}, i.e. L = O(s) for some s ∈ ℕ. Using the isomorphism J^{1}L = ℂ^{n+1} ⊗ O(s − 1), we show easily

H^{0}(X, L ⊗ ⊙^{2}N^{∗}) = \{ \begin{matrix}ℂ^{n+1} ⊗ ⊙^{2}ℂ^{n+1} & for s = 1, \\ ⊙^{2}ℂ^{n+1} & for s = 2, \\ 0 & otherwise, \end{matrix}

H̃^{1}(X, L ⊗ ⊙^{3}N^{∗}) = \{ \begin{matrix}ℂ^{2} ⊗ ⊙^{3}ℂ^{2} & for n = 1, s = 3, \\ ℂ & for n = 1, s = 4, \\ 0 & otherwise, \end{matrix}

which together with Fact 3.1 imply that the only irreducible representations which have ℙ_{n} as the sky and satisfy the criteria (15) and (16) are the entries No.1, 2, 4 and 7 in Table 6.

Case (ii). Any ample line bundle L on X = G(k, ℂ^{n+1}) is of the form

L = [figure: ...; ×; ...; 0; 0; 0; s; 0; 0; 0]

for some s ∈ ℕ. Since

T X = [figure: ...; ×; ...; 1; 0; 0; 0; 0; 0; 1]

one has

⊙^{2}T X = [figure: ...; ×; ...; 2; 0; 0; 0; 0; 0; 2] ⊕ [figure: ...; ×; ...; 0; 1; 0; 0; 0; 1; 0]

and

⊙^{3}T X = [figure: ...; ×; ...; 3; 0; 0; 0; 0; 0; 3]
⊕ [figure: ...; ×; ...; 1; 1; 0; 0; 0; 1; 1] ⊕ [figure: ...; ×; ...; 0; 0; 1; 0; 0; 0; 1; 0; 0]

where the k^{th} node is crossed, k ⩾ 2, and when k = 2 the third summand in the latter formula is assumed to be zero.

Therefore,

H^{0}(X, L ⊗ ⊙^{2}N^{∗}) = H^{0}(X, ⊙^{2}T X ⊗ L^{∗})
= \{ \begin{matrix}Λ^{2}ℂ^{n+1} & for X = G(2, ℂ^{n+1}), s = 1, \\ 0 & otherwise, \end{matrix}

in accordance with the Cartan-Kobayashi-Nagano classification (see Table 5). By the Bott-Borel-Weil theorem,

H^{1}( [figure: …; …; a; b; -s; c; d] )

= \{ \begin{matrix}[figure: …; …; a; 1+b-s; s-2; 1+c-s; d] & if s ⩾ 2, c ⩾ s − 2, d ⩾ s − 1, \\ 0 & otherwise. \end{matrix}

Then one easily finds that

H^{1}(X, ⊙^{2}T X ⊗ L^{∗2}) = \{ \begin{matrix}ℂ & for (X, L) = [figure: ×; 0; 0; 1; 0; 0] \\ 0 & otherwise, \end{matrix}

and

H^{1}(X, ⊙^{3}T X⊗L^{∗2}) = \{ \begin{matrix}[figure: 1; 0; 0; 0; 1] & for (X, L) = [figure: ×; 0; 0; 1; 0; 0] \\ ℂ & for (X, L) = [figure: ×; 0; 0; 0; 1; 0; 0; 0] \\ 0 & otherwise. \end{matrix}

In addition to entry No.3 in Table 6 only the following two representations

𝔤 = { [figure: 0; 0; 1; 0; 0; 0; 0; 0; 1; 0; 0; 0]

may have, by the estimation (19), K(𝔤 ⊕ ℂ) nonvanishing. Moreover, the long exact sequence of (17) allows us to compute K(𝔤) precisely. Indeed, since H^{0}(X, ⊙^{3}T X ⊗ L^{∗2}) = H^{1}(X, ⊙^{2}T X ⊗ L^{∗2}) = ℂ for the first representation and H^{0}(X, ⊙^{3}T X ⊗ L^{∗2}) = H^{1}(X, ⊙^{2}T X ⊗ L^{∗2}) = 0 for the second one, the following piece of the long exact sequence

0 → H^{0}(X, ⊙^{3}T X ⊗ L^{∗2}) → H^{1}(X, ⊙^{2}T X ⊗ L^{∗2})
→ H^{1}(X, L ⊗ ⊙^{3}N^{∗}) → H^{1}(X, ⊙^{3}T X ⊗ L^{∗2}) → 0

implies

K(𝔤 ⊕ ℂ) = K(𝔤) = H^{1}(X, ⊙^{3}T X ⊗ L^{∗2}).

In fact the statement K(𝔤 ⊕ ℂ) = K(𝔤) above requires an additional simple calculation involving the first Bianchi identity, which we postpone to Section 7.

Therefore, the only irreducible representations which have a Grassmanian G(k, ℂ^{n+1}) as the sky and satisfy the criteria (15) and (16) are entries No. 3 and 5 in Table 6.

Case (iii). Let i : Q_{n} ↪ V be the standard embedding, where V is an (n + 1)-dimensional vector space. Denote by [1] the 1-dimensional subspace in ⊙^{2}V^{∗} spanned by the conformal metric [g_{ab}], and by [−1] = [1]^{∗}.

Any ample line bundle L on Q_{n} is of the form i^{∗}(O(s)) for some s ∈ ℕ. The long exact sequence of the extension

0⟶N^{∗}⟶V^{∗} ⊗ (O(1 − s))⟶i^{∗}(O(2 − s))[−1]⟶0

immediately implies

H^{1}(X, L ⊗ ⊙^{3}N^{∗}) = \{ \begin{matrix}space of Weyl tensors & for s = 1, \\ ⊙^{2}V^{∗}[−1] = [−1] ⊕ ⊙_{0}^{2}V^{∗} & for s = 2, \\ 0 & otherwise, \end{matrix}

H^{1}(X, L ⊗ ⊙^{2}N^{∗}) = \{ \begin{matrix}Λ^{2}V^{∗} & for s = 2, \\ V^{∗}[−1] & for s = 3, \\ 0 & otherwise. \end{matrix}

Then in the case s = 2 we have

\begin{aligned}H̃^{1}(X, L ⊗ ⊙^{3}N^{∗}) & = & ker : ⊙^{2}V^{∗}[−1] & ⟶ & Λ^{2}V^{∗} ⊗ ⊙_{0}^{2}V^{∗} \\ & & ϕ_{ab} & ⟶ & ϕ_{ac}g_{bd} + ϕ_{bc}g_{ad} − ϕ_{ad}g_{bc} − ϕ_{bd}g_{ac} \end{aligned}

implying H̃^{1}(X, L ⊗ ⊙^{3}N^{∗}) = [−1] ≃ ℂ.

Taking into account Fact 3.1 we conclude that it is only the classical representations No. 6, exceptional ones No. 9 and No. 11 and the central extensions of the latter two which have the quadric as their sky. Their central extensions have already been ruled out as Berger algebras in Corollary 2.6.

Case (iv). Any ample line bundle L → X is of the form

[figure: …; •; ×; 0; 0; 0; 0; 0; 0; s]

for some positive integer s. Since

T X = [figure: …; •; ×; 0; 1; 0; 0; 0; 0; 0]

one has

⊙^{2}T X = [figure: …; •; ×; 0; 2; 0; 0; 0; 0; 0] ⊕ [figure: …; ×; 0; 0; 0; 1; 0; 0; 0; u]

and

⊙^{3}T X = [figure: …; •; ×; 0; 3; 0; 0; 0; 0; 0]
⊕ [figure: …; ×; 0; 1; 0; 1; 0; 0; 0; u] ⊕ [figure: …; ×; 0; 0; 0; 0; 0; 1; 0; 0; v] ,

where

u = \{ \begin{matrix}1 & for the number of nodes n=5, \\ 0 & otherwise, \end{matrix} , v = \{ \begin{matrix}1 & for n=7, \\ 0 & otherwise. \end{matrix}

Therefore,

H^{0}(X, L ⊗ ⊙^{2}N^{∗}) = H^{0}(X, ⊙^{2}T X ⊗ L^{∗})

= \{ \begin{matrix}[figure: 0; 0; 0; 1; 0] & for n = 5, s = 1, \\ 0 & otherwise, \end{matrix}

in accordance with the Cartan-Kobayashi-Nagano classification (see Table 5).

By the Bott-Borel-Weil theorem

H^{1}( [figure: …; •; ×; a; b; c; d; e; f; -s] )

= \{ \begin{matrix}[figure: …; •; •; •; •; •; •; a; b; c; e+1-s; f; s-2] & for s ⩾ 2, e ⩾ s − 1, \\ 0 & otherwise, \end{matrix}

which immediately implies

H^{1}(X, ⊙^{2}T X ⊗ L^{∗2}) = \{ \begin{matrix}ℂ & for (X, L) = [figure: ×; 0; 0; 0; 0; 0; 1] \\ 0 & otherwise, \end{matrix}

and

H^{1}(X, ⊙^{3}T X ⊗ L^{∗2})

= \{ \begin{matrix}[figure: 0; 1; 0; 0; 0; 0] & for (X, L) = [figure: ×; 0; 0; 0; 0; 0; 1] \\ ℂ & for (X, L) = [figure: ×; 0; 0; 0; 0; 0; 0; 0; 1] \\ 0 & otherwise. \end{matrix}

In addition to entry No. 13 in Table 6, only the following two representations of 𝔰𝔬(2n, ℂ)

𝔤 = [figure: 0; 0; 0; 0; 0; 1] and 𝔤 = [figure: 0; 0; 0; 0; 0; 0; 0; 1]

may have, by estimation (19), K(𝔤 ⊕ ℂ) nonvanishing. Analyzing the long exact sequence of (17) in exactly the same way as in the case (ii) above, one obtains

K(𝔤 ⊕ ℂ) = K(𝔤) = H^{1}(X, ⊙^{3}T X ⊗ L^{∗2}).

Therefore, we are left with only three candidates to holonomies having the sky biholomorphic to type (iv): the entries No. 13, 14 in Table 6 which are known to satisfy the Berger criteria (that K^{1}(𝔤) ≠ 0 for No. 14 will be shown in detail in §8) and the representations

𝔤_{1} =[figure: >; ×; 0; 0; 0; 1] and 𝔤_{2} =[figure: >; ×; 0; 0; 0; 0; 1],

which, by Fact 3.1, may, in principle, have K(𝔤_{i} ⊕ ℂ) ≠ 0. But the Borel-Weil approach can not be used to compute these modules. However, 𝔤_{1} corresponds to the spinor representation of 𝔰𝔭𝔦𝔫(9); hence 𝔤_{1} ⊕ ℂId is not a Berger algebra by Corollary 2.6, and 𝔤_{1} is known not to be a Berger algebra, either [4].

The second case, which corresponds to the spinor representation of 𝔰𝔭𝔦𝔫(11), will be ruled out as a Berger algebra in Lemma 7.3.

Case (v). Any ample line bundle L → X is of the form

[figure: …; <; ×; 0; 0; 0; 0; s]

for some positive integer s.

Since

T X = [figure: …; <; ×; 2; 0; 0; 0; 0],

one has

⊙^{2}T X = [figure: …; <; ×; 4; 0; 0; 0; 0] ⊕ [figure: …; <; ×; 0; 2; 0; 0; 0]

and

⊙^{3}T X = [figure: …; <; ×; 6; 0; 0; 0; 0] ⊕ [figure: …; <; ×; 2; 2; 0; 0; 0] ⊕ [figure: …; <; ×; 0; 0; 2; 0; 0].

By the Bott-Borel-Weil theorem

H^{1}( [figure: …; <; ×; a; b; c; d; -s]) = \{ \begin{matrix}[figure: …; <; a; b; c; 2+d-2s; s-2] & for s ⩾ 2, d ⩾ 2s − 2, \\ 0 & otherwise, \end{matrix}

which implies

H^{1}(X, ⊙^{2}T X ⊗ L^{∗2}) = \{ \begin{matrix}ℂ & for (X, L) = [figure: <; ×; 0; 0; 1] \\ 0 & otherwise, \end{matrix}

and

H^{1}(X, ⊙^{3}T X ⊗ L^{∗2}) = \{ \begin{matrix}[figure: <; 2; 0; 0] & for (X, L) = [figure: <; ×; 0; 0; 1] \\ ℂ & for (X, L) =[figure: <; ×; 0; 0; 0; 0; 1] \\ 0 & otherwise. \end{matrix}

By the estimation (19), only representations

𝔤 = [figure: <; 0; 0; 1] and 𝔤 = [figure: <; 0; 0; 0; 0; 1]

may have K(𝔤) ≠ 0. Moreover, a straightforward analysis of the long exact sequence of (17) gives a precise expression for K(𝔤):

K(𝔤 ⊕ ℂ) = K(𝔤) = H^{1}(X, ⊙^{3}T X ⊗ L^{∗2}).

Therefore, the only irreducible representation which has a submanifold of the type (v) as the sky and satisfies the criteria (15) and (16) is entry 8 in Table 6.

Case (vi). Any ample line bundle L → X is of the form

[figure: ×; s; 0; 0; 0; 0; 0] .

for some positive integer s.

Since

T X = [figure: ×; 0; 0; 0; 0; 0; 1],

one has

⊙^{2}T X = [figure: ×; 2; 0; 0; 0; 0; 0] ⊕ [figure: ×; 1; 0; 0; 0; 1; 0]

and

⊙^{3}T X = [figure: ×; 0; 0; 0; 0; 0; 3] ⊕ [figure: ×; 1; 0; 0; 0; 1; 1] .

Then, by the Bott-Borel-Weil theorem,

H^{1}(X, ⊙^{2}T X ⊗ L^{∗2}) = H^{1}(X, ⊙^{3}T X ⊗ L^{∗2}) = 0

for any s, implying that all irreducible representations 𝔤 of 𝔢 ⊕ ℂ have H^{1,2}(𝔤) = 0, while, in agreement with Table 5, one has

H^{0}(X, ⊙^{2}T X ⊗ L^{∗}) = H^{0}(X, L ⊗ ⊙^{∗}N^{∗2}) = \{ \begin{matrix}[figure: 0; 0; 0; 0; 1; 0] & for s = 1 \\ 0 & otherwise. \end{matrix}

Therefore, the only irreducible representation which has a submanifold of type (vi) as the sky and satisfies the criteria (15) and (16) is entry 10 in Table 6.

Case (vii). Any ample line bundle L → X is of the form

[figure: ×; 0; 0; 0; 0; 0; s; 0]

for some positive integer s.

Since

T X = [figure: ×; 1; 0; 0; 0; 0; 0; 0] ,

one has

⊙^{2}T X = [figure: ×; 2; 0; 0; 0; 0; 0; 0] ⊕ [figure: ×; 0; 0; 0; 0; 1; 0; 0]

and

⊙^{3}T X = [figure: ×; 3; 0; 0; 0; 0; 0] ⊕ [figure: ×; 1; 0; 0; 0; 1; 0; 0] ⊕ [figure: ×; 0; 0; 0; 0; 0; 2; 0] .

Then, by the Bott-Borel-Weil theorem,

H^{1}(X, ⊙^{2}T X ⊗ L^{∗2}) = \{ \begin{matrix}ℂ & for s = 1 \\ 0 & otherwise \end{matrix}

and

H^{1}(X, ⊙^{3}T X ⊗ L^{∗2}) = \{ \begin{matrix}Ad(𝔢_{7}) & for s = 1 \\ 0 & otherwise. \end{matrix}

By estimation (19), only the representation

𝔤 = [figure: 0; 0; 0; 0; 0; 1; 0]

may have K(𝔤) ≠ 0. Again, the long exact sequence of (17) gives a precise expression for K(𝔤):

K(𝔤 ⊕ ℂ) = K(𝔤) = H^{1}(X, ⊙^{3}T X ⊗ L^{∗2}) = Ad(𝔢_{7}).

The statement K(𝔤 ⊕ ℂ) = K(𝔤) requires an additional calculation involving the first Bianchi identities. The necessary details are given in Section 7.

Therefore, the only irreducible representation which has a submanifold of type (v) as the sky and satisfies the criteria (15) and (16) is entry 11 in Table 6.

5. Classification of Segre holonomies

5.1. Segre holonomies. The main result of this section is the following:

Theorem 5.1. Let G be the irreducible holonomy of a torsion-free affine connection which is not locally symmetric. If the semisimple part of G is not simple, then G is one of the groups listed in Table 7.

Table 7: List of Segre holonomies

group G  representation V  restrictions
T_{ℝ} · SL(m, ℝ) · SL(n, ℝ)  ℝ^{m} ⊗ ℝ^{n} ≃ ℝ^{mn}  m ⩾ n ⩾ 2
T_{ℂ} · SL(m, ℂ) · SL(n, ℂ) ℂ^{m} ⊗ ℂ^{n} ≃ ℝ^{2mn}  m ⩾ n ⩾ 2
T_{ℝ} · SL(m, ℍ) · SL(n, ℍ)  ℍ ⊗ ℍ ≃ ℝ^{4mn}  m ⩾ n ⩾ 1
Sp(1) · Sp(p, q)  ℍ^{p+q} ≃ ℝ^{4(p+q)}  p + q ⩾ 2
SL(2, ℝ) · Sp(n, ℝ)  ℝ^{2} ⊗ ℝ^{2n} ≃ ℝ^{4n}  n ⩾ 2
SL(2, ℂ) · Sp(n, ℂ)  ℂ^{2} ⊗ ℂ^{2n} ≃ ℝ^{8n}  n ⩾ 2
SL(2, ℝ) · SO(p, q)  ℝ^{2} ⊗ ℝ^{p+q} ≃ ℝ^{2(p+q)} p + q ⩾ 3
Sp(1) · SO(n, ℍ)  ℍ^{n} ≃ ℝ^{4n}  n ⩾ 2
SL(2, ℂ) · SO(n, ℂ)  ℂ^{2} ⊗ ℂ^{n} ≃ ℝ^{4n}  n ⩾ 3
Notation: T_{𝔽} denotes any connected Lie subgroup of 𝔽^{∗}.

The class of holonomy groups (and the associated geometric structures) studied by the above theorem appear in the literature under different names. For example, the authors of [3], [29] call these almost Grassmanian and the authors of [6] call these paraconformal. In this paper we follow the terminology of Bryant [16] who suggested calling them Segre holonomies. Some applications of Segre structures to high energy physics are discussed in [28], [31].

5.2. Cohomology on reducible homogeneous-rational manifolds. From now on we assume that 𝕏 = X_{1} × X_{2} is a direct product of two compact complex homogeneous-rational manifolds X_{1} and X_{2} and that 𝕃 is an ample holomorphic line bundle on X. Denoting by π_{1} : 𝕏 → X_{1} and π_{2} : 𝕏 → X_{2} the natural projections, we may write 𝕃 = π_{1}^{∗}(L_{1})⊗π_{2}^{∗}(L_{2}) for some uniquely specified ample line bundles L_{1} and L_{2} on X_{1} and X_{2} respectively. We denote ℕ := J^{1}𝕃 and N_{i} := J^{1}L_{i}, i = 1, 2.

Since

0 ⟶ ΩX_{i} ⊗ L_{i} ⟶ N_{i} ⟶ L_{i} ⟶ 0,

one has

0 ⟶ π_{1}^{∗}(Ω^{1}X_{1}) ⊗ 𝕃 + π_{2}^{∗}(Ω^{1}X_{2}) ⊗ 𝕃

⟶ π_{1}^{∗}(N_{1}) ⊗ π_{2}^{∗}(L_{2}) + π_{2}^{∗}(N_{2}) ⊗ π_{1}^{∗}(L_{1}) ⟶ 𝕃 + 𝕃⟶0.

The latter extension combined with

0 ⟶ π_{1}^{∗}(Ω^{1}X_{1}) ⊗ 𝕃 + π_{2}^{∗}(Ω^{1}X_{2}) ⊗ 𝕃 ⟶ ℕ ⟶ 𝕃 ⟶ 0.

implies

0 ⟶ ℕ ⟶ π_{1}^{∗}(N_{1}) ⊗ π_{2}^{∗}(L_{2}) + π_{1}^{∗}(L_{1}) ⊗ π_{2}^{∗}(N_{2}) ⟶ 𝕃 ⟶ 0,

or

(21)  0 ⟶ 𝕃^{∗} ⟶ π_{1}^{∗}(N_{1}^{∗}) ⊗ π_{2}^{∗}(L_{2}^{∗}) + π_{1}^{∗}(N_{1}^{∗}) ⊗ π_{2}^{∗}(L_{2}^{∗}) ⟶ ℕ^{∗} ⟶ 0,

which in turn implies the following two exact sequences (22)

0⟶ \begin{matrix}π_{1}^{∗}(N_{1}^{∗}) ⊗ π_{2}^{∗}(L_{2}^{∗}) \\ + \\ π_{1}^{∗}(L_{1}^{∗}) ⊗ π_{2}^{∗}(N_{2}^{∗}) \end{matrix} ⟶ \begin{matrix}π_{1}^{∗}(L_{1} ⊗ ⊙^{2}N_{1}^{∗}) ⊗ π_{2}^{∗}(L_{2}^{∗}) \\ + \\ π_{1}^{∗}(N_{1}^{∗}) ⊗ π_{2}^{∗}(N_{2}^{∗}) \\ + \\ π_{1}^{∗}(L_{1}^{∗}) ⊗ π_{2}^{∗}(L_{2} ⊗ ⊙^{2}N_{2}^{∗}) \end{matrix} ⟶𝕃 ⊗ ⊙^{2}ℕ^{∗}⟶0

and (23)

0⟶ \begin{matrix}π_{1}^{∗}(⊙^{2}N_{1}^{∗}) ⊗ π_{2}^{∗}(L_{2}^{∗})^{2} \\ + \\ π_{1}^{∗}(L_{1} ⊗ N_{1}^{∗}) ⊗ π_{2}^{∗}(L_{2}^{∗} ⊗ N_{2}^{∗}) \\ + \\ π_{1}^{∗}(L_{1}^{∗})^{2} ⊗ π_{2}^{∗}(⊙^{2}N_{2}^{∗}) \end{matrix} ⟶ \begin{matrix}π_{1}^{∗}(L_{1} ⊗ ⊙^{3}N_{1}^{∗}) ⊗ π_{2}^{∗}(L_{2}^{∗})^{2} \\ + \\ π_{1}^{∗}(⊙^{2}N_{1}^{∗}) ⊗ π_{2}^{∗}(L_{2}^{∗} ⊗ N_{2}^{∗}) \\ + \\ π_{1}^{∗}(L_{1}^{∗} ⊗ N_{1}^{∗}) ⊗ π_{2}^{∗}(⊙N_{2}^{∗}) \\ + \\ π_{1}^{∗}(L_{1}^{∗})^{2} ⊗ π_{2}^{∗}(L_{2} ⊗ ⊙^{3}N_{2}^{∗}) \end{matrix} ⟶𝕃 ⊗ ⊙^{3}ℕ^{∗}⟶0.

Proposition 5.2. Let X be a compact complex homogeneous-rational manifold and L an ample line bundle on X. Then

H^{0}(X, T X ⊗ L^{∗}) = \{ \begin{matrix}ℂ & for (X, L) = (ℂℙ_{1}, O(2)) \\ ℂ^{n+1} & for (X, L) = (ℂℙ_{n}, O(1)), n ⩾ 1 \\ 0 & otherwise. \end{matrix}

Proof. If dim X = 1, i.e. X = ℂℙ^{1}, then the statement follows from the isomorphism T X ≃ O(2).

Asssume now that dim X ⩾ 2. Then, by the Kodaira vanishing theorem, H^{1}(X, L^{∗}) = 0 for any ample line bundle L on X. Applying the Künneth formula to the long exact sequence of (22) with 𝕏 = X × X and 𝕃 = π_{1}^{∗}(L) ⊗ π_{2}^{∗}(L), one easily obtains

H^{0}(𝕏, 𝕃 ⊗ ⊙^{2}ℕ^{∗}) = H^{0}(X, N^{∗}) ⊗ H^{0}(X, N^{∗})
= H^{0}(X, T X ⊗ L^{∗}) ⊗ H^{0}(X, T X ⊗ L^{∗}).

On the other hand, by Theorem 3.3,

H^{0}(𝕏, 𝕃 ⊗ ⊙^{2}ℕ^{∗}) = 𝔤^{(1)},

where 𝔤 is the irreducible representation of

H^{0}(𝕏, 𝕃 ⊗ ℕ^{∗}) ≃ ℂ ⊕ H^{0}(X, T X) ⊕ H^{0}(X, T X)

in V ⊗ V with V = H^{0}(X, L). Table 5 implies that such a 𝔤^{(1)} can be nonzero if and only if H^{0}(X, T X) ≃ 𝔰𝔩(n + 1, ℂ) irreducibly represented in ℂ^{n+1}, i.e. X = ℂℙ_{n}. Then the isomorphism H^{0}(X, L) = ℂ^{n+1} implies L = O(1). Therefore, H^{0}(𝕏, 𝕃 ⊗ ⊙^{2}ℕ^{∗}) with 𝕏 = X × X and 𝕃 = π_{1}^{∗}(L) ⊗ π_{2}^{∗}(L) vanishes unless (X, L) = (ℂℙ_{n}, O(1)) which implies that H^{0}(X, T X ⊗ L^{∗}) vanishes unless (X, L) = (ℂℙ_{n}, O(1)). Finally, the extension

0 ⟶ O(−1) ⟶ ℂ^{n+1} ⊗ O_{ℂℙ_{n}} ⟶ T ℂℙ_{n}(−1) ⟶ 0

implies H^{0}(ℂℙ_{n}, T ℂℙ_{n}(−1)) = ℂ^{n+1} which completes the proof of Proposition 5.2 when dim X ⩾ 2.

Corollary 5.3. Let X be a compact complex homogeneous-rational manifold, L an ample line bundle on X and N = J^{1}L. Then, for any k ⩾ 1,

H^{0} (X, ⊙^{k}N^{∗}) = \{ \begin{matrix}⊙^{k}ℂ^{n+1} & for (X, L) = (ℂℙ_{n}, O(1)), n ⩾ 1 \\ 0 & otherwise. \end{matrix}

Proof. The statement is true for (X, L) = (ℂℙ_{n}, O(1)) since J^{1}O(1) = ℂ^{n+1} ⊗ O_{X}.

The case k = 1 of the required statement follows immediately from Propostion 5.2 and the extension

(24)  0 ⟶ L^{∗}⟶N^{∗} ⟶ T X ⊗ L^{∗}⟶ 0.

The latter also implies

0 ⟶ ⊙^{k} N^{∗}⟶L ⊗ ⊙^{k+1}N^{∗}⟶ ⊙^{k+1} T X ⊗ L^{∗k}⟶ 0

which in turn implies

H^{0}(X, ⊙^{k}N^{∗}) ⊂ H^{0}(X, L ⊗ ⊙^{k+1}N^{∗}).

According to Cartan [17] (see also [38] for another proof), the only irreducible complex Lie subalgebras 𝔤 ⊂ 𝔤𝔩(V) which have 𝔤^{(k)} ≠ 0 for k ⩾ 3 are 𝔤𝔩(m, ℂ), 𝔰𝔩(m, ℂ), 𝔰𝔭(m/2, ℂ) and 𝔰𝔭(m/2, ℂ) ⊕ ℂ standardly represented in ℂ^{m}, m ⩾ 2. The Borel-Weil data (X, L) associated with these four representations are (ℂℙ_{m−1}, O(1)). Therefore, if (X, L) ≠ (ℂℙ_{n}, O(1)), then, by Theorem 3.3, H^{0}(X, L ⊗ ⊙^{k+1}N^{∗}) = 0 for all k ⩾ 3. Hence H^{0}(X, ⊙^{k}N^{∗}) = 0 for all k ⩾ 3. This proves our corollary for k ⩾ 3.

Asume now that k = 2 and denote L̃ := L^{2} and Ñ := J^{1}L̃ ≃ L⊗N. Then

H^{0}(X, L̃ ⊗ ⊙^{2}Ñ^{∗}) = H^{0}(X, ⊙^{2}N^{∗}).

Again, using Theorem 3.3 and Table 5 one concludes that the only irreducibly acting reductive Lie subalgebra 𝔤 ⊂ 𝔤𝔩(V) which has 𝔤^{(1)} ≠ 0 and whose associated pair (X, L̃) is such that L̃ is a square of an ample line bundle on X is 𝔤𝔩(n, ℂ) represented in ⊙^{2}ℂ^{n} with the associated Borel-Weil data (ℂℙ_{n−1}, O(2)). Therefore, H^{0}(X, ⊙^{2}N^{∗}) = 0 for all (X, L) ≠ (ℂℙ_{n}, O(1)). The proof is complete.

5.3. The case 𝕏 = X_{1} × X_{2} with dim X_{i} ⩾ 2. The long exact sequence of (22) implies

(25)  H^{0} (𝕏, 𝕃 ⊗ ⊙^{2}ℕ^{∗}) = H^{0}(X_{1}, N_{1}^{∗}) ⊗ H^{0}(X_{2}, N_{2}^{∗})

while the long exact sequence of (23) contains the following piece

0⟶ \begin{matrix}H^{0} (X_{1}, ⊙^{2}N_{1}^{∗}) ⊗ H^{1} (X_{2}, T X_{2} ⊗ L_{2}^{∗2}) \\ + \\ H^{0} (X_{2}, ⊙^{2}N_{2}^{∗}) ⊗ H^{1} (X_{1}, T X_{1} ⊗ L_{1}^{∗2}) \end{matrix} ⟶H^{1} (𝕏, 𝕃 ⊗ ⊙^{3}ℕ^{∗}) ⟶

(26)  ⟶H^{1} (X_{1}, T X_{1} ⊗ L_{1}^{∗2}) ⊗ H^{1} (X_{2}, T X_{2} ⊗ L_{2}^{∗2}) ⟶ ….

Lemma 5.4. Let G_{i} ⊂ GL(V_{i}), i = 1, 2, be an irreducible complex semisimple matrix Lie group such that the associated Borel-Weil data (X_{i}, L_{i}) satisfy dim X_{i} ⩾ 2. Then G = T_{ℂ} · G_{1} · G_{2} ⊂ GL(V_{1} ⊗ V_{2}) can have K(𝔤) ≠ 0 only if each G_{i} is isomorphic to one of the following representations:

Group:  SL(n, ℂ) Sp(m, ℂ) SO(p, ℂ) G_{2} Spin(7, ℂ)
Representation space:  ℂ^{n}  ℂ^{2m}  ℂ^{p}  ℂ^{7}  ℂ^{8}

where n ⩾ 3, m ⩾ 2 and p ⩾ 4.

Proof. K(𝔤) ≠ 0 if and only if 𝔤^{(1)} ≠ 0 and/or H^{1,2}(𝔤) ≠ 0. By Theorem 3.3, Corollary 5.3 for k = 1 and (25), one has

𝔤^{(1)} = H^{0} (𝕏,𝕃 ⊗ ⊙^{2}ℕ^{∗}) = \{ \begin{matrix}ℂ^{n_{1}+1} ⊗ ℂ^{n_{2}+1} & for (X_{i}, L_{i}) = (ℂℙ_{n_{i}}, O(1)), n_{i} ⩾ 1 \\ 0 & otherwise. \end{matrix}

On the other hand, a glance at Table 3 shows that H^{1}(X, T X ⊗ L^{∗2}) ≠ 0 only if (X, L) = (Q_{n}, j^{∗}O(1)) where Q_{n} is the n-dimensional quadric and j : Q_{n} ↪ ℂℙ_{n+1} is its standard embedding. Then Theorem 3.3, Corollary 5.3 for k = 2 and (26) imply that H^{1,2}(𝔤) can be nonzero only if each pair (X_{i}, L_{i}) is isomorphic either to (ℂℙ_{n}, O(1)) or to (Q_{n}, j^{∗}O(1)).

These observations combined with Fact 3.1 imply that K(𝔤) can be nonzero only for representations listed in Lemma 5.4.

Example 1. Let G be the representation of ℤ_{ℂ} · SL(m, ℂ) · SL(n, ℂ) in the vector space V = V_{m} ⊗ V_{n} where V_{m} and V_{n} are m- and, respectively, n-dimensional complex vector spaces with m, n ⩾ 3. The associated Borel-Weil datum (𝕏, 𝕃) is (ℂℙ_{m−1} × ℂℙ_{n−1}, π_{1}^{∗}(O(1)) ⊗ π_{2}^{∗}(O(1))), implying

𝔤^{(1)} = H^{0}(𝕏, 𝕃 ⊗ ⊙^{2}ℕ^{∗}) = V^{∗}, H^{1,2}(𝔤) = H^{1}(𝕏, 𝕃 ⊗ ⊙^{3}ℕ^{∗}) = 0.

Therefore, K(𝔤) = ∂(𝔤^{(1)}⊗V^{∗}) ≃ V^{∗}⊗V^{∗}. Denoting typical elements of V, V_{m} and V_{n} by v^{a}, v^{A} and vȦ respectively,^{1} and identifying v^{a} ∈ V with its image v^{AȦ} under the isomorphism V = V_{m} ⊗ V_{n}, one may write a typical element R_{abc}^{d} ∈ K(𝔤) ⊂ Λ^{2}V^{∗} ⊗ V^{∗} ⊗ V as

(27)  R_{abc}^{d} ≡ R_{AȦBḂCĊ}^{DḊ} = [δ_{A}^{D}Q_{BḂCȦ} − δ_{B}^{D}Q_{AȦCḂ}] δḊ ; Ċ

+ [δḊ ; ȦQ_{BḂAĊ} − δḊ ; ḂQ_{AȦBĊ}] δ_{C}^{D}

for some Q_{ab} ≡ Q_{AȦBḂ} ∈ V^{∗} ⊗ V^{∗}. Therefore, a torsion-free connection ∇ on an mn-dimensional manifold M with holonomy in ℤ_{ℂ} · SL(m, ℂ) · SL(n, ℂ) has at an arbitrary point x ∈ M the curvature tensor of the form (27) for some Q_{ab}(x) ∈ Ω_{x}M ⊗ Ω_{x}M. It is not hard to show that the second Bianchi identities for ∇,

∇_{e}R_{abc}^{d} + ∇_{b}R_{eac}^{d} + ∇_{a}R_{bec}^{d} = 0,

imply (28)

0 = m (∇_{AȦ}Q_{BḂCĊ} − ∇_{BḂ}Q_{AȦCĊ}) + n (∇_{CĊ}Q_{AȦBḂ} − ∇_{AȦ}Q_{CĊBḂ})
+ (∇_{BḂ}Q_{AĊCȦ} − ∇_{AĊ}Q_{BḂCȦ}) + (∇_{BȦ}Q_{CĊAḂ} − ∇_{CĊ}Q_{BȦAḂ}) .

Example 2. Keeping notation of the preceding paragraph, we consider a subgroup G_{o} ⊂ G which is ℤ_{ℂ} ·SL(m, ℂ)·SO(n, ℂ) represented in V = V_{m} ⊗V_{n} with m, n ⩾ 3. The G_{o}-module K(𝔤_{o}) is a subset of K(𝔤) consisting of all elements R_{abc}^{d} satisfying

R_{AȦBḂCĊ}^{DḊ} g_{ĖḊ} + R_{AȦBḂCĖ}^{DḊ} g_{ĊḊ} = 0,

where g_{ĖḊ} ∈ ⊙^{2}V_{n}^{∗} is the SO(n, ℂ)-invariant quadratic form. Substituting (27) into the above equation, one obtains after elementary algebraic manipulations that

Q_{AȦBḂ} = P_{AB} g_{ȦḂ}

1One may view indices of the type a, A or Ȧ as refering to some fixed basis in a relevant vector space or, alternatively, as abstract labels providing us with a transparent notation for such basic tensor operations as (anti)symmetrization, contraction, etc. (cf. [7], [35]).

for some symmetric tensor P_{AB} ∈ ⊙^{2}V_{m}^{∗}. (Another way to obtain this result is to note that the Borel-Weil datum (𝕏, 𝕃) associated to (G_{o}, V) is (ℂℙ_{m−1} × Q_{n−1}, π_{1}^{∗}(O(1)) ⊗ π_{2}^{∗}(j^{∗}O(1))) implying 𝔤_{o}^{(1)} = H^{0}(𝕏, 𝕃 ⊗ ⊙^{2}ℕ^{∗}) = 0 and K(𝔤_{o}) = H^{1}(𝕏, 𝕃 ⊗ ⊙^{3}ℕ^{∗}) = ⊙^{2}V_{m}^{∗} ⊗ C ⊂ V^{∗} ⊗ V^{∗}, where C is the 1-dimensional subspace of ⊙^{2}V_{n} spanned by g_{ȦḂ}.) Then the second Bianchi identities (28) imply ∇_{a}Q_{bc} = 0 which in turn implies ∇_{m}R_{abc}^{d} = 0. These arguments imply essentially the following:

Lemma 5.5. Let G be the irreducible representation of a subgroup of GL(m, ℂ) · SO(n, ℂ) in the mn-dimensional vector space V_{m} ⊗ V_{n}. If m, n ⩾ 3, then K^{1}(𝔤) = 0.

Example 3. Keeping notation of Example 1, consider a subgroup G_{s} ⊂ G which is ℤ_{ℂ} · SL(m, ℂ) · Sp(n, ℂ) represented in V = V_{m} ⊗ V_{2n} with m ⩾ 3, n ⩾ 2, and note that K(𝔤_{s}) is a subset of K(𝔤) consisting of all elements R_{abc}^{d} satisfying

R_{AȦBḂCĊ}^{DḊ} ε_{ĖḊ} − R_{AȦBḂCĖ}^{DḊ} ε_{ĊḊ} = 0,

where ε_{ĖḊ} ∈ Λ^{2}V_{2n}^{∗} is the Sp(n, ℂ)-invariant symplectic form. Substituting (27) into this equation, one easily finds

Q_{AȦBḂ} = S_{AB} ε_{ȦḂ}

for some antisymmetric tensor S_{AB} ∈ Λ^{2}V_{m}. Then the second Bianchi identities (28) imply ∇_{a}Q_{bc} = 0 which in turn implies ∇_{m}R_{abc}^{d} = 0. We may summarize these arguments as follows.

Lemma 5.6. Let G be the irreducible representation of a subgroup of GL(m, ℂ) · Sp(n, ℂ) in the 2mn-dimensional vector space V_{m} ⊗ V_{2n}. If m ⩾ 3, n ⩾ 2, then K^{1}(𝔤) = 0.

An immediate corollary of Lemmas 5.4–5.6 is the following:

Proposition 5.7. Let G_{i} ⊂ GL(V_{n_{i}}), i = 1, 2, be an irreducible complex semisimple matrix Lie group such that the associated Borel-Weil data (X_{i}, L_{i}) satisfy dim X_{i} ⩾ 2. Then G = ℤ_{ℂ} ·G_{1} ·G_{2} ⊂ GL(V_{n_{1}} ⊗V_{n_{2}}) can have K^{1}(𝔤) ≠ 0 only if G_{1} = SL(n_{1}, ℂ) and G_{2} = SL(n_{2}, ℂ).

5.4. The case 𝕏 = X × ℂℙ_{1} with dim X ⩾ 2. Any ample line bundle on 𝕏 is of the form 𝕃 = π_{1}^{∗}(L) ⊗ π_{2}^{∗}(O(k)) for some ample line bundle L → X and k ⩾ 1. We denote in this subsection V_{n} := H^{0}(X, L), V_{2} := H^{0}(ℂℙ_{1}, O(1)), N := J^{1}L, and 𝔤 stands for the Lie algebra H^{0}(X, L⊗N^{∗})+𝔰𝔩(2, ℂ) represented in V = V_{n} ⊗ V_{2}.

If k ⩾ 2, then the associated matrix Lie group G = exp(𝔤) is an irreducible matrix subgroup of either GL(n, ℂ)SO(p, ℂ) represented in ℂ^{np} for some n, p ⩾ 3 or GL(n, ℂ)Sp(q, ℂ) represented in ℂ^{2nq} for some n ⩾ 3, q ⩾ 2. Then, by Lemmas 5.5 and 5.6, K^{1}(𝔤) = 0.

So we may assume that k = 1.

Proposition 5.8. Let (X, L) be a pair consisting of a compact complex homogeneous-rational manifold X and an ample line bundle L → X. If dim X ⩾ 2, then

𝔤^{(1)} = H^{0}(X, N^{∗}) ⊗ V_{2}^{∗}

and there is an exact sequence of 𝔤-modules

0⟶H^{1,2}(𝔤)⟶ \begin{matrix}H^{0}(X, ⊙^{3}T X ⊗ L^{∗2}) ⊗ Λ^{2}V_{2}^{∗} \\ + \\ H^{1}(X, T X ⊗ L^{∗2}) ⊗ ⊙^{2}V_{2}^{∗} \end{matrix} ⟶H^{1}(X, T X ⊗ L^{∗}) ⊗ V^{∗} ⊗ V_{2}^{∗}.

Proof. Since dim X ⩾ 2, the Kodaira vanishing theorem implies H^{1}(X, L^{∗}) = 0. Then the long exact sequence of

0 ⟶ L^{∗}⟶N^{∗}⟶ T X ⊗ L^{∗}⟶ 0

implies H^{1}(X, N^{∗}) = H^{1}(X, T X ⊗ L^{∗}), while the long exact sequence of (22) with (X_{1}, L_{1}) = (X, L) and (X_{2}, L_{2}) = (ℂℙ_{1}, O(1)) implies

H^{0}(𝕏, 𝕃 ⊗ ⊙^{2}ℕ^{∗}) = H^{0}(X, N^{∗}) ⊗ V_{2}^{∗},

H^{1}(𝕏, 𝕃 ⊗ ⊙^{2}ℕ^{∗}) = H^{1}(X, N^{∗}) ⊗ V_{2}^{∗} = H^{1}(X, T X ⊗ L^{∗}) ⊗ V_{2}^{∗}.

Analogously, the long exact sequence of (23) implies

0 ⟶H^{0}(𝕏,𝕃 ⊗ ⊙^{3}ℕ^{∗})⟶ \begin{matrix}H^{0}(X, ⊙^{2}N^{∗}) ⊗ Λ^{2}V_{2}^{∗} \\ + \\ 0 \end{matrix} ⟶ \begin{matrix}H^{0}(X, L ⊗ ⊙^{3}N^{∗}) ⊗ Λ^{2}V_{2}^{∗} \\ + \\ 0 \end{matrix} ⟶

⟶H^{1}(𝕏,𝕃 ⊗ ⊙^{3}ℕ^{∗})⟶ \begin{matrix}H^{1}(X, ⊙^{2}N^{∗}) ⊗ Λ^{2}V_{2}^{∗} \\ + \\ 0 \end{matrix} ⟶ \begin{matrix}H^{1}(X, L ⊗ ⊙^{3}N^{∗}) ⊗ Λ^{2}V_{2}^{∗} \\ + \\ H^{2}(X, L^{∗} ⊗ N^{∗}) ⊗ ⊙^{2}V_{2}^{∗} \end{matrix} ⟶ ….

Comparing this with the long exact sequence of

(29)  0⟶ ⊙^{2} N^{∗}⟶L ⊗ ⊙^{3}N^{∗}⟶ ⊙^{3} T X ⊗ L^{∗2}⟶0

one obtains

H^{1}(𝕏, 𝕃 ⊗ ⊙^{3}ℕ^{∗}) = H^{0}(X, ⊙^{3}T X ⊗ L^{∗2}) ⊗ Λ^{2}V_{2}^{∗} + H^{1}(X, T X ⊗ L^{∗2}) ⊗ ⊙^{2}V_{2}^{∗}.

Then Theorem 3.3 implies the desired result.

Lemma 5.9. Let G ⊂ GL(V) be an irreducible complex representation of a complex reductive Lie group such that the associated Borel-Weil data are of the form (𝕏 = X × ℂℙ_{1}, 𝕃) with dim X ⩾ 2 and H^{1}(X, T X ⊗ L^{∗2}) = 0. Then K^{1}(𝔤) ≠ 0 if only if G is T_{ℂ} · SL(n, ℂ) · SL(2, ℂ) represented in V = V_{n} ⊗ V_{2}.

Proof. If (X, L) = (ℂℙ_{n}, O(1)), then G has both modules K(𝔤) and K^{1}(𝔤) nonzero only if it is T_{ℂ} · SL(n, ℂ) · SL(2, ℂ) represented in V_{n} ⊗ V_{2}.

Assume now that (X, L) ≠ (ℂℙ_{n}, O(1)). Then, by Proposition 5.9,

K(𝔤) ⊂ H^{0}(X, ⊙^{3}T X ⊗ L^{∗2}) ⊗ Λ^{2}V_{2}^{∗}.

Since H^{0}(X, L⊗⊙^{3}N^{∗}) = 0, the long exact sequence of (29) implies H^{0}(X, ⊙^{3}T X ⊗ L^{∗2}) ⊂ H^{1}(X, ⊙^{2}N^{∗}), while the exact sequence

0⟶ ⊙^{2} N^{∗}⟶N^{∗} ⊗ V_{n}^{∗}⟶Λ^{2}V_{n}^{∗}

implies H^{1}(X, ⊙^{2}N^{∗}) ⊂ Λ^{2}V_{n}^{∗}. Thus K(𝔤) ⊂ Λ^{2}V_{n}^{∗}⊗Λ^{2}V_{2}^{∗} which easily implies K^{1}(𝔤) = 0.

Lemma 5.10. Let G ⊂ GL(V) be an irreducible complex representation of a complex reductive Lie group such that the associated Borel-Weil data are of the form (𝕏 = X ×ℂℙ_{1}, 𝕃) with dim X ⩾ 2 and H^{1}(X, T X ⊗L^{∗2}) ≠ 0. Then K^{1}(𝔤) ≠ 0 only if G is either T_{ℂ} ·SL(n, ℂ)·SL(2, ℂ) or T_{ℂ} ·SO(n, ℂ)·SL(2, ℂ), both represented in V_{n} ⊗ V_{2}.

Proof. It follows from Table 3 that H^{1}(X, T X ⊗ L^{∗2}) ≠ 0 only if (X, L) = (Q_{n}, j^{∗}O(1)) where Q_{n} is the n-dimensional quadric and j : Q_{n} ↪ ℂℙ_{n+1} is its standard embedding. This together with Fact 12 imply that G must be of the form ℤ_{ℂ} · H · SL(2, ℂ) ⊂ 𝔤𝔩(V_{n} ⊗ V_{2}) where H is one of the following representations:

Group H:  SO(n, ℂ) G_{2} Spin(7, ℂ)
Representation space V_{n}:  ℂ^{n}  ℂ^{7}  ℂ^{8}

Since the Borel-Weil data associated to G = ℤ_{ℂ} · SO(n, ℂ) · SL(2, ℂ) represented in V_{n} ⊗ V_{2} are (Q_{n−1} × ℂℙ_{1}, π_{1}^{∗}(j^{∗}O(1)) ⊗ π_{2}^{∗}(O(1)), one has 𝔤^{(1)} = H^{0}(𝕏, 𝕃 ⊗ ⊙^{2}ℕ^{∗}) = 0 and

K(𝔤) = H^{1}(𝕏, 𝕃 ⊗ ⊙^{3}ℕ^{∗}) = Λ^{2}V_{n}^{∗} ⊗ Λ^{2}V_{2}^{∗} + C ⊗ ⊙^{2}V_{2}^{∗}

where C is the 1-dimensional subspace of ⊙^{2}V_{n} spanned by the SO(n, ℂ)- invariant metric g_{AB}. Then a generic element of K(𝔤) must be of the form (cf.

[20])

(30) R_{abc}^{d} = [ε_{ȦḂ}g^{DE}(g_{AB}S_{CE} + g_{AC}S_{BE} + g_{BC}S_{AE}

− g_{AE}S_{BC} − g_{BE}S_{AC}) + Φ_{ȦḂ}(g_{BC}δ_{A}^{D} − g_{AC}δ_{B}^{D}]δḊ ; Ċ

+ [g_{AB}ε_{ȦḂ}ε^{ḊĖ}Φ_{ĊĖ} − S_{AB}(ε_{ḂĊ}δḊ ; Ȧ + ε_{ȦĊ}δḂ ; Ȧ)]δ_{C}^{D}

for some S_{AB} ∈ Λ^{2}V_{n}^{∗} and Φ_{ȦḂ} ∈ ⊙^{2}V_{2}^{∗}.

Let g ⊂ 𝔤𝔩(V) be the Lie algebra of the representation of ℤ_{ℂ} · G_{2} · SL(2, ℂ) (resp. ℤ_{ℂ} · Spin(7, ℂ) · SL(2, ℂ)) in V = ℂ^{7} ⊗ ℂ^{2} (resp. in V = ℂ^{8} ⊗ ℂ^{2}). It is a proper matrix subalgebra of the Lie algebra 𝔤 of the representation of ℤ_{ℂ} · SO(7, ℂ) · SL(2, ℂ) (resp. ℤ_{ℂ} · SO(8, ℂ) · SL(2, ℂ)) in V = ℂ^{7} ⊗ ℂ^{2} (resp. in V = ℂ^{8} ⊗ ℂ^{2}). Then

K(g) ⊂ K(𝔤) = Λ^{2}V_{n}^{∗} ⊗ Λ^{2}V_{2}^{∗} + C ⊗ ⊙^{2}V_{2}^{∗}

and K^{1}(g) ⊂ K^{1}(𝔤). We claim

(31)  K(g) ⊂ Λ^{2}V_{n}^{∗} ⊗ Λ^{2}V_{2}^{∗}.

If not, then a typical element R_{abc}^{d} ∈ K(g) contains a nonzero term Φ_{ȦḂ}(g_{BC}δ_{A}^{D} − g_{AC}δ_{B}^{D})δḊ ; Ċ which easily implies that the image of the map

Λ^{2}V ⟶g

defined by R_{abc}^{d} ∈ g ⊗ Λ^{2}V^{∗} contains Λ^{2}V_{n}^{∗} ≃ 𝔰𝔬(n, ℂ). This contradicts the fact that g is a proper subalgebra of 𝔤.

Finally, it is straightforward to check that the inclusion (31) implies that K^{1}(g) = 0.

Proposition 5.11. Let G ⊂ GL(V) be an irreducible complex representation of a complex reductive Lie group such that the associated Borel-Weil data are of the form (𝕏 = X × ℂℙ_{1}, 𝕃) with dim X ⩾ 2. Then K^{1}(𝔤) ≠ 0 if and only if G is either T_{ℂ} · SL(n, ℂ) · SL(2, ℂ) or SO(n, ℂ) · SL(2, ℂ), both represented in V_{n} ⊗ V_{2}.

Proof. By Lemmas 5.9 and 5.10, one has only to rule out the case ℂ^{∗} · SO(n, ℂ) · SL(2, ℂ). But this follows from R_{abc}^{c} = 0 which itself follows from (30).

5.5. The case 𝕏 = ℂℙ_{1} × ℂℙ_{1}. This is the case of ℤ_{ℂ} · SL(2, ℂ) · SL(2, ℂ) represented in ⊙^{m}V_{2} ⊗ ⊙^{n}V_{2}. In the context of the holonomy classification, this class of representations was studied in [22] and [32] where the following result was established by two different methods.

Proposition 5.12. Let G ⊂ GL(V) be an irreducible complex representation of a complex reductive Lie group such that the associated Borel-Weil data (𝕏, 𝕃) has 𝕏 = ℂℙ_{1} × ℂℙ_{1}. Then K^{1}(𝔤) ≠ 0 only if G is either the representation of T_{ℂ} ·SO(4, ℂ) in ℂ^{4} or the representation of T_{ℂ} ·SO(3, ℂ)·SL(2, ℂ) in ℂ^{6}.

In fact, for the above representation, K(ℂ + 𝔰𝔬(3, ℂ) + 𝔰𝔩(2, ℂ)) = K(𝔰𝔬(3, ℂ) + 𝔰𝔩(2, ℂ)) which means that ℂ^{∗} · SO(3, ℂ) · SL(2, ℂ) can not occur as the holonomy of a torsion-free affine connection.

5.6. Proof of Theorem 5.1. Let G ⊂ GL(V) be an irreducible complex representation of a complex reductive Lie group which can be represented as a tensor product of two or more non-Abelian complex representations. Then, by Propositions 5.7, 5.11 and 5.12, G may occur as the holonomy of a nonmetric torsion-free affine connection only if it is either ℤ_{ℂ} · SL(m, ℂ) · SL(n, ℂ) represented in ℂ^{m} ⊗ ℂ^{n} for m, n ⩾ 2, or SO(l, ℂ) · SL(2, ℂ) represented in ℂ^{l} ⊗ ℂ^{2} for l ⩾ 3.

Let ρ : G → GL(V) be an irreducible representation of a real reductive Lie group G in a real vector space V and let ρ : 𝔤 → 𝔤𝔩(V) be the associated real irreducible representation of the Lie algebra 𝔤 of G. The latter defines naturally a complex representation ρ_{ℂ} : 𝔤_{ℂ} → 𝔤𝔩(V_{ℂ}), where 𝔤_{ℂ} = 𝔤 ⊗ ℂ and V_{ℂ} = V ⊗ ℂ. Then two situations may arise [25]:

(i) The complex representation ρ_{ℂ} : 𝔤_{ℂ} → 𝔤𝔩(V_{ℂ}) is irreducible; in this case we denote ρ_{ℂ} by ρ̃;

(ii) There is a complex vector space W_{ℂ} and an irreducible complex representation ρ′ : 𝔤 → 𝔤𝔩(W_{ℂ}) such that V is the underlying real vector space of W_{ℂ} and ρ is the composition ρ : 𝔤 ⟶^{ρ′} 𝔤𝔩(W_{ℂ})⟶𝔤𝔩(V), where the second arrow is the natural inclusion of the algebra of all complex automorphisms of V into the algebra of all real automorphisms of V. Then the 𝔤_{ℂ}-module V_{ℂ} splits as a direct sum of two irreducible 𝔤_{ℂ}-submodules W_{ℂ} + W_{ℂ} and we denote by ρ̃: 𝔤_{ℂ} → 𝔤𝔩(W_{ℂ}) the restriction of ρ_{ℂ} to one of these.

In both cases, the 𝔤-modules K(ρ(𝔤)) and K^{1}(ρ(𝔤)) are subsets of K(ρ̃(𝔤_{ℂ})) and K^{1}(ρ̃(𝔤_{ℂ})) respectively. In particular, if K(ρ(𝔤)) and K^{1}(ρ(𝔤)) are nonzero, then K(ρ̃(𝔤_{ℂ})) and K^{1}(ρ̃(𝔤_{ℂ})) are nonzero as well.

Assume now that the semisimple part of 𝔤 has at least two non-Abelian ideals. Then the Borel-Weil data associated to the irreducible matrix subalgebra ρ̃(𝔤_{ℂ}) must be of the form (𝕏, 𝕃) = (X_{1} × X_{2}, π_{1}^{∗}(L_{1}) ⊗ π_{2}^{∗}(L_{2})) for some compact complex homogeneous-rational manifolds X_{1} and X_{2} and ample line bundles L_{1} → X_{1} and L_{2} → X_{2}.

Then, by Propositions 5.7, 5.11 and 5.12, ρ(G) ⊂ GL(V) must be of the form ℤ_{ℂ} · G_{1} · G_{2}, where ℤ_{ℂ} is a connected real Lie subgroup of ℂ^{∗} and G_{i} ⊂ GL(V_{i}), i = 1, 2, is one of the following real matrix groups

Group G_{i}:  SL(n, ℂ) SL(n, ℝ) SU(n) SL(m, H)
Representation space V_{i}:  ℝ^{2n}  ℝ^{n}  ℝ^{2n}  ℝ^{4m}
Group G_{i}:  SO(l, ℂ) SO(p, q) SO(n, H)
Representation space V_{i}:  ℝ^{2l}  ℝ^{p+q}  ℝ^{4n}

with n ⩾ 2, m ⩾ 1, l ⩾ 3, p + q ⩾ 3.

Since we know K(ρ̃(𝔤_{ℂ})) explicitly, it is straightforward to check that the only combinations ρ(G) = ℤ_{ℂ} · G_{1} · G_{2} which (i) have K^{1}(ρ(𝔤)) ≠ 0 and (ii) have no proper subgroup G′ ⊂ G with K(ρ(𝔤′)) = K(ρ(𝔤)) are the ones given in the Table 7.

6. Proof of the Main Theorem

6.1. Adjoint representations. In this subsection, we shall rule out the (central extensions of the) adjoint representations of simple Lie groups as candidates for holonomies. In the Borel-Weil language, this is a family of representations which have contact complex homogeneous manifolds as their skies.

Proposition 6.1. Let 𝔤_{s} be a complex simple Lie algebra with rank(𝔤_{s}) ⩾ 3, and let 𝔤 := ℂ ⊕ 𝔤_{s}. Consider the embedding ı : 𝔤 ↪ 𝔤𝔩(𝔤_{s}) via the central extension of the adjoint representation. Then ı(𝔤) ⊂ 𝔤𝔩(𝔤_{s}) is not a Berger algebra.

Proof. Fix a Cartan decomposition 𝔤_{s} = 𝔱 ⊕ ⨁_{α∈Φ} 𝔤_{α}, and fix elements 0 ≠ A_{α} ∈ 𝔤_{α}. Moreover, denote elements of 𝔱 by A_{0}, B_{0}, . . ..

Let us suppose that ı(𝔤) is a Berger group. Then, by Lemma 2.3, K(ı(𝔤)) is a nontrivial 𝔤-module. Also, it is evident that the weights of K(ı(𝔤)) ⊂ Λ^{2}𝔤_{s}^{∗}⊗𝔤 all lie in the root lattice of 𝔱^{∗}. It follows that there must be a weigth ρ of K(ı(𝔤)) which is a root. Let 0 ≠ R ∈ K(ı(𝔤)) be a corresponding weight element.

We write R(x, y) = {x, y} − ω(x, y)Id with {x, y} ∈ 𝔤_{s} and a 2-form ω. Then the Bianchi identity reads

(32) [{x, y}, z] + [{y, z}, x] + [{z, x}, y] = ω(x, y)z + ω(y, z)x + ω(z, x)y.
Step 1. For all A_{0}, B_{0} ∈ 𝔱, {A_{0}, B_{0}} = 0 and ω(A_{0}, ) = 0.
Proof. For (x, y, z) = (A_{0}, B_{0}, A_{−ρ}), (32) reads
[\underbrace{\{A_{0}, B_{0}\}}_{∈𝔤_{ρ}}, A_{−ρ}] + [\underbrace{\{B_{0}, A_{−ρ}\}}_{∈𝔱}, A_{0}] + [\underbrace{\{A_{−ρ}, A_{0}\}}_{∈𝔱}, B_{0}]

= \underbrace{ω(A_{0}, B_{0})}_{=0} A_{−ρ} + ω(B_{0}, A_{−ρ})A_{0} + ω(A_{−ρ}, A_{0})B_{0},

and thus, if [A_{ρ}, A_{−ρ}] ∉ span(A_{0}, B_{0}), then {A_{0}, B_{0}} = 0 and ω(A_{0}, A_{−ρ}) = 0. Since rank(𝔤) ⩾ 3 and ω has weight ρ, the assertion follows.

Step 2. For each root α ≠ −ρ, there is a constant c_{α} such that {A_{0}, A_{α}} = c_{α}(α + ρ)(A_{0})A_{α+ρ} for all A_{0} ∈ 𝔱. We set c_{α} = 0 if α + ρ is not a root.

Proof. The assertion is obvious if α + ρ is not a root. If α + ρ is a root, then from Step 1 and (32) for (x, y, z) = (A_{0}, B_{0}, A_{α}) we get

0 = [\underbrace{\{B_{0}, A_{α}\}}_{∈𝔤_{α+ρ}}, A_{0}] + [\underbrace{\{A_{α}, A_{0}\}}_{∈𝔤_{α+ρ}}, B_{0}]
= −(α + ρ)(A_{0}){B_{0}, A_{α}} + (α + ρ)(B_{0}){A_{0}, A_{α}}.

It follows that for all A_{0} ∈ (α + ρ)^{⊥} we have {A_{0}, A_{α}} = 0, and from there the claim follows.

Step 3. For each root α ≠ −ρ we have {A_{0}, A_{α}} = 0; moreover, ω = 0.

Proof. The first claim is obvious if α + ρ is not a root; thus we assume the contrary and let β := −(α + ρ). Then, using Step 2 and {A_{α}, A_{β}} ∈ 𝔱, we get from (32) for (x, y, z) = (A_{0}, A_{α}, A_{β}) that

−c_{α}β(A_{0})[A_{−β}, A_{β}] + c_{β}α(A_{0})[A_{−α}, A_{α}] = ω(A_{α}, A_{β})A_{0}.

Since α ≠ ±ρ, it follows that α and β are independent, and then we prove the claim using the fact that rank(𝔤) ⩾ 3.

Step 4. {A_{α}, A_{β}} = 0 for all roots α, β ≠ −ρ.

Proof. If α + β + ρ ≠ 0 then {A_{α}, A_{β}} = 0 follows from the previous steps by application of (32) to (x, y, z) = (A_{0}, A_{α}, A_{β}). Thus, we assume now that α + β + ρ = 0.

Let γ ∉ {α, β, α + β} be a root. Again, applying (32) to (x, y, z) = (A_{α}, A_{β}, A_{γ}) and using the previous steps and the preceding observation, we get γ({A_{α}, A_{β}}) = 0 for all such roots γ. The claim follows.

Step 5. For each root α, there is a constant d_{α} such that {A_{−ρ}, A_{α}} = d_{α}A_{α}. Moreover, if α, β ≠ −ρ are roots such that α + β is a root, then d_{α} = −d_{β}. If α ≠ ±ρ, then d_{−α} = −d_{α}.

Proof. Since {A_{−ρ}, A_{α}} ∈ 𝔤_{α}, the existence of d_{α} follows. Applying (32) to (x, y, z) = (A_{−ρ}, A_{α}, A_{β}) yields (d_{α} + d_{β})[A_{α}, A_{β}] = 0. The second assertion follows, while the third is obtained by setting β = −α.

Step 6. d_{α} = 0 for all roots α; i.e. {A_{α}, A_{−ρ}} = 0 for all roots α.

Proof. Let α, β be roots such that α + β is a root and ρ ∉ span(α, β). Then applying Step 5 several times, we get d_{α} = −d_{β} = d_{−(α+β)} = −d_{α+β} = d_{−α} = −d_{α}, and thus, d_{α} = 0 for all α ≠ ±ρ. Next, let α ≠ ±ρ be a root such that α + ρ is a root. Then Step 5 implies that d_{ρ} = −d_{α} = 0. Finally, d_{−ρ} = 0 follows from the definition.

Step 7. R = 0, which yields a contradiction.

Proof. Let α ≠ −ρ be a root. Applying (32) to (x, y, z) = (A_{0}, A_{α}, A_{−ρ}) and using the previous steps yields α({A_{0}, A_{−ρ}}) = 0 for all α ≠ −ρ, and hence {A_{0}, A_{−ρ}} = 0. But now it follows that R = 0.

6.2. Classification of complex Berger algebras. Let 𝔤 ⊂ 𝔤𝔩(V) be an irreducible complex representation, and let 𝔤_{s} denote the semisimple part of 𝔤. For each root α of 𝔤_{s}, we fix 0 ≠ A_{α} ∈ 𝔤_{α}, and let

W_{α} := {weights of A_{α}V }.

Lemma 6.2. Let 𝔤 ⊂ 𝔤𝔩(V) be a Berger algebra. Then for every root α of 𝔤_{s}, there are weights λ_{0},λ_{1} of V such that

(33)  W_{α} ⊂ {λ_{0} + β, λ_{1} + β | β a root or = 0}.

In fact, if R ∈ K(𝔤) is a weight element and if there are weight vectors x_{i} ∈ V of weights λ_{i} for i = 0, 1 such that R(x_{0}, x_{1}) = A_{α}, then (33) holds for these weights.

Proof. We first show the second assertion. Let R ∈ K(𝔤) and x_{i} ∈ V as required. Then, for any y ∈ V, the first Bianchi identity of R ∈ K(𝔤) reads

A_{α}y = R(y, x_{1})x_{0} − R(y, x_{0})x_{1} ∈ span{𝔤x_{0}, 𝔤x_{1}},

i.e. A_{α}V ⊂ span{𝔤x_{0}, 𝔤x_{1}}. Then (33) holds since both A_{α}V and span{𝔤x_{0}, 𝔤x_{1}} are a direct sum of weight spaces, and the weights of the latter are contained in the right-hand side of (33).

To show that such an R exists for all roots, let

S := \{α ∈ Φ | \begin{matrix}there are weight elements R ∈ K(𝔤), x_{0}, x_{1} ∈ V \\ such that R(x_{0}, x_{1}) = A_{α} \end{matrix} \} .

Since K(𝔤) and V are spanned by their weight vectors, it follows that

\underline{𝔤} ⊂ 𝔱_{0} ⊕ ⨁_{α∈S} 𝔤_{α}.

Then, since 𝔤 is Berger, it follows that S = Φ.

Given an irreducible 𝔤 ⊂ 𝔤𝔩(V), we call a weight λ of V extremal if it lies in the orbit of the maximal weight under the Weyl group W. We then get the following slight generalization of Lemma 6.2.

Lemma 6.3. Let 𝔤 ⊂ 𝔤𝔩(V) be a Berger algebra. Then at least one of the following holds.

1. There are extremal weights λ_{0}, λ_{1} and a root α for which (33) holds.

2. There is an affine hyperplane in 𝔱^{∗} which contains all but two extremal weights.

3. There are an extremal weight λ_{0} and a root α such that for all extremal weights λ_{1} with ⟨λ_{1}, α⟩ < 0, λ_{1} − λ_{0} + α is a root or = 0.

Proof. Suppose there is a Berger algebra 𝔤 ⊂ 𝔤𝔩(V) for which none of these conditions holds. Let R ∈ K(𝔤) be a weight element. By Lemma 6.2, the negation of the first condition implies that if x_{i} is an extremal weight vector, i.e. if x_{i} ∈ V_{λ_{i}} with extremal weights λ_{i}, then R(x_{0}, x_{1}) ∉ 𝔤_{α} for any root α; hence R(x_{0}, x_{1}) ∈ 𝔱_{0}. Now let λ_{2} ≠ λ_{0}, λ_{1} be any other extremal weight, and let x_{2} ∈ V_{λ_{2}}. Then the first Bianchi identity for (x_{0}, x_{1}, x_{2}) implies that R(x_{0}, x_{1})x_{2} = 0 for all such x_{2}. But if the second condition from above is violated, then this is impossible unless R(x_{0}, x_{1}) = 0.

Thus, we have that R(x_{0}, x_{1}) = 0 whenever both x_{i} are extremal weight vectors. Now let λ be an arbitrary weight and let x ∈ V_{λ}. If R(x_{0}, x) = A_{α} then the first Bianchi identity for (x_{0}, x_{1}, x) would imply that A_{α}x_{1} ∈ 𝔤 · x_{0}. But if the third condition from above is violated, we can choose λ_{1} such that this is not the case, contradicting R(x_{0}, x) = A_{α}. On the other hand, if R(x_{0}, x) ∈ 𝔱_{0} for all x ∈ V, then by the Bianchi identity for (x_{0}, x_{1}, x) with an extremal weight vector x_{1}, the violation of the second condition above implies that R(x_{0}, x) = 0.

Thus, it follows that R(x_{0}, ) = 0 for all extremal weight vectors x_{0}. Now, let x, y ∈ V be arbitrary and apply the Bianchi identity to the triple (x_{0}, x, y). It follows that R(x, y)x_{0} = 0 for all extremal weight vectors x_{0}. From here, it follows that R = 0, i.e. K(𝔤) = 0, contradicting that 𝔤 ⊂ 𝔤𝔩(V) is Berger.

For the remainder of this section, we assume that 𝔤_{s} is simple. If in addition 𝔤 is not isomorphic to the exceptional Lie algebra 𝔤_{2}, then we have the following conditions:

(34) If α, β are roots then α + 3β is not a root.

(35) If α, β are roots, then | ⟨β, α⟩ | ⩽ 2; if α is a long root then equality holds if and only if α = ±β.

(36) If α is a long root and β is any root, then 2α + β is a root if and only if β = −α.

Proposition 6.4. Let 𝔤 ⊂ 𝔤𝔩(V) be an irreducible Berger algebra such that 𝔤_{s} is simple and not isomorphic to 𝔤_{2}. If α is a root such that there is a root β with ⟨β, α⟩ = 1, then | ⟨λ, α⟩ | ⩽ 3 for all weights λ of V.

Proof. Step 1. If α, β are roots with ⟨β, α⟩ = 1 and if there is a weight λ with ⟨λ, α⟩ ⩽ −4, then ⟨λ, β⟩ ⩾ −2.

Proof. Let α, β, λ be as required, and let λ_{0}, λ_{1} be as in (33). Then λ + kα ∈ W_{α} for k = 1, …, 4, and thus λ + α = λ_{0} + β_{0}, where β_{0} is a root or = 0. Also, since by (34) β_{0} + 3α is not a root,

(37)  \begin{aligned}λ + α & = λ_{0} + β_{0} \\ λ + 4α & = λ_{1} + β_{1} \end{aligned} where β_{i} is a root or = 0 for i = 1, 2.

Suppose that ⟨λ, β⟩ ⩽ −3. Then λ + 3β is a weight, and ⟨λ + 3β, α⟩ ⩽ −1. Therefore, W_{α} ∋ λ + 3β + α = λ_{0} + β_{0} + 3β = λ_{1} + β_{1} + 3(β − α). Now since ⟨β, α⟩ = 1, β − α is a root, hence, by (34), neither β_{0} + 3β nor β_{1} + 3(β − α) are roots, contradicting (33).

Step 2. Suppose that α is a root and λ a weight with ⟨λ, α⟩ ⩽ −4. Then for any root β with ⟨β, α⟩ = 1 we have ⟨α, β⟩ = 1 and ⟨λ, β⟩ = −2.

Proof. We rescale the inner product such that (α, α) = 2. Let α, λ be as requested, and let β be a root with ⟨β, α⟩ = 1. Then α − β is another root with ⟨α − β, α⟩ = 1. Thus, we can apply Step 1 to both β and α − β:

\begin{matrix}(λ, β) & ⩾ −(β, β), \\ (λ, α − β) & ⩾ −(α − β, α − β) = −(β, β). \end{matrix}

Adding these, we have (λ, α) ⩾ −2(β, β). But ⟨β, α⟩ = 1 implies that (β, β) ⩽ 2, and, by hypothesis, (λ, α) ⩽ −4; hence all inequalities are equalities. In particular, (β, β) = (α, α); thus ⟨α, β⟩ = ⟨β, α⟩ = 1.

Step 3. With the hypothesis from the proposition, ⟨λ, α⟩ ⩾ −3.

Proof. Let α, β be roots with ⟨β, α⟩ = 1, and suppose there is a weight λ with ⟨λ, α⟩ ⩽ −4. Choose β_{0}, β_{1} as in (37). By Step 2, we have that ⟨λ, β⟩ = −2, hence λ + 2β is a weight, and ⟨λ + 2β, α⟩ ⩽ −2. Therefore, W_{α} ∋ λ + 2β + 2α = λ_{0} + β_{0} + α + 2β. If β_{0} + α + 2β were a root or = 0, then applying (34) twice and using ⟨α, β⟩ = 1 from Step 2, we would get

2 ⩾ ⟨β_{0} + α + 2β, β⟩ = ⟨β_{0}, β⟩ + 5 ⩾ 3,

which is impossible. Thus, since W_{α} ∋ λ + 2β + 2α = λ_{1} + β_{1} + 2(β − α),

(33) implies that β_{1} + 2(β − α) must be a root. This means that β_{1} is a root with ⟨α − β, β_{1}⟩ = 2 for every root β with ⟨β, α⟩ = 1. But then, replacing β by α − β, we also have ⟨β, β_{1}⟩ = 2, and thus ⟨α, β_{1}⟩ = ⟨α − β, β_{1}⟩ + ⟨β, β_{1}⟩ = 4 which is impossible.

Application of Step 3 to both α and −α concludes the proof of the proposition.

Proposition 6.5. Let 𝔤 ⊂ 𝔤𝔩(V) be an irreducible Berger algebra such that 𝔤_{s} is simple, not isomorphic to 𝔤_{2} and rank(𝔤_{s}) > 1. Moreover, assume that the root system of long roots of 𝔤_{s} is irreducible. Then for every weight λ and every long root α, | ⟨λ, α⟩ | ⩽ 2.

Note that the additional hypothesis only excludes those 𝔤_{s} whose Dynkin diagram is of type C_{n}; i.e., we assume 𝔤_{s} ≅ 𝔰̸𝔭(n, ℂ).

Proof. Since for a long root α there exists a root β with ⟨β, α⟩ = 1 if rank(𝔤_{s}) > 1, Proposition 6.4 implies that | ⟨λ, α⟩ | ⩽ 3 if α is a long root. Suppose that | ⟨λ, α⟩ | = 3. After replacing α by its negative if necessary, we assume without loss of generality that ⟨λ, α⟩ = −3. Let λ_{0}, λ_{1} be weights satisfying (33).

Case 1. λ_{i} ≠ λ + 2α for i = 0, 1. By hypothesis, λ + kα ∈ W_{α} for k = 1, 2, 3. Then λ + α = λ_{0} + β_{0} with β_{0} ≠ −α. Thus, by (36), β_{0} + 2α is not a root, and we have

(38)  \begin{aligned}λ + α & = λ_{0} + β_{0} \\ λ + 3α & = λ_{1} + β_{1} \end{aligned} where β_{i} is a root or = 0 for i = 1, 2.

Let β be a long root with ⟨β, α⟩ = 1, which exists since the root system of long roots is irreducible. After replacing β by α − β if necessary, we may assume that ⟨λ, β⟩ ⩽ −2. Then λ + 2β is a weight with ⟨λ + 2β, α⟩ = −1, and hence W_{α} ∋ λ + 2β + α = λ_{0} + β_{0} + 2β = λ_{1} + β_{1} + 2(β − α). Since ⟨β, α⟩ = 1, β − α is a long root. Thus, (33) and (36) imply that either β = −β_{0} or β = α − β_{1}. In other words, the only possible long roots β with ⟨β, α⟩ = 1 are −β_{0}, α + β_{0}, α − β_{1} and β_{1}. If rank(𝔤_{s}) ⩾ 4 then there are more than four long roots β with ⟨β, α⟩ = 1, which yields a contradiction.

Next, suppose that rank(𝔤_{s}) = 3. Then, under the above hypothesis, the root system of long roots of 𝔤_{s} is A_{3}. Thus for any long root α, there are exactly four long roots β with ⟨β, α⟩ = 1, and therefore, ⟨−β_{0}, α⟩ = ⟨β_{1}, α⟩ = 1. Now let R ∈ K(𝔤) be a weight element of weight ρ such that R(x_{0}, x_{1}) = A_{α} for weight vectors x_{i} ∈ V_{λ_{i}}. Then ρ = λ_{0} + λ_{1} − α, and hence, ⟨ρ, α⟩ = 0.

Let x ∈ V_{λ} and x′ ∈ V_{λ+2α}. Then R(x_{0}, x) ∈ 𝔤_{α+λ−λ_{1}} = 𝔤_{2α−β_{1}} = 0, and R(x_{1}, x) ∈ 𝔤_{α+λ−λ_{0}} = 𝔤_{−β_{0}}. Then the Bianchi identity for (x_{0}, x_{1}, x) and the fact that A_{α}x ≠ 0 imply that 0 ≠ R(x_{1}, x) ∈ g_{−β_{0}}. Similarly, the Bianchi identity for (x_{0}, x_{1}, x′) implies that 0 ≠ R(x_{0}, x′) ∈ 𝔤_{β_{1}}.

Case 2. λ_{0} = λ + 2α. Let β as before, and again assume without loss of generality that ⟨λ, β⟩ ⩽ −2. Then W_{α} ∋ λ + 2β + α = λ_{0} − α + 2β, and since −α + 2β is not a root, it follows from (33) that

λ + α + 2β = λ_{1} + β_{1}, β_{1} a root or = 0.

Let γ be a long root with ⟨γ, α⟩ = ⟨γ, β⟩ = 1.

Suppose that ⟨λ, β − γ⟩ ⩽ −2. Then λ+2(β−γ)+kα ∈ W_{α} for k = 1, 2, 3. Now

\begin{aligned}λ + 2(β − γ) + α & = λ_{0} − α + 2(β − γ) = & λ_{1} + β_{1} − 2γ, \\ λ + 2(β − γ) + 3α & = λ_{0} + α + 2(β − γ) = & λ_{1} + β_{1} + 2(α − γ). \end{aligned}

But by (36), ±α + 2(β − γ) is not a root or = 0, and not both β_{1} − 2γ and β_{1} + 2(α − γ) can be a root or = 0, violating (33). Thus, ⟨λ, β − γ⟩ ⩾ −1. Since ⟨λ, β⟩ ⩽ −2, this implies, in particular, that ⟨λ, γ⟩ ⩽ −1.

Suppose that ⟨λ, γ⟩ = −1. Then, by the above, ⟨λ, β⟩ = −2, and hence λ + (β − γ) + kα ∈ W_{α} for k = 1, 2, 3. Now,

\begin{aligned}λ + (β − γ) + α & = λ_{0} − α + (β − γ) = & λ_{1} + β_{1} − β − γ, \\ λ + (β − γ) + 3α & = λ_{0} + α + (β − γ) = & λ_{1} + β_{1} − β − γ + 2α. \end{aligned}

Since ⟨β − γ, α⟩ = 0 and these are long roots, ±α + (β − γ) is not a root. Therefore, by (33), both β_{1} − β − γ and β_{1} − β − γ + 2α must be roots, implying, by (36), that β_{1} − β − γ = −α. But then 1 = ⟨α, γ⟩ = − ⟨β_{1}, γ⟩ + 3, implying by (35) that β_{1} = γ; i.e., α = β, which is impossible.

Thus, ⟨λ, γ⟩ ⩽ −2, and therefore, W_{α} ∋ λ + 2γ + α = λ_{0} − α + 2γ = λ_{1}+β_{1}−2(β−γ). Now, since −α+2γ is neither a root nor = 0, β_{1}−2(β−γ) must be a root or = 0 by (33), and (36) implies that β_{1} = β−γ; i.e., λ_{1} = λ+α+β+γ. But this means that there is at most one long root γ with ⟨γ, α⟩ = ⟨γ, β⟩ = 1, namely γ = β − β_{1}. If rank(𝔤_{s}) ⩾ 4 then there is more than one such γ, which yields a contradiction.

Next, suppose that rank(𝔤_{s}) = 3. Let R ∈ K(𝔤) be a weight element of weight ρ such that R(x_{0}, x_{1}) = A_{α} for weight vectors x_{i} ∈ V_{λ_{i}}. Then again, ρ = λ_{0} + λ_{1} − α, and hence, ⟨ρ, α⟩ = 0 follows.

Let x ∈ V_{λ+β} and x′ ∈ V_{λ+2β} such that A_{α}x, A_{α}x′ are both ≠ 0. Then an analysis similar to Case 1 implies that 0 ≠ R(x_{0}, x′) ∈ 𝔤_{β−γ}, and either 0 ≠ R(x_{0}, x) ∈ 𝔤_{−γ} or 0 ≠ R(x_{1}, x) ∈ 𝔤_{β−α}.

From these two cases, the proposition follows for rank(𝔤) ⩾ 4. If rank(𝔤) = 3, then in either case, if R ∈ K(𝔤) is a weight element of weight ρ such that R(x_{0}, x_{1}) = A_{α} for weight vectors x_{i} ∈ V_{λ_{i}} and a long root α then ⟨ρ, α⟩ = 0. But then, in the first case above, it follows that ⟨ρ, α⟩ = ⟨ρ, β_{0}⟩ = ⟨ρ, β_{1}⟩ = 0, and hence ρ = 0. Treating the second case similarly we conclude the following:

For every weight element R ∈ K(𝔤) of weight ρ ≠ 0 and any x_{i} ∈ V_{λ_{i}}, either R(x_{0}, x_{1}) ∈ 𝔱_{0} or R(x_{0}, x_{1}) is a root vector of a short root.

But now a straightforward analysis shows that this is impossible; i.e., the only weight of K(𝔤) is ρ = 0; i.e., K(𝔤) is a trivial 𝔤-module, and we get a contradiction from Lemma 2.3.

In the case rank(𝔤) = 2, i.e., if the root system is of type A_{2}, the statement is shown by a direct and straightforward analysis.

Proposition 6.6. Let 𝔤 ⊂ 𝔤𝔩(V) be as in Proposition 6.4, such that 𝔤_{s} ≅ 𝔰̸𝔬(7, ℂ) and rank(𝔤_{s}) > 2. Suppose there is a weight λ and a (not necessarily long) root α with | ⟨λ, α⟩ | ⩾ 2. Then one of the following must hold.

1. | ⟨λ, β⟩ | ⩽ 1 for all long roots β with ⟨α, β⟩ = 0, or

2. There is a unique long root γ with ⟨α, γ⟩ = 0 and ⟨λ, γ⟩ ⩾ 2, and the root system {long roots β | ⟨α, β⟩ = 0} contains {±γ} as a direct summand.

We shall split the proof of this proposition into two lemmas.

Lemma 6.7. Let 𝔤, V, λ and α be as in Proposition 6.6. Then the set

{long roots γ | ⟨α, γ⟩ = 0, ⟨λ, γ⟩ ⩾ 2}

contains at most one element.

Proof. After replacing α by −α if necessary, we assume without loss of generality that ⟨λ, α⟩ ⩽ −2. Let λ_{0}, λ_{1} be as in (33). Suppose there are two distinct long roots γ_{1}, γ_{2} with ⟨α, γ_{i}⟩ = 0 and ⟨λ, γ_{i}⟩ ⩾ 2 for i = 1, 2. If γ_{1} + γ_{2} were a root then ⟨λ, γ_{1} + γ_{2}⟩ ⩾ 4, contradicting Proposition 6.4. It follows that ⟨γ_{1}, γ_{2}⟩ ∈ {0, 1}.

Case 1. ⟨γ_{1}, γ_{2}⟩ = 0. In this case, we have that λ + kα − rγ_{1} − sγ_{2} ∈ W_{α} for k = 1, 2, r, s = 0, 1, 2. Thus, by (33),

λ + α = λ_{0} + β_{0}, where β_{0} is a root or = 0.

After possibly switching γ_{1} and γ_{2}, we assume without loss of generality that β_{0} ≠ γ_{2}, i.e. β_{0} − 2γ_{2} is not a root. Therefore, by (33),

λ + α − 2γ_{2} = λ_{1} + β_{1}, where β_{1} is a root or = 0.

Next, we consider W_{α} ∋ λ+α−2γ_{1} = λ_{0} +β_{0} −2γ_{1} = λ_{1} +β_{1} −2γ_{1} +2γ_{2}. Since ⟨β_{1} − 2γ_{1} + 2γ_{2}, γ_{2}⟩ = ⟨β_{1}, γ_{2}⟩ + 4 ⩾ 2 with equality if and only if β_{1} = −γ_{2} by (35), it follows that β_{1} − 2γ_{1} + 2γ_{2} is not a root. Thus, by (33) and (36), β_{0} = γ_{1}, i.e. λ_{0} = λ + α − γ_{1}.

Now consider W_{α} ∋ λ + 2α − 2γ_{1} = λ_{0} + α − γ_{1} = λ_{1} + β_{1} + α − 2γ_{1} + 2γ_{2}. Since ⟨γ_{1}, α⟩ = 0, α − γ_{1} is not a root, hence β_{1} + α − 2γ_{1} + 2γ_{2} must be a root or = 0 by (33), and this yields a contradiction in a similar way.

Case 2. ⟨γ_{1}, γ_{2}⟩ = 1. In this case, we have that λ + kα − rγ_{1} − sγ_{2} ∈ W_{−α} for k = 1, 2, r, s = 0, 1, 2 with r + s ⩽ 2. Thus, by (33),

λ + α = λ_{0} + β_{0}, where β_{0} is a root or = 0.

Again, without loss of generality we may assume β_{0} ≠ γ_{2}, and therefore,

λ + α − 2γ_{2} = λ_{1} + β_{1}, where β_{1} is a root or = 0.

Next, we consider W_{α} ∋ λ+α−2γ_{1} = λ_{0} +β_{0} −2γ_{1} = λ_{1} +β_{1} +2(γ_{2} −γ_{1}). By (33) and (36), either β_{0} = γ_{1} or β_{1} = γ_{1}−γ_{2}. If β_{0} = γ_{1}, i.e. λ_{0} = λ+α−γ_{1}, then W_{α} ∋ λ + 2α = λ_{0} + α + γ_{1} = λ_{1} + β_{1} + α + 2γ_{2}. But α + γ_{1} is not a root as ⟨γ_{1}, α⟩ = 0, and β_{1} + α + 2γ_{2} is seen not to be a root or = 0 in a similar way as before. Hence, we get a contradiction from (33). The case β_{1} = γ_{1} − γ_{2} is ruled out in a similar way, and this proves the lemma.

Lemma 6.8. Let 𝔤, V, λ and α as before, and suppose the set in Lemma 6.7 is nonempty. Then there is no long root δ other than ±γ with ⟨α, δ⟩ = 0 and ⟨γ, δ⟩ ≠ 0, unless 𝔤_{s} ≅ 𝔰𝔬(7, ℂ), and α is a short root.

Proof. The claim is obvious if the root system of long roots is reducible (i.e. 𝔤_{s} ≅ 𝔰𝔭(n, ℂ)) or if rank(𝔤) ⩽ 3. Thus, without loss of generality we may assume that the premises of Proposition 6.5 are satisfied, and therefore we may assume that ⟨λ, α⟩ = − ⟨λ, γ⟩ = −2. As usual, let λ_{0}, λ_{1} be the weights satisfying (33).

Suppose there is a long root δ ≠ ±γ with ⟨α, δ⟩ = 0 and ⟨γ, δ⟩ ≠ 0. After replacing δ by −δ if necessary, we may assume that ⟨γ, δ⟩ = 1. Then γ − δ is another such root, and because of Lemma 6.7, we have both ⟨λ, δ⟩ ⩽ 1 and ⟨λ, γ − δ⟩ ⩽ 1, i.e. ⟨λ, δ⟩ = 1. It follows that

{λ + kα − rγ, λ + kα − δ | k = 1, 2, r = 0, 1, 2} ⊂ W_{α}.

Case 1. λ_{i} ≠ λ + α − γ for i = 0, 1. As in previous cases, (33) yields

\begin{aligned}λ + α & = λ_{0} + β_{0}, \\ λ + α − 2γ & = λ_{1} + β_{1}, \end{aligned} where β_{0} ≠ γ and β_{1} ≠ −γ are roots or = 0.

Now, W_{α} ∋ λ+2α−2γ = λ_{0}+β_{0}+α−2γ = λ_{1}+β_{1}+α. But ⟨β_{0} + α − 2γ, γ⟩ = ⟨β_{0}, γ⟩ − 4 ⩽ −3, since β_{0} ≠ γ, and thus β_{0} + α − 2γ is not a root. Therefore, β_{1} + α must be a root or = 0.

Next, for k = 1, 2, W_{α} ∋ λ + kα − δ = λ_{0} + β_{0} + (k − 1)α − δ = λ_{1} + β_{1} + (k − 1)α + 2γ − δ. But ⟨β_{1} + (k − 1)α + 2γ − δ, γ⟩ = ⟨β_{1}, γ⟩ + 3 ⩾ 2, since β_{1} ≠ −γ. Hence, if β_{1} + (k − 1)α + 2γ − δ was a root it would have to equal γ; i.e., β_{1} = −(k − 1)α + (δ − γ). If k = 2, this is impossible since ⟨δ − γ, α⟩ = 0. Also, if k = 1 this would contradict β_{1} + α being a root or = 0. Therefore, both β_{0} − δ and β_{0} + α − δ must be roots or = 0.

Finally, W_{α} ∋ λ + 2α = λ_{0} + β_{0} + α = λ_{1} + β_{1} + α + 2γ. We exclude that β_{1} + α + 2γ is a root or = 0 as before; hence β_{0} + α is a root or = 0.

Since β_{0}, β_{0} + α, β_{0} − δ, β_{0} + α − δ are all roots or = 0, we conclude that β_{0} ∉ {0, −α, δ}, and from there it follows that ⟨β_{0}, δ⟩ = 1 for all long roots δ with ⟨α, δ⟩ = 0, ⟨γ, δ⟩ = 1. However, if such a δ exists, then γ − δ is another such root; hence 1 = ⟨β_{0}, γ − δ⟩ = ⟨β_{0}, γ⟩ − ⟨β_{0}, δ⟩, and hence ⟨β_{0}, γ⟩ = 2, contradicting that β_{0} ≠ γ.

Case 2. λ_{0} = λ + α − γ. In this case, since γ + α is not a root, it follows that

λ + 2α = λ_{1} + β_{1}, where β_{1} is a root or = 0.

Moreover, λ + 2α − 2γ = λ_{0} + α − γ = λ_{1} + β_{1} − 2γ. Since α − γ is not a root,

(36) implies that β_{1} = γ, i.e.

λ_{1} = λ_{0} + α.

Thus, using Lemma 6.2 and the fact that the Weyl group acts transitively on the set of long roots, we conclude:

(39)

If x, y ∈ V are weight vectors of weight λ, μ respectively, and if R ∈ K(𝔤) is a weight element such that R(x, y) = A_{β} for some long root β, then λ − μ must be a long root.

Now, by Lemma 6.2, there is a weight element R ∈ K(𝔤) such that R(x_{0}, x_{1}) = A_{α} where x_{i} ∈ V are weight vectors of weights λ_{i} for i = 0, 1. Let x_{2} be a weight vector of weight λ − δ. Then R(x_{0}, x_{2}) ∈ 𝔤_{α+γ−δ} = 0, since α+γ −δ is not a root, and R(x_{1}, x_{2}) ∈ 𝔤_{γ−δ}; however, since γ −δ is a long root, and λ_{1} − (λ − δ) = 2α − γ + δ is not a root, (39) implies that R(x_{1}, x_{2}) = 0. But now the Bianchi identity for x_{0}, x_{1}, x_{2} reads

A_{α}x_{2} = 0 for all x_{2} ∈ V_{λ−δ},

contradicting that λ − δ + α is a weight.

It is now clear that Proposition 6.6 follows immediately from Lemmas 6.7 and 6.8.

Proposition 6.9. Let 𝔤 ⊂ 𝔤𝔩(V) be an irreducible complex Lie subalgebra whose semisimple part is simple and has rank at least three. Then 𝔤 is a Berger algebra only if either its sky is biholomorphic to a compact hermiteansymmetric manifold, or if 𝔤 is (the central extension of ) the image of one of the following representations:

[figure: (i); 1; 1; 0; (ii); …; <; 0; 1; 0; 0; 0; (iii); >; 0; 0; 1; 0]
(iv) [figure: >; 0; 0; 0; 1] (v) [figure: 0; 1; 0; 0; 0; 0] (vi) [figure: 0; 0; 0; 0; 0; 0; 1] .

Proof. The list of irreducible hermitean symmetric spaces is given at the beginning of Section 4. We give the proof for each type of the Dynkin diagram of G.

1. Type A_{n}. In this case, the root system is Φ = {α_{i,j} := θ_{i} − θ_{j} | i ≠ j ∈ {1, …, n + 1} }, and the positive roots are Φ^{+} = {α_{i,j} | i < j}. The dominant weight of G can be represented in a unique way as λ_{0} = c_{1}θ_{1} + … + c_{n}θ_{n} with integers c_{1} ⩾ … c_{n} ⩾ 0. For convenience, we set c_{n+1} = 0. Note that due to the symmetry of the root system A_{n} we may assume without loss of generality that c_{1} − c_{2} ⩾ c_{n}. From Proposition 6.5 it follows that ⟨λ_{0}, α_{1,n+1}⟩ = c_{1} ⩽ 2, i.e. c_{1} = 1 or 2. If c_{1} = 1 then the representation of G on V is a fundamental representation, and its sky is hermitean symmetric. Thus, we may assume that c_{1} = 2. Since the system of roots perpendicular to α_{1,n+1} is A_{n−2} which is irreducible and does not contain A_{1} as a summand for n ⩾ 4, it follows from Proposition 6.6 that ⟨λ_{0}, α_{2,n}⟩ = c_{2} − c_{n} ⩽ 1 if n ⩾ 4. If n = 3 and c_{2} − c_{n} = 2 then λ_{0} = 2(θ_{1} + θ_{2}), and then the sky of G is hermitean symmetric. If c_{2} − c_{n} = 0 then, since c_{1} − c_{2} ⩾ c_{n}, either λ_{0} = 2θ_{1} or λ_{0} = 2θ_{1} + θ_{2} + … + θ_{n}. In the first case, the sky of G is hermitean symmetric, while in the second case G acts via the adjoint representation and hence is not Berger by Proposition 6.1. Thus, we may assume that c_{2} − c_{n} = 1 and, because of c_{1} − c_{2} ⩾ c_{n}, we have c_{2} = 1, c_{n} = 0. In other words,

λ_{0} = 2θ_{1} + θ_{2} + … + θ_{k}, for some k ∈ {2, …, n − 1}.

The Weyl group of A_{n} is the permutation group S_{n+1} which acts by permutation of the indices of θ_{1}, …, θ_{n+1}. Thus, the extremal weights are all elements of the form λ = 2θ_{m_{0}} + θ_{m_{1}} + θ_{m_{k}} for distinct indices m_{0}, …, m_{k}. From here, it is now straightforward to show that the last two conditions of Lemma 6.3 are violated, while the first one holds only if n = 3 and λ_{0} = 2θ_{1} + θ_{2}. This corresponds precisely to (i) above.

2. Type B_{n}. The root system is Φ = {±θ_{i}, ±θ_{i} ± θ_{j} | i < j, i = 1, …, n}, and the positive roots are Φ^{+} = {θ_{i}, θ_{i} ±θ_{j} | i < j, i = 1, …, n}. The maximal weight is given by λ_{0} = c_{1}θ_{1} + … + c_{n}θ_{n} with c_{1} ⩾ … ⩾ c_{n} ⩾ 0, where either all c_{k} are integers, or all c_{k} are half-integers. From Proposition 6.5 it follows that ⟨λ_{0}, θ_{1} + θ_{2}⟩ = c_{1} + c_{2} ⩽ 2. If c_{2} = 0, then obviously the sky of G is hermitean symmetric. Thus, we may assume that c_{2} > 0, and hence c_{1} ⩽ \frac{3}{2}.

Suppose that c_{1} = \frac{3}{2}. Then λ_{0} = θ_{1} + \frac{1}{2}(θ_{1} + … + θ_{n}). In this case, the extremal weights are of the form ±θ_{k} + \frac{1}{2}(±θ_{1} ± … ± θ_{n}). From here, one sees easily that the last two conditions of Lemma 6.3 are violated, while the first one holds only if n ⩽ 3. However, a more careful analysis shows that the case n = 3 can be ruled out for Berger groups as well.

Next, suppose that c_{1} = 1. If c_{2} = 0 then the corresponding sky is hermitean symmetric; thus we assume that c_{2} = 1, i.e.,

λ_{0} = θ_{1} + … + θ_{k}.

If k = 2 then this is the adjoint representation, which is not Berger; if k = n then the corresponding sky is hermitean symmetric. Thus, we assume that 3 ⩽ k ⩽ n − 1 and, in particular, that n ⩾ 4. It is now straightforward to show that none of the conditions in Lemma 6.3 hold; thus these groups are not Berger.

Finally, suppose that c_{1} = \frac{1}{2}. Then the corresponding representation is the spinor representation whose sky is hermitean symmetric.

3. Type C_{n}. The root system is Φ = {±2θ_{i}, ±θ_{i} ± θ_{j} | i < j, i = 1, …, n}, and the positive roots are Φ^{+} = {2θ_{i}, θ_{i}±θ_{j} | i < j, i = 1, …, n}. The maximal weight is given by λ_{0} = c_{1}θ_{1} + … + c_{n}θ_{n} with integers c_{1} ⩾ … ⩾ c_{n} ⩾ 0. From Proposition 6.4 it follows that ⟨λ_{0}, θ_{1} + θ_{2}⟩ = c_{1} + c_{2} ⩽ 3. If c_{2} = 0, then obviously the sky of G is hermitean symmetric. Thus, we may assume that c_{2} > 0, and hence c_{1} ⩽ 2.

Suppose that c_{1} = 2. Then λ_{0} = 2θ_{1} + θ_{2} + … + θ_{k}, with 2 ⩽ k ⩽ n. The extremal weights are then all weights of the form ±2θ_{m_{1}} ± θ_{m_{2}} ± … ± θ_{m_{k}} for some pairwise distinct indices m_{1}, …, m_{k}. It is now easy to see that the conditions in Lemma 6.3 are violated if n ⩾ 3; hence these subgroups cannot be Berger.

Next, suppose that c_{1} = 1; hence λ_{0} = θ_{1}+…+θ_{k} for some k ∈ {1, …, n}. The sky of this representation is hermitean symmetric if k = 1 or k = n; hence we assume that 2 ⩽ k ⩽ n − 1. Once again, the extremal weights are of the form ±θ_{m_{1}} ± … ± θ_{m_{k}} for some pairwise distinct indices m_{1}, …, m_{k}, and then straightforward calculation shows that the conditions in Lemma 6.3 are violated if k ⩾ 3.

Thus, we assume that k = 2, and this is precisely (ii) from above.

4. Type D_{n}. The root system is Φ = {±θ_{i} ± θ_{j} | i < j, i = 1, …, n}, and is hence contained in the root system B_{n}. Thus, the investigation of this case is very similar to that of the case B_{n}. The details are omitted.

5. Type F_{4}. Suppose that the maximal weight λ is given by [figure: >; c_{1}; c_{2}; c_{3}; c_{4}].

The maximal short root of F_{4} is given by α = [figure: >; 1; 2; 3; 2], and for β =

[figure: >; 0; 0; 0; 1] we have ⟨β, α⟩ = 1. Thus, Proposition 6.4 implies that ⟨λ, α⟩ = 2c_{1} + 4c_{2} + 3c_{3} + 2c_{4} ⩽ 3. Thus, c_{2} = 0 and exactly one of c_{1}, c_{3}, c_{4} is equal to 1. However, the case c_{1} = 1 corresponds to the adjoint representation which is not Berger by Proposition 6.1, and the two remaining cases are precisely (iii) and (iv).

6. Type E_{6}. Let λ be given by [figure: c_{1}; c_{2}; c_{3}; c_{4}; c_{5}; c_{6}]. The maximal root of E_{6}

is given by[figure: 1; 2; 3; 2; 1; 2], and the root system orthogonal to the maximal

root is A_{5} with maximal root [figure: 1; 1; 1; 1; 1; 0]. It follows from Propositions

6.5 and 6.6 that c_{1} +2c_{2} +3c_{3} +2c_{4} +c_{5} +2c_{6} ⩽ 2 and c_{1} +…+c_{5} ⩽ 1. If c_{6} = 1 then all other c_{i} = 0, and this describes the adjoint representation which is not Berger by Proposition 6.1. If c_{1} = 1 or c_{5} = 1, then all other c_{i} = 0, hence the sky of G is hermitean symmetric. Thus, the only possibility left is (v).

7. Type E_{7} or E_{8}. These cases are dealt with in complete analogy to E_{6}.

6.3. Skies of complex Berger algebras.

Theorem 6.10. Let 𝔤 ⊂ 𝔤𝔩(V) be an irreducible complex Lie subalgebra whose semisimple part is simple. Then 𝔤 is a Berger algebra only if its sky is biholomorphic to a compact hermitean-symmetric manifold.

Proof. By Proposition 6.9, it suffices to show that the representations given there are not Berger algebras, and to consider the case where rank(𝔤_{s}) = 2.

Type A_{3} (i). We have the associated Borel-Weil data (X, L) = × × 1 1 0 There is a natural projection X⟶X_{0} = [figure: ×] whose fibre X_{v} is ℙ_{2}. The short exact sequence (6) takes the form

T X = × × -1 1 1 + ν^{∗}([figure: ×; 1; 0; 1]).

By Lemma 2.7(i),

H^{1}(X, ⊙^{2}T X ⊗ L^{∗2}) ⊂ H^{1}(× × -2 0 2 ) = 0

and

H^{1}(X, ⊙^{3}T X ⊗ L^{∗2}) ⊂ H^{1}(× × -5 1 3 ) + H^{1}([figure: ×; -4; 0; 2] ⊗ [figure: ×; -1; 0; 1]) = 0.

Then, by the estimation (19), H^{1}(X, L ⊗ ⊙^{3}N^{∗}) = 0 implying K(𝔤 ⊕ ℂ) = 0.

Type A_{2}. Let (X, L) = × × m n with m ⩾ 2 and n ⩾ 1. There is a fibration ν : X → X_{0} = × . The short exact sequence (6) takes the form

T X = × × -1 2 + ν^{∗}(× 1 1 ),

implying

⊙^{2}T X ⊗ L^{∗2} = × × -2-2m 4-2n + × × -1-2m 2-2n ⊗ ν^{∗}(× 1 1 ) + × × -2m -2n⊗ ν^{∗}(× 2 2 )

and

⊙^{3}T X ⊗ L^{∗2} = × × -3-2m 6-2n + × × -2-2m 4-2n⊗ ν^{∗}(× 1 1 )
+ × × -1-2m 2-2n⊗ ν^{∗}(× 2 2 )
+ × × -2m -2n⊗ ν^{∗}(× 3 3 ).

Applying the direct image functor ν_{∗}^{i} and using the formula

× a b ⊗ × c d = ⨁_{0⩽i⩽b} × a+c+i-1 c+d-2i , b ⩽ d,

one easily finds that H^{1}(X, ⊙^{2}T X ⊗ L^{∗2}) = 0 for all m ⩾ 2, n ⩾ 1 and

H^{1}(X, ⊙^{3}T X ⊗ L^{∗2}) ⊂ \{ \begin{matrix}H^{1}(× -3 2 ) & for m = 2, n = 1 \\ 0 & otherwise. \end{matrix}

Hence, by (19), we get an estimation

K(𝔤 ⊕ ℂ) ⊂ \{ \begin{matrix}[figure: 1; 0] & for m = 2, n = 1 \\ 0 & otherwise. \end{matrix}

This leaves the cases where m = 2, n = 1 and m = n = 1. However, a direct analysis using arguments similar to those from subsection 7.1. shows that these representations do not yield Berger algebras. We omit further details.

Type B_{2}. The Borel-Weil data associated with a representation [figure: >; s; t], s, t ⩾ 1, are (X, L) =× >× s t . There is a projection ν : X → X_{0} = × > with the associated short exact sequence (6) given by

T X = × >× -1 2 + ν^{∗}(× > 0 2 ).

Then

⊙^{2}T X ⊗ L^{∗2} = × >× -2-2s 4-2t + × >× -1-2s 2-2t⊗ ν^{∗}(× > 0 2 ) + × >× -2s -2t ⊗ ν^{∗}(⊙^{2}× > 0 2 ),

implying H^{1}(⊙^{2}T X ⊗L^{∗2}) = 0 for all s, t ⩾ 1. A similar analysis of ⊙^{3}T X ⊗L^{∗2} produces an estimation

H^{1}(X, ⊙^{3}T X ⊗ L^{∗2}) ⊂ \{ \begin{matrix}H^{1}(× > -3 4 ) & for m = n = 1, \\ 0 & otherwise. \end{matrix}

Hence, by (19), we get an estimation

K(𝔤 ⊕ ℂ) ⊂ \{ \begin{matrix}[figure: >; 1; 0] & for m = n = 1 \\ 0 & otherwise. \end{matrix}

Finally, the case m = n = 1 can be analyzed by methods similar to those of of subsection 7.1. It turns out that 𝔤 ⊕ ℂ is not a Berger algebra in this case either. We omit the details.

Type C_{n} (ii), n ⩾ 3. The Borel-Weil data associated with the fundamental

representation of the second node are (X, L) = [figure: …; ×; <; 0; 1; 0; 0; 0; 0]. By (5),

T X =[figure: …; ×; <; 1; -1; 1; 0; 0; 0] + [figure: …; ×; <; 2; 0; 0; 0; 0; 0],

implying

⊙^{2}T X⊗L^{∗2} = [figure: …; ×; <; 2; -2; 2; 0; 0; 0; ⊕; …; ×; <; 0; -1; 0; 0; 0; 0; ⊕; …; ×; <; 0; -2; 0; 1; 0; 0] + [figure: …; ×; <; 3; -3; 1; 0; 0; 0; ⊕; …; ×; <; 1; -2; 1; 0; 0; 0] + [figure: …; ×; <; 4; -2; 0; 0; 0; 0; ⊕; …; ×; <; 0; 0; 0; 0; 0; 0]

where the summand [figure: …; ×; <; 0; -2; 0; 1; 0; 0] must be set to zero in the case n = 3. All the irreducible bundles on the right-hand side have the first cohomology vanishing except H^{1}( [figure: …; ×; <; 1; -2; 1; 0; 0; 0]) = ℂ which implies an estimation H^{1}(X, ⊙^{2}T X ⊗ L^{∗2}) ⊂ ℂ. In fact, the long exact sequence

0 → H^{0}([figure: …; ×; <; 0; 0; 0; 0; 0; 0]) → H^{1}([figure: …; ×; <; 1; -2; 1; 0; 0; 0])
→ H^{1}(X, ⊙^{2}T X ⊗ L^{∗2}) → 0

implies the vanishing of H^{1}(X, ⊙^{2}T X ⊗L^{∗2}). A similar analysis of ⊙^{3}T X ⊗L^{∗2} produces an estimation

H^{1}(X, ⊙^{3}T X ⊗ L^{∗2}) ⊂ H^{1}([figure: …; ×; <; 1; -2; 1; 0; 0; 0]) = ℂ

which, by (19), implies K(𝔤⊕ℂ) ⊂ H^{1}(X, L⊗⊙^{3}N^{∗}) ⊂ ℂ which in turn implies K^{1}(𝔤 ⊕ ℂ) = 0.

Type F_{4} (iv). The associated Borel-Weil data are (X, L) = [figure: ×; >; 0; 0; 0; 1].

By (5), T X = [figure: ×; >; 0; 0; 1; -1] + [figure: ×; >; 1; 0; 0; 0] , implying

⊙^{2}T X ⊗ L^{∗2} = [figure: ×; >; 0; 0; 2; -4; ⊕; ×; >; 0; 0; 0; -1] + [figure: ×; >; 1; 0; 1; -3; ⊕; ×; >; 0; 0; 1; -2] + \begin{matrix} & & & × & > & 2 & 0 & 0 & -2 \\ ⊕ \\ [figure: ×; >; 0; 0; 0; 0] . \end{matrix}

All the irreducible bundles on the right-hand side have the first cohomology vanishing except [figure: ×; >; 0; 0; 1; -2] whose first cohomology is ℂ. Therefore, H^{1}(X, ⊙^{2}T X ⊗ L^{∗2}) ⊂ ℂ. In fact, the long exact sequence

0 → H^{0}([figure: ×; >; 0; 0; 0; 0]) → H^{1}([figure: ×; >; 0; 0; 1; -2]) → H^{1}(X, ⊙^{2}T X ⊗ L^{∗2}) → 0

implies the vanishing of H^{1}(X, ⊙^{2}T X ⊗L^{∗2}). A similar analysis of ⊙^{3}T X ⊗L^{∗2} produces an estimation

H^{1}(X, ⊙^{3}T X ⊗ L^{∗2}) ⊂ H^{1}([figure: ×; >; 0; 0; 1; -2]) = ℂ

which, by (19), implies K(𝔤⊕ℂ) ⊂ H^{1}(X, L⊗⊙^{3}N^{∗}) ⊂ ℂ which in turn implies K^{1}(𝔤 ⊕ ℂ) = 0.

Type F_{4} (iii). The associated Borel-Weil data are (X, L) = [figure: ×; >; 0; 0; 1; 0]. By (5),

T X = [figure: ×; >; 1; 0; -1; 1] + [figure: ×; >; 0; 1; -2; 2] + [figure: ×; >; 0; 0; 0; 1] + [figure: ×; >; 1; 0; 0; 0].

By this formula for T X, it is straighforward but very tedious to obtain the associated (non-direct) decompositions into irreducible homogeneous bundles E_{i} for ⊙^{2}T X ⊗ L^{∗2} and ⊙^{3}T X ⊗ L^{∗2}, and to observe that all irreducible summonds E_{i} have the first cohomology zero implying H^{1}(X, ⊙^{2}T X ⊗ L^{∗2}) = H^{1}(X, ⊙^{2}T X ⊗ L^{∗2}) = 0. By (19), this implies H^{1}(X, L ⊗ ⊙^{3}N^{∗}) = 0 which in turn implies K(𝔤 ⊕ ℂ) = 0.

Type E_{6} (v). The associated Borel-Weil data are (X, L) = [figure: ×; 0; 1; 0; 0; 0; 0]. By (5),

T X = [figure: ×; 1; -1; 0; 1; 0; 0] + [figure: ×; 0; 0; 0; 0; 0; 1],

implying

⊙^{2}T X ⊗ L^{∗2} = [figure: ×; 2; -4; 0; 2; 0; 0; ⊕; ×; 2; -3; 0; 0; 0; 1; ⊕; ×; 0; -3; 1; 0; 1; 0] + [figure: ×; 1; -3; 0; 1; 0; 1; ⊕; ×; 1; -2; 0; 0; 1; 0] + [figure: ×; 0; -2; 0; 0; 0; 2].

All irreducible bundles on the right-hand side of the above equality have the first cohomology vanishing, implying H^{1}(X, ⊙^{2}T X ⊗ L^{∗2}) = 0. Analogously, one obtains H^{1}(X, ⊙^{3}T X ⊗ L^{∗2}) = 0. By (19), H^{1}(X, L ⊗ ⊙^{3}N^{∗}) = 0 implying K(𝔤 ⊕ ℂ) = 0.

Type E_{7} (vi). The associated Borel-Weil data are

(X, L) = [figure: ×; 0; 0; 0; 0; 0; 0; 1].

By (5),

T X = [figure: ×; 0; 0; 0; 1; 0; 0; -1] + [figure: ×; 1; 0; 0; 0; 0; 0; 0],

implying

⊙^{2}T X ⊗ L^{∗2} = [figure: ×; 0; 0; 0; 2; 0; 0; -4; ⊕; ×; 0; 1; 0; 0; 0; 1; -3] + [figure: ×; 1; 0; 0; 1; 0; 0; -3; ⊕; ×; 0; 0; 0; 0; 1; 0; -3] + [figure: ×; 2; 0; 0; 0; 0; 0; -2].

Since every irreducible bundle on the right-hand side of the above equality has the first cohomology vanishing, one obtains H^{1}(X, ⊙^{2}T X ⊗L^{∗2}) = 0. A similar analysis of ⊙^{2}T X ⊗ L^{∗2} results in H^{1}(X, ⊙^{3}T X ⊗ L^{∗2}) = 0. Then, by (19), H^{1}(X, L ⊗ ⊙^{3}N^{∗}) = 0 implying K(𝔤 ⊕ ℂ) = 0.

Type G_{2}. The Borel-Weil data associated with the representation [figure: >; s; 0] are (X, L) = ×> s 0 . By (5), T X = ×> -1 3 + ×> 1 0 , implying

⊙^{2}T X = [figure: ×; >; -2; 6; ⊕; ×; >; 0; 2] + ×> 0 3 + ×> 2 0

and

⊙^{3}T X = [figure: ×; >; -3; 9; ⊕; ×; >; -1; 5; ⊕; ×; >; 0; 3] + [figure: ×; >; -1; 6; ⊕; ×; >; 1; 2] + ×> 1 3 + ×> 3 0 .

Thus H^{1}(X, ⊙^{2}T X ⊗ L^{∗2}) = 0 for any s ⩾ 1 and

H^{1}(X, ⊙^{3}T X ⊗ L^{∗2}) ⊂ \{ \begin{matrix}ℂ & for s = 1, \\ 0 & otherwise. \end{matrix}

Now, K(𝔤 ⊕ ℂ) ⊂ H^{1}(X, L ⊗ ⊙^{3}N^{∗}) ⊂ ℂ implying K^{1}(𝔤 ⊕ ℂ) = 0.

Next we consider the representation [figure: >; s; t], s, t ⩾ 1, which has the associated Borel-Weil data (X, L) = × ×> s t . There is a projection ν : X → X_{0} = ×> . One has T X = × ×> -1 2 + ν^{∗}(T X_{0}) with T X_{0} = ×> -1 3 + ×> 1 0 . Hence

⊙^{2}T X =  × ×> -2 4 +  × ×> -1 2 ⊗ ν^{∗}(T X_{0}) + ν^{∗}(T X_{0}).

Computing ν_{∗}^{i}(⊙^{2}T X ⊗ ⊗L^{∗2}), i = 0, 1, one easily finds

H^{1}(X, ⊙^{2}T X ⊗ L^{∗2}) ⊂ \{ \begin{matrix}H^{1}( ×> -1-2s 0 ⊗ T X_{0}) & for any s and t = 1, \\ 0 & otherwise. \end{matrix}

Since H^{1}( ×> -1-2s 0 ⊗T X_{0}) = 0, we obtain H^{1}(X, ⊙^{3}T X⊗L^{∗2}) = 0 for any s, t ⩾ 1. Analogous computation shows that H^{1}(X, ⊙^{3}T X ⊗L^{∗2}) vanishes for all s, t ⩾ 1 as well. Hence, by (19), H^{1}(X, L ⊗ ⊙^{3}N^{∗}) = 0 implying K(𝔤 ⊕ C) = 0.

The proof of Theorem 6.10 is completed.

6.4. Proof of the Main Theorem. Suppose 𝔤 ⊂ 𝔤𝔩(V) is a real Lie subalgebra which occurs as the irreducible holonomy of a torsion-free affine connection which is not locally symmetric. Then 𝔤 is a Berger algebra by Proposition 2.2. By Theorem 5.1, we may assume that the semisimple part of 𝔤 is simple. Moreover, by Proposition 2.4, it follows that K(𝔤_{ℂ}) ≠ 0. Therefore, 𝔤_{ℂ} must be one of the entries of Table 6. So, in order to complete the proof of the Main Theorem, one first determines all possible real forms of real type of the entries of Table 6, and then investigates which real forms of the complex Lie algebras with (𝔤_{ℂ})^{(1)} ≠ 0 (cf. Table 5) satisfy Berger’s criteria.

For the entries 1–3, 6, 7, 9 and 12, this has been carried out by Berger [8], Bryant [14], [16] and McClean [16]; all admissible real forms of entries 4, 10 and 13 have also been studied by Bryant [15], [16]. Finally, the entries 5, 8, 11 and 14 satisfy (𝔤_{ℂ})^{(1)} = 0, and thus Table 3 contains all real forms of real type for these entries. That all of these do occur as holonomies will be shown in Section 7.

This argument immediately implies the following corollary, completing the classification of complex Berger algebras:

Corollary 6.11. Let 𝔤 ⊂ 𝔤𝔩(V) be an irreducible complex Lie subalgebra whose semisimple part is simple. Then 𝔤 is a Berger algebra if and only it is an entry of Table 7.

Since all our arguments above relied essentially on the assumption dim K(𝔤) > 1, we obtained actually a stronger result than the one formulated in the above corollary:

Theorem 6.12. Let 𝔤 ⊂ 𝔤𝔩(V) be an irreducible complex semisimple Lie subalgebra. Then dim K(𝔤 ⊕ ℂ) > 1 if and only if 𝔤 is (the semisimple part of ) an entry of Table 6 or a complex entry of Table 7.

7. Existence and supersymmetry of exotic holonomies

7.1. Existence theorem. Let V be a 2n-dimensional vector space over 𝔽 = ℝ or ℂ equipped with a non-degenerate symplectic form ω. Denote by Sp(n, 𝔽) the subgroup of GL(V) which preserves ω and by 𝔰𝔭(n, 𝔽) the Lie algebra of Sp(n, 𝔽).

Since 𝔰𝔭(n, 𝔽) ≃ ⊙^{2}V^{∗}, for any irreducible subalgebra 𝔤 ⊂ 𝔰𝔭(n, 𝔽) one has a nonzero 𝔤-invariant symmetric map

⊙^{2}V ⟶ 𝔤
u ⊗ v ⟶ u ∘ v,

which is unique up to nonzero scalar factor.

Let B : ⊙^{2}𝔤 → 𝔽 be the Killing form of 𝔤.

Lemma 7.1. The products ω : Λ^{2}V → 𝔽 and ∘ : ⊙^{2}V → 𝔤 satisfy, for some nonzero constants λ, μ ∈ 𝔽 , the identities

(40)  ω(Au, v) = λB(A, u ∘ v),
(41) B(u ∘ v, s ∘ t) − B(u ∘ t, s ∘ v) = μ(2ω(u, s)ω(v, t)
+ω(u, t)ω(v, s) + ω(u, v)ω(s, t)),

for all A ∈ 𝔤 and all u, v, s, t ∈ V, if and only if the pair (G, V) is an entry of the table of Theorem C.

Proof. Both identities are trivially satisfied for 𝔤 = 𝔰𝔭(n, 𝔽).

The first identity holds for an arbitrary 𝔤 ⊂ 𝔰𝔭(n, 𝔽), since the Killing form B on 𝔤 coincides with the restriction of the Killing form on 𝔰𝔭(n, 𝔽) which is proportional to the metric ω ⊗ ω induced on ⊙^{2}V^{∗} from V^{∗} ⊗ V^{∗}.

A straightforward calculation shows that the second identity (41) is equivalent to saying that, for any fixed A ∈ 𝔤, the map

(42)  ρ_{A} : Λ^{2}V ⟶ 𝔤
u ⊗ v ⟶ μ ω(u, v) A − u ∘ (Av) + v ∘ (Au)

defines an element of K(𝔤), i.e. lies in the kernel of the composition

𝔤 ⊗ Λ^{2}V^{∗}⟶V ⊗ V^{∗} ⊗ Λ^{2}V^{∗}⟶V ⊗ Λ^{3}V^{∗}.

Therefore, if (41) is satisfied, then K(𝔤) contains an irreducible summand isomorphic to Ad(𝔤). So 𝔤 can only be a representation listed in the table of Theorem C.

The 𝔰𝔭(n, 𝔽)-module K(𝔰𝔭(n, 𝔽)) contains an irreducible submodule Ad(𝔰𝔭(n, 𝔽)). Since the above identities do hold for 𝔤 = 𝔰𝔭(n, 𝔽), the elements of this submodule are given explicitly by the formula (42). As follows from Theorem 6.12, the only irreducible symplectic subgroups 𝔤 ⊂ 𝔰𝔭(n, 𝔽) which have dim K(𝔤) > 1 are the ones given in the table of Theorem C. They all have K(𝔤) ≃ Ad(𝔤) so that their elements must be given explicitly by the same formula (42). Then the Bianchi identities for maps (42) imply (41).

For 𝔤 = 𝔢_{7}^{ℂ} represented in ℂ^{56} the identities (40) and (41) were first established by Adams [1] via a reduction from 𝔢_{8}^{ℂ}. We refer to [21] for details.

Having obtained in the proof of Lemma 7.1 an explicit structure of K(𝔤) for all proper subgroups 𝔤 of 𝔰𝔭(n, 𝔽) listed in the table of Theorem C, we can show easily that a generic element of K^{1}(𝔤) ⊂ 𝔤 ⊗ V^{∗} ⊗ Λ^{2}V^{∗} is of the form

V ⊗ Λ^{2}V ⟶ 𝔤
s ⊗ u ⊗ v ⟶ μ ω(u, v) (s ∘ w) − (u ∘ (s ∘ w))v + (v ∘ (s ∘ w))u

for some fixed w ∈ V ≃ V^{∗}. This establishes the isomorphism K^{1}(𝔤) = V^{∗}.

Lemma 7.2. The 𝔤-module S(𝔤) = ⊙^{2}𝔤 ⊗ Λ^{2}V^{∗} ∩ 𝔤 ⊗ K(𝔤) contains a one-dimensional submodule if and only if G is one of the entries of the table of Theorem C.

Proof. First note that the 𝔤-module K(𝔤) ⊗ 𝔤 ≃ 𝔤 ⊗ 𝔤 has only one 1dimensional 𝔤-submodule. So, if there is an invariant element ϕ in S(𝔤), it is unique up to a nonzero scalar factor. Since S(𝔤) ⊂ V ⊗ K^{1}(𝔤) such an element gives rise to 𝔤-equivariant maps

(43)  ϕ′ : 𝔤^{∗} ⟶ K(𝔤),
(44)  ϕ′′ : V^{∗} ⟶ K^{1}(𝔤),

implying, in particular, that Ad(𝔤) ⊂ K(𝔤). Thus, if S(𝔤) has a one-dimensional submodule, then 𝔤 must be in the table of Theorem C.

Let us now construct a nonzero 𝔤-invariant element ϕ ∈ S(𝔤) for every entry of that table. Formula (42) gives no choice but to consider the following element of 𝔤 ⊗ K(𝔤) as a candidate to ϕ:

ϕ(C, D, u, v) = μ ω(u, v) B(C, D) + B(u ∘ (Cv), D) − B(v ∘ (Cu), D),

where C, D ∈ 𝔤, u, v ∈ V and we identify 𝔤 = 𝔤^{∗} via the Killing form. Clearly, this element is 𝔤-invariant. Since B(u ∘ (Cv), D) = λ ω(Du, Cv), we have

ϕ(C, D, u, v) = μ ω(u, v) B(C, D) + λ ω(Du, Cv) − λ ω(Dv, Cu)
= μ ω(u, v) B(C, D) − λ ω(Cv, Du) − λ ω(Dv, Cu),

which makes it evident that ϕ ∈ (⊙^{2}𝔤⊗Λ^{2}V^{∗})∩(𝔤⊗K(𝔤)). That ϕ′ : 𝔤^{∗}⟶K(𝔤) is an isomorphism follows from the very definition of ϕ. Since ϕ′′_{2} : V^{∗}⟶K^{1}(𝔤) is evidently nonzero, it is an isomorphism as well.

Proof of Theorem A. It was shown in [20] that all statements of Theorem A hold for those 𝔤 ⊂ 𝔤𝔩(V) whose associated module S(𝔤) contains an invariant element ϕ for which the corresponding maps (43) and (44) are isomorphisms.

Since such an element ϕ was given above for all representations listed in the table of Theorem C, and thus, in particular, for the new series of exotic holonomies, Theorem A follows.

This argument also shows that the statements of Theorem A are true for all torsion-free affine connections ∇ with “generic” symplectic holonomy 𝔰𝔭(n, 𝔽) satisfying the condition that the curvature tensor R^{∇} takes values in the submodule Ad(𝔰𝔭(n, 𝔽)) ⊂ K(𝔰𝔭(n, 𝔽)).

Finally, we show the following result.

Lemma 7.3. If 𝔰𝔭𝔦𝔫(11, ℂ) ⊂ 𝔤𝔩(V) with V ≅ ℂ^{16} is the image of the spinor representation, then K(𝔰𝔭𝔦𝔫(11, ℂ)) = K(𝔰𝔭𝔦𝔫(11, ℂ) ⊕ ℂ Id) = 0.

Proof. Since 𝔰𝔭𝔦𝔫(11, ℂ) ⊂ 𝔰𝔭𝔦𝔫(12, ℂ) ⊂ 𝔤𝔩(V), it follows that K(𝔰𝔭𝔦𝔫(11, ℂ)⊕ℂ Id) ⊂ K(𝔰𝔭𝔦𝔫(12, ℂ)⊕ℂ Id) = K(𝔰𝔭𝔦𝔫(12, ℂ)) ≅ Ad(𝔰𝔭𝔦𝔫(12)) by the above result. In particular, K(𝔰𝔭𝔦𝔫(11, ℂ) ⊕ ℂ Id) = K(𝔰𝔭𝔦𝔫(11, ℂ)).

But now, a calculation shows that there is no nonzero element A ∈ 𝔰𝔭𝔦𝔫(12, ℂ) such that ρ_{A}(u, v) ∈ 𝔰𝔭𝔦𝔫(11, ℂ) for all u, v ∈ V with ρ_{A} as in (42). Thus, K(𝔰𝔭𝔦𝔫(11, ℂ)) = 0.

7.2. Supersymmetry. A (smooth, analytic, etc.) supermanifold of dimension (m|n) is a locally ringed space (M, O_{M}) with the following properties [29]:

(i) the structure sheaf O_{M} = O_{M,0} ⊕ O_{M,1} is a sheaf of ℤ_{2}-graded supercommutative rings; (ii) M_{red} = (M, O_{M,red} := O_{M}/[O_{M,1} + O_{M,1}^{2}]) is a (smooth, analytic, etc.) classical manifold of dimension m; (iii) O_{M} is locally isomorphic to the exterior algebra Λ(E) of a locally free O_{M,red}-module E of rank n. If ϕ : Λ(E) → O_{M} is any such local isomorphism, x̄^{1}, …, x̄^{m} are local coordinates on M_{red} and θ̄^{1}, …, θ̄^{n} are free local generators of E, then the set of m + n sections

x^{1} = ϕ(x̄^{1}), …, x^{m} = ϕ(x̄^{m}), θ^{1} = ϕ(θ̄^{1}), …, θ^{n} = ϕ(θ̄^{n})

of the structure sheaf O_{M} form a local coordinate system on M. Any local function f on M can be expressed as a polynomial in anticommuting odd coordinates θ^{α}, α = 1, …, n,

f(x, θ) = ∑_{k=0}^{n} ∑_{α_{1},...,α_{k}=1}^{n} f_{α_{1}...α_{k}}(x)θ^{α_{1}} ⋯ θ^{α_{k}}

whose coefficients f_{α_{1}...α_{k}}(x) are classical (smooth, analytic, etc.) functions of the commuting variables x^{a}, a = 1, …, m.

When a need arises to use odd constants in the structure sheaf of a supermanifold M, one simply replaces M by its relative version, i.e. by a submersion of supermanifolds π : M → S whose typical fibre is M. Then (odd) constants are just (odd) elements of π^{−1}(O_{S}). Such a replacement is made often tacitly — the necessary changes are routine; cf. [29], [30].

A SUSY-structure on an (m|n)-dimensional supermanifold M is, by definition, a rank 0|n locally split subsheaf T_{1} ⊂ T M such that the associated Frobenius form

\begin{aligned}Φ : & Λ^{2}T_{1} & ⟶ & T_{0} := T M/T_{1} \\ & X ⊗ Y & ⟶ & [X, Y] mod T_{1} \end{aligned}

is surjective.

Examples. (i) A (1|1)-dimensional supermanifold M with a SUSY-struct-ure is the same as a SUSY_{1}-curve [30]. (ii) A SUSY-structure on a (3|2)- dimensional supermanifold is equivalent to N = 1 superconformal supergravity in 3 dimensions [33]. (iii) A SUSY-structure on a (4|4)-dimensional supermanifold with T_{1} being a direct sum of two integrable rank (0|2) distributions T_{l} and T_{r} is the same as an N = 1 superconformal supergravity in 4 dimensions [29].

Let V be a 2n-dimensional vector space over 𝔽 = ℝ or ℂ equipped with a nondegenerate symplectic form ω and 𝔤 ⊆ 𝔰𝔭(n, 𝔽) be an irreducible Lie subalgebra. Define M𝔤 as a linear supermanifold 𝔤 ⊕ Π(V^{∗}), where Π is the parity change functor [29]. Then, as usual for linear (super)manifolds, there is an isomorphism i : [𝔤⊕Π(V^{∗})]⊗O_{M} → T M𝔤 so that T M comes equipped with a free subsheaf i(Π(V^{∗}) ⊗ O_{M}) of rank 0|n. Let (e_{a}), a = 1, …, m = dim 𝔤, be a basis in the vector space 𝔤 and (e_{α}), α = 1, …, 2n, be a basis in the vector space Π(V^{∗}), and let (x^{a}) and (θ^{α}) be the associated commuting and, respectively anticommuting linear coordinates. Then the system (x^{a}, θ^{α}) is a global coordinate system on M𝔤. Since i(Π(V^{∗}) ⊗ O_{M}) is freely generated by ∂_{α} = ∂/∂θ^{α}, the associated Frobenius form is zero, implying that this subsheaf is not a SUSY-structure on M. This can be corrected by the following natural map

\begin{matrix}Φ : Λ^{2}(Π(V^{∗})) & ⟶^{≃} & ⊙^{2}V^{∗} & ⟶ & 𝔤 \\ e_{α}∧e_{β} & ⟶ & Π(e_{α}) ⊙ Π(e_{β}) & ⟶ & ϕ_{αβ}^{a}e_{a}, \end{matrix}

which gives rise to a monomorphism

\begin{matrix}j : Π(V^{∗}) ⊗ O_{M} & ⟶ & T M \\ e_{α} & ⟶ & ∂_{α} + ϕ_{αβ}^{a}θ^{β}∂_{a}, \end{matrix}

where ∂_{a} = ∂/∂x^{a} and summation over repeated indices is assumed.

Lemma 7.4. The subsheaf T_{1} = j(Π(V^{∗})) ⊂ T M𝔤 is a SUSY-structure on M𝔤.

Proof. The subsheaf T_{1} is freely generated by the vector fields D_{α} = ∂_{α} + ϕ_{αβ}^{b}θ^{β}∂_{b}. A simple calculation shows [D_{α}, D_{β}] = 2ϕ_{αβ}^{d}∂_{d} implying that the Frobenius form of T_{1} is equal to Φ and is, therefore, nondegenerate.

The SUSY-structure on M𝔤 gives rise to a group G of SUSY-translations x^{a}⟶x^{a} + ϕ_{αβ}^{a}ϵ^{α}θ^{β} which mix even and odd coordinates, ϵ^{α} being some odd constants. A function f on M𝔤 is said to be SUSY-invariant if it is infinitesimally invariant under G, i.e. if it satisfies the differential equation D_{α}f = ∂_{α}f.

Proof of Theorem C. Let f be a nonzero 𝔤- and SUSY-invariant polynomial of order two in θ^{α} and of order at least two in x^{a}; i.e.,

f = ∑_{k⩾2} ϕ_{a_{1}···a_{k} μν}x^{a_{1}} … x^{a_{k}}θ^{μ}θ^{ν}

for some ϕ_{a_{1}···a_{k} μν} ∈ ⊙^{k}𝔤 ⊗ Λ^{2}(V^{∗}). An easy calculation shows that f satisfies the equation D_{α}f = ∂_{α}f if and only if ϕ_{a_{1}···a_{k} μν} ∈ ⊙^{k}𝔤⊗Λ^{2}(V^{∗})∩⊙^{k−1}𝔤⊗K(𝔤). Then, since f is 𝔤-invariant, K(𝔤) must contain 𝔤-submodules ⊙^{k−1}𝔤, k ⩾ 2. By Theorem 6.12, 𝔤 can only be an entry of the table of Theorem C. As Lemma 7.2 shows, every such entry 𝔤 does admit a nonzero invariant element ϕ_{a_{1}a_{2} μν} ⊂ S(𝔤) so that the set of 𝔤- and SUSY-invariant functions of the required type on the associated supermanifold M𝔤 is nonempty.

8. Classification of complex homogeneous-rational manifolds X and ample line bundles L → X such that H^{1}(X, T X ⊗ L^{∗}) ≠ 0

Proof of Theorem B. Statement (i) of Theorem B follows from Proposition 5.2. Let us prove statement (ii).

If X is reducible, say X = X_{1} × X_{2} and L = π_{1}^{∗}(L_{1}) ⊗ π_{2}^{∗}(L_{2}), then

H^{1}(X, T X ⊗ L^{∗}) = H^{0}(X_{1}, T X_{1} ⊗ L_{1}^{∗}) ⊗ H^{1}(X_{2}, L_{2}^{∗})
+ H^{0}(X_{2}, T X_{2} ⊗ L_{2}^{∗}) ⊗ H^{1}(X_{1}, L_{1}^{∗}).

This together with statement (i) implies that in the class of reducible X only two bottom lines in Table 3 contribute to the list of all (X, L) with H^{1}(X, T X ⊗ L^{∗}) ≠ 0.

Assume from now on that X is irreducible. Though the tangent bundle T X is homogeneous, it is not irreducible in general; even worse, since the parabolic P is not reductive, T X is not a direct sum of irreducible homogeneous vector bundles. This makes a naive idea of computing H^{∗}(X, T X ⊗ L^{∗}) by the straightforward application of the Bott theorem impractical.

Consider the Atiyah exact sequence

(45)  0⟶ Q⟶ 𝔤 ⊗ O_{X}⟶ T X⟶ 0,

where Q = G ×_{Ad} 𝔭. Since the central term of this extension is a trivial vector bundle and H^{i}(X, L^{∗}) = 0 for 0 ⩽ i ⩽ dim X − 1, we have, in case dim X ⩾ 3,

H^{1}(X, T X ⊗ L^{∗}) = H^{2}(X, Q ⊗ L^{∗}).

An exact sequence of 𝔭-modules

0 ⟶ 𝔫̃ ⟶ 𝔭 ⟶ 𝔭/𝔫̃ ⟶ 0,

where 𝔫̃ = 𝔫 \ ∑_{α∈Φ𝔭^{+}} 𝔤_{α}, gives rise to an exact sequence

0 ⟶ Ω^{1}X ⟶ Q ⟶ S ⟶ 0

of homogeneous vector bundles, where S = G ×_{Ad} 𝔭/𝔫̃ and where the isomorphism G ×_{Ad} 𝔫̃ ≃ Ω^{1}X was used. According to Nakano [34], for any compact complex manifold X and any positive line bundle L on X the groups H^{i}(X, Ω^{1}X ⊗L^{∗}) = 0 vanish for all i ⩽ dim X −2. Then, in the case dim X ⩾ 5, the long exact sequence of the latter extension implies

H^{2}(X, Q ⊗ L^{∗}) = H^{2}(X, S ⊗ L^{∗})

which in turn implies

H^{1}(X, T X ⊗ L^{∗}) = H^{2}(X, S ⊗ L^{∗}).

The advantage of working with S instead of T X is that S can always be decomposed into a direct sum of irreducible homogeneous subbundles.

The Lie algebra 𝔰 = 𝔭/𝔫̃ is reductive and the adjoint representation of 𝔭 on 𝔰 is semisimple. Under the adjoint representation 𝔰 → 𝔤𝔩(𝔰) the Lie algebra 𝔰 decomposes into a direct sum of its ideals

𝔰 = ξ_{1} + … ξ_{k} + 𝔰_{1} + … + 𝔰_{m}

where ξ_{j}, j = 1, …, rankX, lie in the center of 𝔰 and the non-Abelian ideals 𝔰_{i}, i = 1, …, m, are simple. Then, by Bott’s theorem,

H^{2}(X, S ⊗ L^{∗}) = ⨁_{j=1}^{k} J^{2}(−λ) + ⨁_{i=1}^{m} J^{2}(μ_{i} − λ),

where λ is the weight of L and μ_{1}, …, μ_{m} are the maximal roots of the simple ideals 𝔰_{1}, …, 𝔰_{m} [11], [39]. Since, for dim X ⩾ 2, J^{2}(−λ) = H^{2}(X, L^{∗}) = 0, we obtain the following:

Lemma 8.1. If dim X ⩾ 5, then H^{1}(X, T X ⊗ L^{∗}) = ⨁_{i=1}^{m} J^{2}(μ_{i} − λ).

There are seven irreducible compact complex homogeneous-rational manifolds X with dim X ⩽ 4: projective spaces ℂℙ_{k} for k = 1, 2, 3, 4, quadrics Q_{3}, Q_{4} and the complete flag manifold F (1, 2; ℂ^{3}). It is elementary to check that Theorem B is true for this family.

We assume from now on that X is an irreducible complex homogeneousrational manifold with dim X ⩾ 5.

Let us introduce the following notation: If Γ is a connected subgraph of the Dynkin diagram for X, then the number of simple roots {α_{j} | j ∈ I} in this graph is denoted by |Γ|; if ω is an integral weight such that ω(H_{α_{j}}) ⩽ 0 for all α_{j} ∈ Γ, then we write ω|_{Γ} ⩽ 0.

Lemma 8.2. H^{1}(X, T X ⊗ L^{∗}) = 0 for any ample line bundle L on X if at least one of the following conditions is satisfied:

(i) rankX ⩾ 3;

(ii) rankX = 2 and the crossed nodes are adjacent;

(iii) rankX = 2, the crossed nodes are not adjacent and, for each maximal root μ_{i} of the simple ideal 𝔰_{i}, i = 1, …, m, at least one crossed node is contained in a connected subgraph Γ of the Dynkin diagram for X such that |Γ| ⩾ 2 and μ_{i}|_{Γ} ⩽ 0.

(iv) rankX = 1 and, for each maximal root μ_{i} of the simple ideal 𝔰_{i}, i = 1, …, m, the crossed node is contained in a connected subgraph Γ of the Dynkin diagram for X such that |Γ| ⩾ 3 and μ_{i}|_{Γ} ⩽ 0.

Proof. (i) Let λ be the weight of L. Since L is ample, the coefficient of λ over each crossed node is a negative integer (its coefficient over each uncrossed node is, of course, zero). Then (−λ + μ_{i} + η)(H_{α_{j}}) ⩽ −λ(H_{α_{j}}) + 1 ⩽ 0 for all crossed nodes α_{j} and all i ∈ {1, …, m}. If the number of crossed nodes is greater than or equal to 3, then either −λ + μ_{i} + η is singular or ind(−λ + μ_{i} + η) ⩾ 3. Whence ⨁_{i=1}^{m} J^{2}(μ_{i} − λ) = 0 and the statement follows from Lemma 8.1.

(ii) If α_{j} and α_{j+1} are adjacent crossed nodes, then α_{j} + α_{j+1} is a positive root and one has (−λ + μ_{i} + η)(H_{α_{j}}) ⩽ 0, (−λ + μ_{i} + η)(H_{α_{j+1}}) ⩽ 0 and hence (−λ + μ_{i} + η)(H_{α_{j}+α_{j+1}}) ⩽ 0. Thus either −λ + μ_{i} + η is singular or ind(−λ + μ_{i} + η) ⩾ 3 for all i and the statement follows from Lemma 8.1.

(iii) and (iv) If Γ′ is a connected subgraph of the Dynkin diagram for X, then the sum of all simple roots in Γ′ is a positive root [12]. Under the conditions stated in (iii) and (iv), one easily finds at least three positive roots α_{j} such that (−λ + μ_{i} + η)(H_{α_{j}}) ⩽ 0 for all i ∈ {1, …, m}. Then, again, either −λ + μ_{i} + η is singular or ind(−λ + μ_{i} + η) ⩾ 3 implying J^{2}(−λ + μ_{i}) = 0. Thus the statement follows from Lemma 8.1.

Therefore, we can restrict our attention to the cases rankX = 1, 2.

The case rank X = 2. It clear from items (ii) and (iii) of Lemma 8.2 that H^{1}(X, T X ⊗ L^{∗}) ≠ 0 only for those X which have the semisimple part 𝔰′ of the parabolic algebra 𝔭 simple, i.e. the number m of simple ideals of 𝔰′ is 1. Therefore, the crossed nodes must be located at the ends of the Dynkin diagram for X. Inspecting the list of maximal roots (4) leaves one with the following three candidates to the list of all (X, L) with H^{1}(X, T X ⊗ L^{∗}) ≠ 0:

(1) (X, L) = … × • • • × s 0 0 0 t for some s,t⩾ 1. Computing σ_{1} ∘σ_{n}(−λ+ μ_{1} + η) − η as shown in the following diagram

−λ + μ_{1} =  … × • • • × -1-s 1 0 1 -1-t ⟶^{+η}  … × • • • × -s 2 1 2 -t
⟶^{σ_{1}∘σ_{n}}  … × • • • × s 2-s 1 2-t t ⟶^{−η}  … × • • • × s-1 1-s 0 1-t t-1

one concludes

H^{1}(X, T X ⊗ L^{∗}) = J^{2}(−λ + μ_{1}) = \{ \begin{matrix}ℂ & s=t=1 \\ 0 & otherwise. \end{matrix}

(2) (X, L) =[figure: …; ×; >; ×; s; 0; 0; 0; t] for some s,t⩾ 1. Then the graph
−λ + μ_{1} = [figure: …; ×; >; ×; -1-s; 1; 0; 1; -2-t; ⟶^{+η}; …; ×; >; ×; -s; 2; 1; 2; -1-t]
⟶^{σ_{1}∘σ_{n}} [figure: …; ×; >; ×; s; 2-s; 1; 1-t; t+1]
⟶^{−η} [figure: …; ×; >; ×; s-1; 1-s; 0; -t; t]

implies H^{1}(X, T X ⊗ L^{∗}) = J^{2}(−λ + μ_{1}) = 0 for all s,t⩾ 1.

(3) (X, L) =[figure: …; ×; ×; 0; 0; 0; 0; 0; s; t] for some s,t⩾ 1. In this case

μ_{1} =[figure: …; ×; ×; 1; 0; 0; 0; 1; -1; -1] .

The only element of the Weyl group W of length 2 which can, in principle, map −λ + μ_{1} + η to a strictly dominant weight in Λ^{++} is σ_{n−1} ∘ σ_{n}. However, a computation as above shows that

σ_{n−1} ∘ σ_{n}(−λ + μ_{1} + η) − η = [figure: …; ×; ×; 1; 0; 0; 0; -s-t; s-1; t-1]

which implies H^{1}(X, T X ⊗ L^{∗}) = J^{2}(−λ + μ_{1}) = 0 for all s,t⩾ 1.

The case rankX = 1. The number m of simple ideals of the semisimple part of the parabolic algebra 𝔭 can, in principle, be equal to 1, 2 or 3. The case m = 3, however, is ruled out by Lemma 8.2(iv). If m = 2, then, by Lemma 8.2(iv), at least one of the ideals must be isomorphic to 𝔰𝔩(2, ℂ). Therefore, the crossed node must be either an end node (for m = 1) or the node adjacent to an end node (for m = 2) of the Dynkin diagram for X. Inspecting the list of maximal roots (4) excludes all but the following candidates to (X, L) with H^{1}(X, T X ⊗ L^{∗}) ≠ 0:

(1) (X, L) = (ℂℙ_{n}, O(s)) = … × • • • s 0 0 0 (n ⩾ 5 nodes) for some s ⩾ 1. The odd dimensional projective space has another representation as

(ℂℙ_{2n−1}, O(s)) = [figure: …; ×; <; s; 0; 0; 0; 0] (n ⩾ 3 nodes).

The long exact sequence

0 ⟶ O(−s) ⟶ ℂ^{n+1} ⊗ O(1 − s) ⟶ T X ⊗ L^{∗} ⟶ 0

implies H^{1}(X, T X ⊗ L^{∗}) = 0 for all s ⩾ 1.

(2) (X, L) = [figure: …; ×; 0; s; 0; 0; 0] for some s ⩾ 1. There are two maximal roots

μ_{1} = [figure: …; ×; 2; -1; 0; 0; 0] , μ_{2} = [figure: …; ×; 0; -1; 1; 0; 1] .

That J^{2}(−λ + μ_{1}) = 0 for all s ⩾ 1 follows from the proof of Lemma 8.2(iv), while

σ_{1} ∘ σ_{2}(−λ + μ_{2} + η) − η = [figure: …; ×; s-2; 0; 1-s; 0; 1]

implies J^{2}(−λ + μ_{2}) = 0 for all s ⩾ 1 as well. Thus H^{1}(X, T X ⊗ L^{∗}) = 0 for all s ⩾ 1.

(3) (X, L) = [figure: …; ×; >; s; 0; 0; 0; 0] (n ⩾ 3 nodes) for some s ⩾ 1. Note that for n = 3 this pair is biholomorphic to × > 0 s [39] .

The maximal root is

μ_{1} = {[figure: …; ×; >; -1; 0; 1; 0; 0; n ⩾ 4 nodes; ×; >; -1; 0; 2; n = 3 nodes.]

Then an easy computation shows

σ_{2} ∘ σ_{1}(−λ + μ_{1} + η) − η = {[figure: …; ×; >; 0; s-2; 2-s; 0; 0; n ⩾ 4 nodes; ×; >; 0; s-2; 4-2s; n = 3 nodes]

which implies

H^{1}(X, T X ⊗ L^{∗}) = J^{2}(−λ + μ_{1}) = \{ \begin{matrix}ℂ & s = 2 \\ 0 & otherwise. \end{matrix}

(4) (X, L) = [figure: ×; >; 0; s; 0] for some s ⩾ 1. The maximal roots are
μ_{1} = [figure: ×; >; 2; -1; 0] , μ_{2} = [figure: ×; >; 0; -1; 2] .

Then

σ_{3} ∘ σ_{2}(−λ + μ_{1} + η) − η = [figure: ×; >; 2-s; -s; 2s-2] ,
σ_{1} ∘ σ_{2}(−λ + μ_{1} + η) − η = [figure: ×; >; s-2; 0; 2-2s] ,

implying H^{1}(X, T X ⊗ L^{∗}) = J^{2}(−λ + μ_{1}) + J^{2}(−λ + μ_{2}) = 0 for all s ⩾ 1.

(5) (X, L) = [figure: …; >; ×; 0; 0; 0; 0; s] (n nodes) for some s ⩾ 1. This pair is biholomorphic to the following one [39]:

(X, L) = [figure: …; ×; 0; 0; 0; 0; 0; 0; s] (n + 1 nodes).
Then, by Lemma 8.2(iv), H^{1}(X, T X ⊗ L^{∗}) = 0 for all s ⩾ 1.

(6) (X, L) = [figure: …; ×; >; 0; 0; 0; s; 0] for some s ⩾ 1. The maximal roots are

μ_{1} = [figure: …; ×; >; 1; 0; 1; -1; 0] , μ_{2} = [figure: …; ×; >; 0; 0; 0; -1; 2] .

The proof of Lemma 8.2(iv) implies J^{2}(−λ + μ_{2} + η) = 0 for all s ⩾ 1, while

σ_{n} ∘ σ_{n−1}(−λ + μ_{1} + η) − η = [figure: …; ×; >; 1; 0; 1; -s; 2s-2]

implies J^{2}(−λ + μ_{1} + η) = 0 for all s ⩾ 1. Whence, by Lemma 8.1, H^{1}(X, T X ⊗ L^{∗}) = 0 for all s ⩾ 1.

(7) (X, L) = [figure: …; ×; <; 0; s; 0; 0; 0; 0] (n ⩾ 3 nodes) for some s ⩾ 1. The maximal roots are

μ_{1} = [figure: …; ×; <; 2; -1; 0; 0; 0; 0] , μ_{2} = [figure: …; ×; <; 0; -2; 2; 0; 0; 0].

The vanishing of J^{2}(−λ + μ_{1}) for all n ⩾ 4, s ⩾ 1 follows from the proof of Lemma 8.2(iv). That this module vanishes for n = 3, s ⩾ 1 follows from a simple calculation:

σ_{3} ∘ σ_{2}(−λ + μ_{1} + η) − η = [figure: ×; <; 2-s; 1-s; s-2] .

Analogously, one finds

σ_{1} ∘ σ_{2}(−λ + μ_{2} + η) − η = [figure: …; ×; <; s-1; 0; 1-s; 0; 0; 0]

implying J^{2}(−λ + μ_{2}) = 0 for all s ⩾ 2 and J^{2}(−λ + μ_{2}) = ℂ for s = 1. Therefore, by Lemma 8.1,

H^{1}(X, T X ⊗ L^{∗}) = \{ \begin{matrix}ℂ & s = 1 \\ 0 & otherwise. \end{matrix}

(8) (X, L) = [figure: …; ×; s; 0; 0; 0; 0; 0; 0] (n ⩾ 5 nodes) for some s ⩾ 1. The maximal root is

μ_{1} = [figure: …; ×; -1; 0; 1; 0; 0; 0; 0]

and an easy calculation shows that

σ_{2} ∘ σ_{1}(−λ + μ_{1} + η) − η = [figure: …; ×; 0; s-2; 2-s; 0; 0; 0; 0] .

Therefore,

H^{1}(X, T X ⊗ L^{∗}) = J^{2}(−λ + μ_{1}) = \{ \begin{matrix}ℂ & s = 2 \\ 0 & otherwise. \end{matrix}

(9) (X, L) = [figure: ×; >; 0; s; 0; 0] for some s ⩾ 1. The maximal roots are

μ_{1} = [figure: ×; >; 2; -1; 0; 0] , μ_{2} = [figure: ×; >; 0; -1; 1; 1] .

From the proof of Lemma 8.2(iv) it follows that J^{2}(−λ + μ_{1}) = 0 for all s ⩾ 1. The only element of the Weyl group of length 2 which can, in principle, make −λ + μ_{2} + η strictly dominant is σ_{2} ∘ σ_{1}. Since

σ_{2} ∘ σ_{1}(−λ + μ_{2} + η) − η = [figure: ×; >; s-1; 0; 1-2s; 1]

the module J^{2}(−λ + μ_{2}) vanishes for all s ⩾ 1. Therefore, H^{1}(X, T X ⊗ L^{∗}) = 0 for all s ⩾ 1.

(10) (X, L) = [figure: ×; >; 0; 0; s; 0] for some s ⩾ 1. The maximal roots are

μ_{1} = [figure: ×; >; 1; 1; -3; 0] , μ_{2} = [figure: ×; >; 0; 0; -1; 2] .

From the proof of Lemma 8.2(iv) it follows that J^{2}(−λ + μ_{2}) = 0 for all s ⩾ 1. Since

σ_{4} ∘ σ_{3}(−λ + μ_{1} + η) − η = [figure: ×; >; 1; -s; 0; s-1]

the module J^{2}(−λ + μ_{2}) vanishes for all s ⩾ 1. Therefore, H^{1}(X, T X ⊗ L^{∗}) = 0 for all s ⩾ 1.

(11) (X, L) = [figure: ×; >; 0; 0; 0; s] for some s ⩾ 1. The maximal root is
μ_{1} = [figure: ×; >; 0; 1; 0; -2] .

Since

σ_{3} ∘ σ_{4}(−λ + μ_{1} + η) − η = [figure: ×; >; 0; 1-s; s-1; 0]

we obtain

H^{1}(X, T X ⊗ L^{∗}) = J^{2}(−λ + μ_{1}) = \{ \begin{matrix}ℂ & s = 1 \\ 0 & otherwise. \end{matrix}

(12) (X, L) = ×> s 0 for some s ⩾ 1. The maximal root is μ_{1} = ×> -1 2 .

Hence

σ_{2} ∘ σ_{1}(−λ + μ_{1} + η) − η = ×> 3-2s 3s-4,

implying H^{1}(X, T X ⊗ L^{∗}) = J^{2}(−λ + μ_{1}) = 0 for all s ⩾ 1.

Theorem B is proved.

Acknowledgements. It is a pleasure to thank D. Alekseevski, L. Berard-Bergery, N. Hitchin, Yu.I. Manin, H. Pedersen, H.-B. Rademacher and K.P. Tod for valuable discussions and comments. Hospitality of MPI in Bonn and ESI in Vienna (SM), University of Glasgow (LS), ICMS in Edinburgh (both of us) during the work on different stages of the project is gratefully acknowledged. The authors are also grateful to the British-German Council for partial financial support (grant No. 877).

Glasgow University , Glasgow, UK E-mail address: sm@maths.gla.ac.uk Mathematisches Institut, Universität Leipzig, Leipzig, Germany E-mail address: schwachh@mathematik.uni-leipzig.de

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(Received January 24, 1997)

(Revised December 11, 1998)