Bounds for the Anticanonical Bundle of a

Homogeneous Projective Rational Manifold

Dennis Snow

Received: February 26, 2004

Communicated by Thomas Peternell

Abstract. The following bounds for the anticanonical bundle K_{X}^{∗} = det T_{X} of a complex homogeneous projective rational manifold X of dimension n are established:

3^{n} ≤ dim H^{0}(X, K_{X}^{∗}) ≤ \binom{2n + 1}{n} and 2^{n}n! ≤ deg K_{X}^{∗} ≤ (n + 1)^{n}

with equality in the lower bounds if and only if X is a flag manifold and equality in the upper bounds if and only if X is complex projective space. None of these bounds holds for general Fano manifolds.

2000 Mathematics Subject Classification: Primary 14M17; Secondary 14M15, 32M10

The homogeneous compact complex manifolds X that admit an equivariant embedding in projective space are precisely the quotients X = G/P where G is a semisimple complex Lie group and P is a parabolic subgroup. Moreover, any such quotient is rational and has a very ample anticanonical bundle, K_{X}^{∗} = det T_{X}. In particular, X is a Fano manifold. Various bounds have been established for the numerical invariants of K_{X}^{∗} when X is a general Fano manifold, see [6, 8, 9, 10]. For example, there exists a constant c(n) that depends only on n = dim X such that deg K_{X}^{∗} ≤ c(n)^{n}. In this article we establish the following bounds when X = G/P:

3^{n} ≤ dim H^{0}(X, K_{X}^{∗}) ≤ \binom{2n + 1}{n} and 2^{n}n! ≤ deg K_{X}^{∗} ≤ (n + 1)^{n}

with equality in the lower bounds if and only if X is a flag manifold (i.e., P is a Borel subgroup of G), and equality in the upper bounds if and only if X is complex projective space, P^{n}.

These bounds do not hold for general Fano manifolds. For example, let X = P(O_{P^{n−1}} ⊕ O_{P^{n−1}}(n − 1)). Then X is a P^{1}-bundle over P^{n−1}, π : X → P^{n−1}, and K_{X}^{∗} = π^{∗}O_{P^{n−1}}(1)⊗ξ^{2} where ξ is the tautological line bundle on X whose restriction to any fiber P^{1} of π gives ξ|_{P^{1}} ≅ O_{P^{1}}(1). It follows that X is a Fano manifold with dim H^{0}(X, K_{X}^{∗}) = n + \binom{2n−1}{n−1} + \binom{3n−2}{n−1} and deg K_{X}^{∗} = ((2n − 1)^{n} − 1)/(n − 1). An example where the lower bounds do not hold is given by X = S × (P^{1})^{n−2} where S is a del Pezzo surface. In the homogeneous case there are well-known formulas from representation theory that can be used to calculate dim H^{0}(X, K_{X}^{∗}) and deg K_{X}^{∗} exactly. However, these formulas, which are products of rational numbers indexed by the roots of the group, do not easily lend themselves to comparison with expressions in n = dim X. The point of this paper is to overcome this difficulty. The bounds are proved by first reducing to the case of simple Lie groups and then showing for each classical type that the known formulas can be broken up into subproducts of certain simple sequences. These subproducts are shown to satisfy inequalities that can be combined to yield the desired inequalities for the full product. The bounds for the exceptional types are verified through exhaustive calculations. The above upper bounds can be trivially extended to any homogeneous compact complex manifold X = G/H. The sections of K_{X}^{∗} define an equivariant map of X to projective space that coincides with the normalizer fibration G/H → G/N, N = N_{G}(H^{0}), [1, p.79]. Since the base Y = G/N is a homogeneous projective rational manifold, the upper bounds hold for Y and hence for X. For a homogeneous projective rational manifold X, the dimension of the holomorphic automorphism group, dim Aut(X) = dim H^{0}(X, T_{X}), never exceeds n(n+2). In fact, this bound holds when X is any homogeneous compact Kähler manifold [5]. However, there are homogeneous compact complex manifolds for which dim Aut(X) grows exponentially in n, see [12]. In [13], the above estimate for dim H^{0}(X, K_{X}^{∗}) plays an important role in establishing the following bound for the non-Kähler case: dim Aut(X) ≤ n^{2}−1+\binom{2n−1}{n−1} ∼ O(2^{2n−1}/\sqrt{(n − 1)π}).

1 Roots and Weights

In this section we introduce some notation and well-known facts about semisimple Lie groups [2, 7], and recall a formula for finding the weight μ_{X} associated to the line bundle K_{X}^{∗} when X = G/P [4, 11]. Let G be a semisimple complex Lie group and let T be a maximal torus of G. Let Lie(G) and Lie(T) be the corresponding Lie algebras. Let Φ ⊂ Lie(T)^{∗} denote the roots of G with respect to T and let {α_{1}, …, α_{ℓ}} be a system of simple roots. Let Φ^{+} denote the subset of positive roots—those that are positive integral combinations of the simple roots. For any root α ∈ Φ, let e_{α} ∈ Lie(G) be the corresponding root vector: [x, e_{α}] = α(x)e_{α} for all x ∈ Lie(T). Let λ_{1}, …, λ_{ℓ} be the fundamental dominant weights of G—those weights defined by ⟨λ_{i}, α_{j}⟩ = 2(λ_{i}, α_{j})/(α_{j}, α_{j}) = δ_{ij} where ( , ) denotes the Killing form.

Any weight μ ∈ Lie(T)^{∗} can be written μ = ∑_{i=1}^{ℓ}⟨μ, α_{i}⟩λ_{i}. A Borel subgroup is a maximal solvable subgroup of G, and any such subgroup is conjugate to the subgroup B generated by T and the root groups exp Ce_{α}, for all α ∈ −Φ^{+}. Let P be a parabolic subgroup of G, that is, a subgroup containing a Borel subgroup. We may assume that P contains B. Let P = R·S be a Levi decomposition of P where R is a maximal solvable normal subgroup of P and S is semisimple. We let Φ_{P} denote the subsystem of roots of S and let Φ_{P}^{+} = Φ_{P} ∩ Φ^{+}. Let I denote the subset of indexes, I ⊂ {1, …, ℓ}, such that Φ_{P}^{+} ∩ {α_{1}, …, α_{ℓ}} = {α_{i}}_{i∈I}. The conjugacy class of P is uniquely determined by I and any such choice of indexes is associated to a parabolic subgroup of G. Let X = G/P, and define Φ_{X}^{+} = Φ^{+} \ Φ_{P}^{+}. Since T_{X} is generated at the identity coset by the root vectors e_{α} for α ∈ Φ_{X}^{+}, the anticanonical bundle K_{X}^{∗} = det T_{X}, n = dim X, is the homogeneous line bundle associated to the weight

μ_{X} = ∑_{α∈Φ_{X}^{+}} α

The weight μ_{X} is dominant: ⟨μ_{X}, α_{i}⟩ > 0 for i ∉ I, and ⟨μ_{X}, α_{i}⟩ = 0 for i ∈ I. In particular, K_{X}^{∗} is a very ample line bundle and μ_{X} is orthogonal to the roots Φ_{P}^{+}. If P = B, X is called a flag manifold. We now recall a simple formula for calculating the coefficients ⟨μ_{X}, α_{i}⟩ of μ_{X}, see [11]: A set of indexes J is called connected if the subdiagram of the Dynkin diagram of G corresponding to the simple roots α_{j}, j ∈ J, is connected. An index i is said to be adjacent to J if i ∉ J and J_{0} ∪ {i} is connected for some connected component J_{0} of J. The set of indexes adjacent to J is denoted by ∂J. The number of elements in J is denoted |J|.

Definition 1 Let J be a connected set of indexes. For i ∉ ∂J define ν_{i}(J) = 0. For i ∈ ∂J define ν_{i}(J) to be the number next to the appropriate diagram below. The black nodes correspond to J and the white node corresponds to i. Symmetry of Dynkin diagrams is tacitly assumed.

∘ • ⋯ • •  |J|,
∘ • ⋯ •>•  2|J| − 1, • • ⋯ •>∘  2|J|,
∘ • ⋯ •<•  2|J|,  • • ⋯ •<∘  |J|,
∘ • ⋯ • {• \atop •} • 2|J| − 2, • • ⋯ • {• \atop •} ∘ 2|J| − 2,
∘ • {• \atop •} • •  10,  • • {• \atop ∘} • •  9,
∘ • • {• \atop •} • •  16,  • • • {• \atop •} • ∘  15,
• • • {• \atop ∘} • •  12,  ∘ • • • {• \atop •} • • 27,
• • • • {• \atop •} • ∘ 21,  • • • • {• \atop ∘} • • 15,
∘ •>• •  6,  • •>• ∘  9,
∘<•  3,  •<∘  1,

For an arbitrary set of indexes I define ν_{i}(I) = ν_{i}(I_{1}) + ⋯ + ν_{i}(I_{p}) where I_{1}, …, I_{p} are the connected components of I.

Proposition 1 ([11]) Let X = G/P where G is a semisimple complex Lie group and P is a parabolic subgroup defined by a set of indexes I. Let μ_{X} be the weight of the anticanonical bundle K_{X}^{∗} of X. Then

μ_{X} = ∑_{i∉I}(2 + ν_{i}(I))λ_{i}

2 Estimating Products

We now prove some estimates for various products that appear in the proof of the main theorem.

Lemma 1 For any non-negative integers s and t,

\binom{2t + 1}{t}\binom{2s + 1}{s} ≤ \binom{2(t + s) + 1}{t + s}  (1)
\frac{(t + 1)^{t}}{t!} · \frac{(s + 1)^{s}}{s!} ≤ \frac{(t + s + 1)^{t+s}}{(t + s)!}  (2)

with equality if and only if t or s is 0.

Proof. The inequalities are obviously equalities when s or t is 0. So we assume t, s > 0 and show strict inequalities hold for (1) and (2) by fixing s and applying induction on t. They are easily seen to hold for t = 1. Let g(t) = \binom{2t+1}{t} (resp., (t+1)^{t}/t!), and let f(t) = g(t+1)/g(t) = 4−2/(t+2) (resp., [1+1/(t+1)]^{t+1}), an increasing function of t > 0. By the induction hypothesis,

g(t + 1)g(s) = f(t)g(t)g(s) < f(t)g(t + s) < f(t + s)g(t + s) = g(t + s + 1)

□

Definition 2 Let t and s be positive integers. A simple sequence (of length s) is a set S of rational numbers of the form

S = S(t, s) = \{\frac{3t + s − 1 + i}{t + i} | 0 ≤ i ≤ s − 1\}

The shifted sequence of S(t, s) is

S′ = S′(t, s) = \{\frac{3t + s − 1 + i}{t + i} − 1 | 0 ≤ i ≤ s − 1\}

The products of the numbers in S and S′ are denoted by

ΠS = ∏_{i=0}^{s−1} \frac{3t + s − 1 + i}{t + i} = \binom{3t + 2s − 2}{s}/\binom{t + s − 1}{s}

ΠS′ = ∏_{i=0}^{s−1} \frac{2t + s − 1}{t + i} = \frac{(2t + s − 1)^{s}(t − 1)!}{(t + s − 1)!}

Documenta Mathematica 9 (2004) 251–263

Lemma 2 Let S(t, s) be a simple sequence and let S′(t, s) be the shifted sequence of S(t, s).

1. If the first and last elements of S(t, s) are removed, the remaining set is the simple sequence S(t + 1, s − 2).

2. ΠS(t, s) and ΠS′(t, s) are decreasing in t. In particular,

3^{s} = lim_{t→∞} ΠS(t, s) ≤ ΠS(t, s) ≤ ΠS(1, s) = \binom{2s + 1}{s}
2^{s} = lim_{t→∞} ΠS′(t, s) ≤ ΠS′(t, s) ≤ ΠS′(1, s) = \frac{(s + 1)^{s}}{s!}

Proof. The first assertion is immediate from the definition. To prove the second assertion, let f(t) = ΠS(t, s) and let m = [(s − 1)/2] be the least integer ≤ (s − 1)/2. Then, for t > 0,

\frac{d}{dt} log f(t) = ∑_{i=0}^{s−1} \frac{2i − (s − 1)}{(3t + s − 1 + i)(t + i)}

= ∑_{i=0}^{m} −\frac{s − 1 − 2i}{(3t + s − 1 + i)(t + i)} + \frac{s − 1 − 2i}{(3t + 2s − 2 − i)(t + s − 1 − i)} ≤ 0

and hence f is decreasing. Now let g(t) = ΠS′(t, s) and define h(t) = g(t + 1)/g(t) = [1 + 2/(2t + s − 1)]^{s}t/(t + s). Then

\frac{d}{dt} log h(t) = \frac{s(s^{2} − 1)}{t(t + s)((2t + s)^{2} − 1)} ≥ 0

so h is increasing and approaches 1 as t → ∞. Therefore, g is decreasing. □

3 Bounds for K_{X}^{∗}

Theorem 1 Let X be a homogeneous projective rational manifold of dimension n. Then

3^{n} ≤ dim H^{0}(X, K_{X}^{∗}) ≤ \binom{2n + 1}{n} and 2^{n}n! ≤ deg K_{X}^{∗} ≤ (n + 1)^{n}

with equality in the lower bounds if and only if X is a flag manifold and equality in the upper bounds if and only if X = P^{n}.

Proof. Write X = G/P where G is a semisimple Lie group and P is a parabolic subgroup. Let I be the subset of indexes that defines P, and let I_{1}, …, I_{m} be the connected components of I. Let μ_{X} be the weight of the anticanonical bundle as given in Proposition 1 so that H^{0}(X, K_{X}^{∗}) is the irreducible representation of G with highest weight μ_{X} = ∑_{i∉I}(2 + ν_{i}(I))λ_{i}. Set δ = (1/2) ∑_{α>0} α = λ_{1} + ⋯ + λ_{ℓ}. By the Weyl dimension formula [7],

h = dim H^{0}(X, K_{X}^{∗}) = ∏_{α∈Φ_{X}^{+}} \frac{(μ_{X} + δ, α)}{(δ, α)}  (3)

and the degree of K_{X}^{∗} is given by [4]

d = deg K_{X}^{∗} = n! ∏_{α∈Φ_{X}^{+}} \frac{(μ_{X}, α)}{(δ, α)}  (4)

Let G_{1}, …, G_{r} be the simple factors of G. Then X = X_{1} × ⋯ × X_{r} where X_{i} = G_{i}/G_{i} ∩ P. Let n = dim X, n_{i} = dim X_{i}, h_{i} = dim H^{0}(X_{i}, K_{X_{i}}^{∗}) and d_{i} = deg K_{X_{i}}^{∗}, 1 ≤ i ≤ r. If 3^{n_{i}} ≤ h_{i} ≤ \binom{2n_{i}+1}{n_{i}} and 2^{n_{i}}n_{i}! ≤ d_{i} ≤ (n_{i} + 1)^{n_{i}}, 1 ≤ i ≤ r, then the above formulas along with Lemma 1 imply 3^{n} ≤ h = h_{1} ⋯ h_{r} ≤ ∏_{i=1}^{r} \binom{2n_{i}+1}{n_{i}} ≤ \binom{2n+1}{n} and 2^{n}n! ≤ d = n! (d_{1}/n_{1}!) ⋯ (d_{r}/n_{r}!) ≤ n! ∏_{i=1}^{r}(n_{i} + 1)^{n_{i}}/n_{i}! ≤ (n + 1)^{n}, since n = n_{1} + ⋯ + n_{r}. We may therefore assume that G is simple. The theorem can be verified by direct calculation for each of the exceptional simple Lie groups and their finite number of conjugacy classes of parabolic subgroups. While the details are too lengthy to include in this article, the results can be summarized as follows. The minimum of h is 3^{n} and is achieved only for Borel subgroups. The maximum of h is always strictly less than \binom{2n+1}{n}. In fact, the minimum of \binom{2n+1}{n}/h over all parabolic subgroups for each type is greater than 3.11 for E_{6}, 9.96 for E_{7}, 758.2 for E_{8}, 3.24 for F_{4}, and 1.22 for G_{2}, and this minimum is achieved for the maximal parabolic subgroups defined by I = {2, …, ℓ}, or {1, 2, 3} for F_{4} (the simple roots are indexed from left to right in the diagrams shown in Definition 1). The proof for the classical types A_{ℓ}, B_{ℓ}, C_{ℓ} and D_{ℓ} is accomplished by showing that the product (3) can be written as a product of simple sequences S_{1}, …, S_{σ}. For, if we know that h = ΠS_{1} ⋯ ΠS_{σ}, it follows from (4) that d = n! ΠS′_{1} ⋯ ΠS′_{σ}, and by Lemmas 1 and 2, we obtain 3^{n} ≤ h ≤ ∏_{i=1}^{σ} \binom{2|S_{i}|+1}{|S_{i}|} ≤ \binom{2n+1}{n} and 2^{n}n! ≤ d ≤ n! ∏_{i=1}^{σ}(|S_{i}| + 1)^{|S_{i}|}/|S_{i}|! ≤ (n + 1)^{n}, since n = |S_{1}| + ⋯ + |S_{σ}|. We now prove that such a decomposition of (3) is possible for each simple classical type. Type A_{ℓ}: Let μ_{X} + δ = m_{1}λ_{1} + ⋯ + m_{ℓ}λ_{ℓ}, where m_{i} = ⟨μ_{X}, α_{i}⟩ + 1, 1 ≤ i ≤ ℓ, and let ϵ_{1}, …, ϵ_{ℓ+1} denote the standard orthonormal basis of R^{ℓ+1}. The simple roots for type A_{ℓ} are α_{i} = ϵ_{i} − ϵ_{i+1}, 1 ≤ i ≤ ℓ and the positive roots are α_{i} +⋯+α_{j−1} = ϵ_{i} −ϵ_{j}, 1 ≤ i < j ≤ ℓ+1. The dimension formula (3) becomes h = ∏ a_{ij} where a_{ij} = (m_{i} + ⋯ + m_{j−1})/(j − i) and the product is taken over all indexes i < j that are not both in same connected component of I.

According to Proposition 1, the coefficients m_{1}, …, m_{ℓ} are given by

m_{i} = \{ \begin{matrix}1 & if i ∈ I \\ 3 + |I_{ν}| & if i ∈ ∂I_{ν} for some ν \\ 3 + |I_{ν}| + |I_{ν+1}| & if i ∈ ∂I_{ν} ∩ ∂I_{ν+1} for some ν \\ 3 & otherwise \end{matrix}  (5)

An example is given by the list of numbers at the top of Figure 1 (the indexes in I correspond to black nodes). The numbers in the product h = ∏ a_{ij} can be arranged into rectangular arrays as follows. Let i_{1} < ⋯ < i_{k} be an ordered list of those indexes i not in I and set i_{0} = 0, i_{k+1} = ℓ + 1. For 1 ≤ p ≤ q ≤ k, define R_{pq} = {a_{ij} | i_{p−1} < i ≤ i_{p}, i_{q} < j ≤ i_{q+1}}, as illustrated in Figure 1.

Figure 1: Type A_{ℓ} decomposition

3 5 1 1 8 1 1 1 6 ∘ ∘ • • ∘ • • • ∘

\frac{3}{1}  \frac{8}{2}  \frac{9}{3}  \frac{10}{4}  \frac{18}{5}  \frac{19}{6}  \frac{20}{7}  \frac{21}{8}  \frac{27}{9}
\frac{5}{1}  \frac{6}{2}  \frac{7}{3}  \frac{15}{4}  \frac{16}{5}  \frac{17}{6}  \frac{18}{7}  \frac{24}{8}
\frac{10}{3}  \frac{11}{4}  \frac{12}{5}  \frac{13}{6}  \frac{19}{7}
\frac{9}{2}  \frac{10}{3}  \frac{11}{4}  \frac{12}{5}  \frac{18}{6}
\frac{8}{1}  \frac{9}{2}  \frac{10}{3}  \frac{11}{4}  \frac{17}{5}

\frac{9}{4}

\frac{8}{3}

\frac{7}{2}

\frac{6}{1}

Then h is the product of the numbers in all the rectangular arrays R_{pq}, 1 ≤ p ≤ q ≤ k. Each R_{pq} consists of rational numbers whose numerators and denominators both increase by 1 in each row and column, starting in the lower left corner, a_{i_{p}(i_{q}+1)}. From (5) it follows that a_{i_{p}(i_{q}+1)} has the form (3t+s−1)/t where t = i_{q} − i_{p} + 1 and s is the number of rows + columns − 1 = (i_{p} − i_{p−1}) + (i_{q+1} − i_{q}) − 1. Therefore, R_{pq} may be decomposed into simple sequences, R_{pq} = S_{0}∪…∪S_{σ} where S_{0} = {a_{ij} | i = i_{p−1}+1 or j = i_{q}+1} = S(t, s) is the set of numbers in the left column and the top row of R_{pq}, and S_{i} = S(t+i, s−2i) is obtained by removing the lower left and top right number from S_{i−1}, 1 ≤ i ≤ σ, as illustrated in Figure 2.

Figure 2: Type A_{ℓ} rectangular array

\frac{10}{3}  \frac{11}{4}  \frac{12}{5}  \frac{13}{6}
\frac{9}{2}  \frac{10}{3}  \frac{11}{4}  \frac{12}{5}
\frac{8}{1}  \frac{9}{2}  \frac{10}{3}  \frac{11}{4}

Repeating this procedure for each rectangular array R_{pq}, 1 ≤ p ≤ q ≤ k, shows that for type A_{ℓ} the product (3) can be decomposed into a product of simple sequences. Type B_{ℓ}: We now show the same type of decomposition is possible for type B_{ℓ} by embedding the appropriate numbers into a diagram for type A_{2ℓ−1}. We again write μ_{X} + δ = m_{1}λ_{1} + ⋯ + m_{ℓ}λ_{ℓ}, m_{i} = ⟨μ_{X}, α_{i}⟩ + 1, 1 ≤ i ≤ ℓ, and let ϵ_{1}, …, ϵ_{ℓ} denote the standard orthonormal basis of R^{ℓ}. The simple roots are α_{i} = ϵ_{i} − ϵ_{i+1}, 1 ≤ i ≤ ℓ − 1, and α_{ℓ} = ϵ_{ℓ}. The positive roots are α_{i} + ⋯ + α_{j−1} = ϵ_{i} − ϵ_{j}, α_{i} + ⋯ + α_{j−1} + 2α_{j} + ⋯ + 2α_{ℓ} = ϵ_{i} + ϵ_{j}, 1 ≤ i < j ≤ ℓ, and α_{i} + ⋯ + α_{ℓ} = ϵ_{i}, 1 ≤ i ≤ ℓ. The dimension formula (3) becomes h = ∏ a_{ij} ×∏ b_{ij} where a_{ij} = (m_{i} +⋯+m_{j−1})/(j −i), 1 ≤ i < j ≤ ℓ, b_{ij} = (m_{i} +⋯+m_{j−1} +2m_{j} +⋯+2m_{ℓ−1} +m_{ℓ})/(2ℓ−i−j +1), 1 ≤ i ≤ j ≤ ℓ. To avoid trivial factors, these products should be taken over i, j not in the same connected component of I, although in the following arguments it is convenient to include all terms. Define Î = {i | i ∈ I or 2ℓ − i ∈ I}. Then Î defines a parabolic subgroup P̂ of a simple group Ĝ of type A_{2ℓ−1}. Let X̂ = Ĝ/P̂. By Proposition 1, the coefficients of μ_{X} + δ appear as the first half of the coefficients of μX̂ + δ, see Figure 3.

Figure 3: Conversion of type B_{ℓ} to type A_{2ℓ−1}

m_{1} m_{2}  m_{ℓ−1} m_{ℓ} m_{ℓ−1}  m_{2} m_{1}

• • ⋯ • • • ⋯ • •

For a fixed i the product h_{i} = ∏ a_{ij} × ∏ b_{ij} can be arranged as

\frac{m_{i}}{1} · \frac{m_{i} + m_{i+1}}{2} ⋯ \frac{m_{i} + ⋯ + m_{ℓ}}{ℓ − i + 1} · \frac{s_{i} + m_{ℓ−1}}{ℓ − i + 2} ⋯ \frac{s_{i} + m_{ℓ−1} + ⋯ + m_{i}}{2(ℓ − i) + 1}

where s_{i} = m_{i} + ⋯ + m_{ℓ}. Therefore, the non-trivial terms in h correspond to the numbers in the upper left half of the rectangular arrays R_{pq} defined for type A_{2ℓ−1}. These triangular arrays can clearly be broken up into simple sequences, see Figure 4, showing that h is a product of simple sequences.

Figure 4: Type B_{ℓ} decomposition

1 1 1 9 1 1 1 9 1 1 1

• • • > ∘ ⟶ • • • ∘ • • •

\frac{12}{4}  \frac{13}{5}  \frac{14}{6}  \frac{15}{7}
\frac{11}{3}  \frac{12}{4}  \frac{13}{5}
\frac{10}{2}  \frac{11}{3}

\frac{9}{1}

Type C_{ℓ}: The proof for this case is almost identical to type B_{ℓ}. The simple roots are α_{i} = ϵ_{i} − ϵ_{i+1}, 1 ≤ i ≤ ℓ − 1, and α_{ℓ} = 2ϵ_{ℓ}. The positive roots are α_{i} + ⋯ + α_{j−1} = ϵ_{i} − ϵ_{j}, α_{i} + ⋯ + α_{j−1} + 2α_{j} + ⋯ + 2α_{ℓ−1} + α_{ℓ} = ϵ_{i} + ϵ_{j}, 1 ≤ i < j ≤ ℓ, and 2α_{i} + ⋯ + 2α_{ℓ} + α_{ℓ} = 2ϵ_{i}, 1 ≤ i ≤ ℓ. The dimension formula (3) becomes h = ∏ a_{ij} × ∏ b_{ij} where a_{ij} = (m_{i} + ⋯ + m_{j−1})/(j − i), 1 ≤ i < j ≤ ℓ, b_{ij} = (m_{i} + ⋯ + m_{j−1} + 2m_{j} + ⋯ + 2m_{ℓ})/(2ℓ − i − j + 2), 1 ≤ i ≤ j ≤ ℓ. Define Î = {i | i ∈ I or 2ℓ − i + 1 ∈ I}. Then Î defines a parabolic subgroup P̂ of a simple group Ĝ of type A_{2ℓ}. Let X̂ = Ĝ/P̂. By Proposition 1, the coefficients of μ_{X} + δ appear as the first half of the coefficients of μX̂ + δ, see Figure 5.

Figure 5: Conversion of type C_{ℓ} to type A_{2ℓ}

m_{1} m_{2}  m_{ℓ−1} m_{ℓ} m_{ℓ} m_{ℓ−1}  m_{2} m_{1}

• • ⋯ • • • • ⋯ • •

For a fixed i the product h_{i} = ∏ a_{ij} × ∏ b_{ij} can be arranged as

\frac{m_{i}}{1} · \frac{m_{i} + m_{i+1}}{2} ⋯ \frac{m_{i} + ⋯ + m_{ℓ}}{ℓ − i + 1} · \frac{s_{i} + m_{ℓ}}{ℓ − i + 2} ⋯ \frac{s_{i} + m_{ℓ} + ⋯ + m_{i+1}}{2(ℓ − i) + 1}

where s_{i} = m_{i} + ⋯ + m_{ℓ}. Therefore, the non-trivial terms in h correspond to the numbers in the upper left half (above the diagonal) of the rectangular arrays R_{pq} defined for type A_{2ℓ}. These triangular arrays can be broken up into simple sequences as before, see Figure 6, showing that h is a product of simple sequences. Type D_{ℓ}: The proof for this case must be handled somewhat differently than the previous two cases. The simple roots are α_{i} = ϵ_{i} − ϵ_{i+1}, 1 ≤ i ≤ ℓ − 1, and α_{ℓ} = ϵ_{ℓ−1} + ϵ_{ℓ}. The positive roots are α_{i} + ⋯ + α_{j−1} = ϵ_{i} − ϵ_{j}, 1 ≤ i < j ≤ ℓ,

Figure 6: Type C_{ℓ} decomposition

1 1 1 6 1 1 1 6 6 1 1 1 • • • < ∘ ⟶ • • • ∘ ∘ • • •

\frac{9}{4}  \frac{15}{5}  \frac{16}{6}  \frac{17}{7}
\frac{8}{3}  \frac{14}{4}  \frac{15}{5}
\frac{7}{2}  \frac{13}{3}

\frac{6}{1}

α_{i} + ⋯ + α_{j−1} + 2α_{j} + ⋯ + 2α_{ℓ−2} + α_{ℓ−1} + α_{ℓ} = ϵ_{i} + ϵ_{j}, 1 ≤ i < j ≤ ℓ − 1, and α_{i} + ⋯ + α_{ℓ−2} + α_{ℓ} = ϵ_{i} + ϵ_{ℓ}, 1 ≤ i ≤ ℓ − 2. The dimension formula (3) becomes h = ∏ a_{ij} × ∏ b_{ij} × ∏ c_{i} where a_{ij} = (m_{i} + ⋯ + m_{j−1})/(j − i), 1 ≤ i < j ≤ ℓ, b_{ij} = (m_{i}+⋯+m_{j−1}+2m_{j}+⋯+2m_{ℓ−2}+m_{ℓ−1}+m_{ℓ})/(2ℓ−i−j), 1 ≤ i < j ≤ ℓ − 1, c_{i} = (m_{i} + ⋯ + m_{ℓ−2} + m_{ℓ})/(ℓ − i), 1 ≤ i ≤ ℓ − 2, and c_{ℓ−1} = m_{ℓ}. By symmetry of the Dynkin diagram, we may assume m_{ℓ−1} ≤ m_{ℓ}. We first assume m_{ℓ−1} = m_{ℓ}. Define Î = {i | i ∈ I or 2ℓ − i − 1 ∈ I (and i > ℓ)}. Then Î defines a parabolic subgroup P̂ of a simple group Ĝ of type A_{2ℓ−2}. Let X̂ = Ĝ/P̂. By Proposition 1, the coefficients of μ_{X} + δ appear as the first half of the coefficients of μX̂ + δ, see Figure 7.

Figure 7: Conversion of type D_{ℓ} to type A_{2ℓ−2}

m_{1} m_{2}  m_{ℓ−1} m_{ℓ} m_{ℓ−2}  m_{2} m_{1}

• • ⋯ • • • ⋯ • •

For a fixed i the product h_{i} = ∏ a_{ij} × ∏ b_{ij} can be arranged as

\frac{m_{i}}{1} · \frac{m_{i} + m_{i+1}}{2} ⋯ \frac{m_{i} + ⋯ + m_{ℓ}}{ℓ − i + 1} · \frac{s_{i} + m_{ℓ−2}}{ℓ − i + 2} ⋯ \frac{s_{i} + m_{ℓ−2} + ⋯ + m_{i+1}}{2(ℓ − i) − 1}

where s_{i} = m_{i} + ⋯ + m_{ℓ}. Therefore, the non-trivial terms in ∏ h_{i} correspond to the numbers in the upper left half (above the diagonal) of the rectangular arrays R_{pq} defined for type A_{2ℓ−2}. These triangular arrays can be broken up into simple sequences as before, see Figure 8. The numbers in the remaining product, ∏ c_{i}, are easily seen to form a product of simple sequences by Proposition 1. Therefore, the full product h is a product of simple sequences. We now assume m_{ℓ−1} < m_{ℓ}. In this case, the product h is organized in a slightly different way. For fixed i, the previous product h_{i} is split into two

Figure 8: Type D_{ℓ} decomposition, m_{ℓ−1} = m_{ℓ}

1 8 1 1 1 8 1 1 1 1 8 1

• ∘ {• \atop •} • ⟶ • ∘ • • • • ∘ •

1

\frac{9}{2}  \frac{10}{3}  \frac{11}{4}  \frac{12}{5}  \frac{13}{6}  \frac{21}{7} \frac{8}{1}  \frac{9}{2}  \frac{10}{3}  \frac{11}{4}  \frac{12}{5}  \frac{10}{3}  \frac{11}{4}

terms with c_{i} inserted at the beginning of the second term:

\frac{m_{i}}{1} · \frac{m_{i} + m_{i+1}}{2} ⋯ \frac{m_{i} + ⋯ + m_{ℓ−1}}{ℓ − i}

\frac{m_{i} + ⋯ + m_{ℓ−2} + m_{ℓ}}{ℓ − i} · \frac{s_{i}}{ℓ − i + 1} · \frac{s_{i} + m_{ℓ−2}}{ℓ − i + 2} ⋯ \frac{s_{i} + m_{ℓ−2} + ⋯ + m_{i+1}}{2(ℓ − i) − 1}

Therefore, the non-trivial terms in the product h come from two arrays, the first corresponding to the numbers in the rectangular arrays R_{pq} defined for type A_{ℓ−1} and the second corresponding to the numbers in the upper half of certain rectangular arrays R_{pq} defined for type A_{2ℓ−3}. As before, these rectangular and triangular arrays can be broken up into simple sequences, see Figure 9, and hence the product h is a product of simple sequences.

Figure 9: Type D_{ℓ} decomposition, m_{ℓ−1} < m_{ℓ}

6 1 1 1  6 1 1 1  7 1 1

∘ • {• \atop ∘} • ⟶ ∘ • • • ⋯ ∘ • •

7

\frac{6}{1}  \frac{7}{2}  \frac{8}{3}  \frac{9}{4}  \frac{15}{4}  \frac{16}{5}  \frac{17}{6}  \frac{18}{7} \frac{9}{3}  \frac{10}{4}  \frac{11}{5} \frac{8}{2}  \frac{9}{3} \frac{7}{1}

Documenta Mathematica 9 (2004) 251–263

It remains to show that equality is obtained only in the designated cases. From Lemmas 1 and 2 is is clear that if h = 3^{n} then all the simple sequences making up h must have length one and each consists of the number 3. Consequently, m_{i} = 3 for 1 ≤ i ≤ ℓ, so that μ_{X} = 2δ, and, by Proposition 1, X is a flag manifold. Likewise, if h = \binom{2n+1}{n}, then h must be the product of just one simple sequence, h = S(1, n). By Proposition 1, this situation occurs either in type A_{n} when m_{1} = n + 2 and m_{i} = 1, 2 ≤ i ≤ n (or m_{n} = n + 2 and m_{i} = 1, 1 ≤ i ≤ n − 1), or in type C_{ℓ} when n = 2ℓ − 1, m_{1} = n + 2 = 2ℓ + 1, and m_{i} = 1, 2 ≤ i ≤ ℓ. In both of these cases the underlying manifold is projective space, P^{n}. If the degree is d = 2^{n}n! (resp. (n + 1)^{n}), then from (3) and (4), h = 3^{n} (resp. \binom{2n+1}{n}), and the same argument applies. □

References

[1] Akhiezer, D., Lie Group Actions in Complex Analysis, Vieweg, Braunschweig, 1995.

[2] Borel, A., Linear Algebraic Groups, 2nd ed., Benjamin, New York, 1991.

[3] Borel, A., Hirzebruch, F., Characteristic classes and homogeneous spaces, part I, Amer. J. Math. 80 (1958), 459–538.

[4] Borel, A., Hirzebruch, F., Characteristic classes and homogeneous spaces, part II, Amer. J. Math. 81 (1959), 315–381.

[5] Borel, A., Remmert, R., Über kompakte homogene Kählersche Mannigfaltigkeiten, Math. Ann. 145 (1962), 429–439.

[6] Clemens, H., Ran, Z., A new method in Fano geometry, Internat. Math. Res. Notices 2000, no.10, 527–549.

[7] Fulton, W., Harris, J., Representation Theory, Springer, Berlin, Heidelberg, New York, 1991.

[8] Iskovskikh, V., Prokhorov, Y., Fano Varieties, Algebraic Geometry V, Encyclopaedia of Mathematical Sciences, Vol. 47, Springer, Berlin, Heidelberg, New York, 1999.

[9] Kollár, J., Rational Curves on Algebraic Varieties, Springer, Berlin, Heidelberg, New York, 1996.

[10] Kollár, J., Miyaoka, Y., Mori, S., Rational Connectedness and Boundedness of Fano Manifolds, J. Diff. Geom. 36 (1992), 765–779.

[11] Snow, D., The nef value of homogeneous line bundles and related vanishing theorems, Forum Math. 7 (1995), 385–392.

[12] Snow, D., Winkelmann, J., Compact complex homogeneous manifolds with large automorphism groups, Invent. Math. 134 (1998), 139–144.

[13] Snow, D., A bound for the dimension of the automorphism group of a homogeneous compact complex manifold, to appear Proc. Amer. Math. Soc.

Dennis Snow Department of Mathematics University of Notre Dame Notre Dame, Indiana 46556 snow.1@nd.edu 264

Documenta Mathematica 9 (2004)