Duality and Integrability

on Contact Fano Manifolds

Jarosław Buczyński ^{1}

Received: February 2 2010 Revised: June 8, 2010

Communicated by Thomas Peternell

Abstract. We address the problem of classification of contact Fano manifolds. It is conjectured that every such manifold is necessarily homogeneous. We prove that the Killing form, the Lie algebra grading and parts of the Lie bracket can be read from geometry of an arbitrary contact manifold. Minimal rational curves on contact manifolds (or contact lines) and their chains are the essential ingredients for our constructions.

2010 Mathematics Subject Classification: Primary: 14M17; Secondary: 53C26, 14M20, 14J45; Keywords and Phrases: complex contact manifold, Fano variety, minimal rational curves, adjoint variety, Killing form, Lie bracket, Lie algebra grading

1 Introduction

In this article we are interested in the classification of contact Fano manifolds. We review the relevant definitions in §2. So far the only known examples of contact Fano manifolds are obtained as follows. For a simple Lie group G consider its adjoint action on ℙ(𝔤), where 𝔤 is the Lie algebra of G. This action has a unique closed orbit X and this X has a natural contact structure. In this situation X is called a projectivised minimal nilpotent orbit, or the adjoint variety of G. By the duality determined by the Killing form, equivalently we can consider the coadjoint action of G on ℙ(𝔤^{∗}) and X is isomorphic to the unique closed orbit in ℙ(𝔤^{∗}).

1Dedicated in memory of Marcin Hauzer.

Documenta Mathematica 15 (2010) 821–841

Conjecture 1.1 (LeBrun, Salamon). If X is a Fano complex contact manifold, then X is the adjoint variety of a simple Lie group G.

This problem originated with a problem in Riemannian geometry. In [Ber55] a list of all possible holonomy groups of simply connected Riemannian manifolds is given. The existence problem for all the cases has been solved locally. Also compact non-homogeneous examples with most of the possible holonomy groups were constructed with the unique exception of the quaternion-Kähler manifolds. It is conjectured that the compact quaternion-Kähler manifolds must be homogeneous (see [LeB95] and references therein).

Conjecture 1.2 (LeBrun, Salamon). Let M be a compact quaternion-Kähler manifold. Then M is a homogeneous symmetric space (more precisely, it is one of the Wolf spaces — see [Wol65]).

The relation between the conjectures is given by the construction of a twistor space. The S^{2}-bundle of complex structures on tangent spaces to a quaternion-Kähler manifold M is called the twistor space of M. If M is compact, it has positive scalar curvature, and then the twistor space X has a natural complex structure and is a contact Fano manifold with a Kähler-Einstein metric. In particular, the twistor space of a Wolf space is an adjoint variety. Hence Conjecture 1.1 implies Conjecture 1.2. Conversely, if X is a contact Fano manifold with Kähler-Einstein metric, then it is a twistor space of a quaternion-Kähler manifold — see [LeB95]. In order to study the non-homogeneous contact manifolds (potentially nonexistent) it is natural to assume Pic X ≃ ℤ and further that X is not isomorphic to a projective space. This only exludes the adjoint varieties of types A and C (see §2 for more details). With this assumption, we take a closer look at three pieces of the homogeneous structure on adjoint varieties: the Killing form B on 𝔤, the Lie algebra grading 𝔤 = 𝔤_{−2} ⊕ 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1} ⊕ 𝔤_{2} (see [LM02, §6.1] and references therein) and a part of the Lie bracket on 𝔤. Understanding the underlying geometry allows us to define the appropriate generalisations of these notions on arbitrary contact Fano manifolds. An essential building block for our constructions is the notion of a contact line (or simply line) on X. These contact lines were studied by Kebekus [Keb01],

[Keb05] and Wiśniewski [Wiś00]. Also they are an instance of minimal rational curves, which are studied extensively. The geometry of contact lines was the original motivation to study Legendrian subvarieties in projective space (see [Bucz09] for an overview and many details). We briefly review the subject of lines on contact Fano manifolds in §3.1. The key ingedient is the construction of a family of divisors D_{x} parametrised by points x ∈ X (see §3.3). These divisors are swept by pairs of intersecting contact lines, one of which passes through x. In other words, set theoretically D_{x} is the set of points of X, which can be joined with x using at most 2 intersecting contact lines. The idea to study these loci comes from Wiśniewski [Wiś00] where he observed, that (under an additional minor assumption) these loci contain some non-trivial divisorial components and he studied the intersection numbers of certain curves on X with the divisorial components. Here we prove all the components of D_{x} are divisorial and draw conclusions from that observation going into a different direction than those of [Wiś00].

Theorem 1.3. Let X be a contact Fano manifold with Pic X ≃ ℤ and assume X is not isomorphic to a projective space. Then the locus D_{x} ⊂ X swept by the pairs of intersecting contact lines, one of which passes through x ∈ X is of pure codimension 1 and thus D_{x} determines a divisor on X. Let ⟨D⟩ ⊂ H^{0}(O(D_{x})) be the linear system spanned by these divisors. Let ϕ: X → ℙ⟨D⟩^{∗} be the map determined by the linear system ⟨D⟩ and let ψ: X → ℙ⟨D⟩ be the map x ↦ D_{x}. Then:

(i) both ϕ and ψ are regular maps.

(ii) there exists a unique up to scalar non-degenerate bilinear form B on ⟨D⟩, which determines an isomorphism ℙ⟨D⟩^{∗} ≃ ℙ⟨D⟩ making the following diagram commutative:

\xymatrix{ & ℙ⟨D⟩^{∗} \ar[dd]^{≃} \\ X \ar[ur]^{ϕ} \ar[dr]^{ψ} \\  & ℙ⟨D⟩.}

(iii) The bilinear form B is either symmetric or skew-symmetric.

(iv) If X ⊂ ℙ(𝔤^{∗}) is the adjoint variety of simple Lie group G, then ⟨D⟩ = 𝔤 and B is the Killing form on 𝔤.

With the notation of the theorem, after fixing a pair of general points x, w ∈ X there are certain natural linear subspaces of ⟨D⟩, which we denote ⟨D⟩_{−2}, ⟨D⟩_{−1}, ⟨D⟩_{0}, ⟨D⟩_{1} and ⟨D⟩_{2} (see §5 for details).

Theorem 1.4. If X ⊂ ℙ(𝔤^{∗}) is the adjoint variety of a simple Lie group G with Pic X ≃ Z and X not isomorphic to a projective space, then there exists a choice of a maximal torus of G and a choice of order of roots of 𝔤, such that ⟨D⟩_{i} = 𝔤_{i} for every i ∈ {−2, −1, 0, 1, 2}, where 𝔤 = 𝔤_{−2} ⊕ 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1} ⊕ 𝔤_{2} is the Lie algebra grading of 𝔤.

Finally, if X is the adjoint variety of G, then there is a rational map

[·, ·]: X × X −−→ ℙ(𝔤),

which is the Lie bracket on 𝔤 (up to projectivisation). Also there is a divisor D ⊂ X × X, such that for general (x, z) ∈ D the Lie bracket [x, z] is in X. We recover this bracket restricted to D for general contact manifolds:

Theorem 1.5. For X and D_{x} and in Theorem 1.3, let D ⊂ X × X be the divisor consisting of pairs (x, z) ∈ X × X, such that z ∈ D_{x}. There exists a rational map [·, ·]^{D}: D −−→ X, such that [x, z]^{D} = y, where y is an intersection point of a pair of contact lines that join x and z. In particular, this intersection point y and the pair of lines are unique for general pair (x, z) ∈ D. Moreover, if X is the adjoint variety of a simple Lie group G, then [·, ·]^{D} is the restricion of the Lie bracket.

In §2 we introduce and motivate our assumptions and notation. In §3 we review the notion of contact lines and their properties. We continue by studying certain types of loci swept by those lines and calculate their dimensions. In particular we prove there Theorem 3.6, which is a part of results summarised in Theorem 1.3. We also study the tangent bundle to D_{x} as a subspace of T X. In §4 we study the duality of maps ϕ and ψ introduced in Theorem 1.3 together with the consequences of this duality. This section is culminated with the proof of Theorem 1.3. In §5 we generalise the Lie algebra grading to arbitrary contact manifolds and prove Theorem 1.4. In §6 we prove that certain lines are integrable with respect to a special distribution on D_{x} and we apply this to prove Theorem 1.5.

Acknowledgements

The author is supported by Marie Curie International Outgoing Fellowship. The author would like to thank the following people for many enlightening discussions: Jun-Muk Hwang, Stefan Kebekus, Joseph M. Landsberg, Jarosław Wiśniewski and Fyodor L. Zak. Comments of Laurent Manivel helped to improve the presentation in the paper.

2 Preliminaries

Throughout the paper all our projectivisations ℙ are naive. This means, if V is a vector space, then ℙV = (V \ 0)/ℂ^{∗}, and similarly for vector bundles. A complex manifold X of dimension 2n + 1 is contact if there exists a vector subbundle F ⊂ T X of rank 2n fitting into an exact sequence:

0 → F → T X →^{θ} L → 0

such that the derivative dθ ∈ H^{0}(⋀_{2}F^{∗} ⊗L) of the twisted form θ ∈ H^{0}(T^{∗}X ⊗ L) is nowhere degenerate. In particular, dθ_{x} is a symplectic form on the fibre of contact distribution F_{x}. See [Bucz09, §E.3 and Chapter C] and references therein for an overview of the subject. A projective manifold X is Fano, if its anticanonical divisor K_{X}^{∗} = ⋀_{dim X}T X is ample.

If X is a projective contact manifold, then by Theorem of Kebekus, Peternell, Sommese and Wiśniewski [KPSW00] combined with a result by Demailly [Dem02], X is either a projectivisation of a cotangent bundle to a smooth projective manifold or X is a contact Fano manifold, with Pic X ≃ ℤ. In the second case, since K_{X} ≃ (L^{∗})^{⊗(n+1)}, by [KO73], either X ≃ ℙ^{2n+1} or Pic X = ℤ · [L]. Here we are interested in the case X ≄ ℙ^{2n+1}. Thus our assumption spelled out below only exclude some well understood cases (the projectivised cotangent bundles and the projective space) and they agree with the assumptions of Theorems 1.3, 1.4 and 1.5.

Notation 2.1. Throughout the paper X denotes a contact Fano manifold with Pic X generated by the class of L, where L = T X/F and F ⊂ T X is the contact distribution on X. We also assume dim X = 2n + 1.

From Theorem of Ye [Ye94] it follows that n ≥ 2. We will also consider the homogeneous examples of contact manifolds (i.e. the adjoint varieties). Thus we fix notation for the Lie group and its Lie algebra.

Notation 2.2. Throughout the paper G denotes a simple complex Lie group, not of types A or C (i.e. not isomorphic to SL_{n} nor Sp_{2n} nor their discrete quotients). Further 𝔤 is the Lie algebra of G. Thus 𝔤 is one of 𝔰𝔬_{n} (types B and D), or one of the exceptional Lie algebras 𝔤_{2}, 𝔣_{4}, 𝔢_{6}, 𝔢_{7} or 𝔢_{8}.

The contact structure on ℙ^{2n−1} = ℙ(ℂ^{2n}) is determined by a symplectic form ω on ℂ^{2n}. The precise relation between the contact and symplectic structures is decribed for instance in [Bucz09, §E.1] (see also [LeB95, Ex. 2.1]). In particular, for all x ∈ X, the projectivisation of a fibre of the contact distribution ℙF_{x} comes with a natural contact structure. Let M be a projective contact manifold (in our case M = X with X as in Notation 2.1 or M = ℙ^{2n−1}). A subvariety Z ⊂ M is Legendrian, if for all smooth points z ∈ Z the tangent space T_{z}Z is contained in the contact distribution of M and Z is of pure dimension \frac{1}{2}(dim M − 1). Recall from [Har95, Lecture 20] or [Mum99, III.§3,§4] the notion of tangent cone. For a subvariety Z ⊂ X, and a point x ∈ Z let τ_{x}Z ⊂ T_{x}X be the tangent cone of Z at x. In this article we will only need the following elementary properties of the tangent cone:

• τ_{x}Z is an affine cone (i.e. it is invariant under the standard action of ℂ^{∗} on T_{x}X).

• dim_{x} Z = dim τ_{x}Z and thus if Z is irreducible, then dim Z = dim τ_{x}Z.

• If x ∈ Z_{1} ⊂ Z_{2} ⊂ X, then τ_{x}Z_{1} ⊂ τ_{x}Z_{2}.

• If Z is smooth at x, then τ_{x}Z = T_{x}Z.

Since τ_{x}Z is a cone, let ℙτ_{x}Z ⊂ ℙT_{x}X be the corresponding projective variety.

3 Loci swept out by lines

A rational curve l ⊂ X is a contact line (or simply a line) if deg L|_{l} = 1. Let RatCurves^{n}(X) be the normalised scheme parametrising rational curves on X, as in [Kol96, II.2.11]. Let Lines(X) ⊂ RatCurves^{n}(X) be the subscheme parametrising lines. Then every component of Lines(X) is a minimal component of X in the sense of [HM04]. We fix H ≠ ∅ a union of some irreducible components of Lines(X). By a slight abuse of notation, from now on we say l is a (contact) line if and only if l ∈ H. For simplicity, the reader may choose to restrict his attention to one of the extreme cases: either to the case H = Lines(X) (and thus be consistent with [Wiś00] and the first sentence of this section) or to the case where H is one of the irreducible components of Lines(X) (and thus be consistent with [Keb01, Keb05]). In general it is expected that Lines(X) (with X as in Notation 2.1) is irreducible and all the cases are the same.

3.1 Legendrian varieties swept by lines

We denote by C_{x} ⊂ X the locus of contact lines through x ∈ X. Let 𝒞_{x} := ℙτ_{x}C_{x} ⊂ ℙ(T X). Note that with our assumptions both C_{x} and 𝒞_{x} are closed subsets of X or ℙ(T_{x}X) respectively. The following theorem briefly summarises results of [Keb05] and earlier:

Theorem 3.1. With X as in Notation 2.1 let x ∈ X be any point. Then:

(i) There exist lines through x, in particular C_{x} and 𝒞_{x} are non-empty.

(ii) C_{x} is Legendrian in X and 𝒞_{x} ⊂ ℙ(F_{x}) and 𝒞_{x} is Legendrian in ℙ(F_{x}).

(iii) If in addition x is a general point of X, then 𝒞_{x} is smooth and each irreducible component of 𝒞_{x} is linearly non-degenerate in ℙ(F_{x}). Further C_{x} is isomorphic to the projective cone over 𝒞_{x} ⊂ ℙ(F_{x}), i.e. C_{x} ≃ 𝒞̃_{x} ⊂ ℙ(F_{x} ⊕ ℂ), in such a way that lines through x are mapped bijectively onto the generators of the cone and restriction of L to C_{x} via this isomorphism is identified with the restriction of O_{ℙ(F_{x}⊕ℂ)}(1) to 𝒞̃_{x}. In particular all lines through x are smooth and two different lines intersecting at x will not intersect anywhere else, nor they will share a tangent direction.

Proof. Part (i) is proved in [Keb01, §2.3]. The proof of (ii) is essentially contained in [KPSW00, Prop. 2.9]. Explicit statements are in [Keb01, Prop. 4.1] for C_{x} and in [Wiś00, Lemma 5] for 𝒞_{x}. Also [HM99] may claim the authorship of this observation, since the proof in the homogeneous case is no different than in the general case. Assume x ∈ X is a general point. The statements of (iii) are basically [Keb05, Thm 1.1], which however assumes (in the statement) that H is irreducible. This is never used in the proof, with the exception of the argument for the irreducibility of C_{x} — see however Remark 3.2. Thus 𝒞_{x} is smooth and C_{x} is isomorphic to the cone over 𝒞_{x} as claimed. Each irreducible component 𝒞_{x} is non-degenerate in ℙF_{x} by [Keb01, Thm 4.4] — again the statement is only for 𝒞_{x}, not for its components, however the proof stays correct in this more general setup. In particular, [Keb01, Lemma 4.3] implies that C_{x} polarised by L|_{C_{x}} is not isomorphic with a linear subspace with polarised by O(1). Thus the other results of this theorem give alternate (but more complicated) proof of that generalised non-degeneracy.

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Remark 3.2. Note that (assuming H is irreducible) Kebekus [Keb05] also stated that C_{x} and 𝒞_{x} are irreducible for general x. However it was observed by Kebekus himself together with the author that there is a gap in the proof. This gap is on page 234 in Step 2 of proof of Proposition 3.2 where Kebekus claims to construct “a well defined family of cycles” parametrised by a divisor D^{0}. This is not necessarily a well defined family of cycles: Condition (3.10.4) in [Kol96, §I.3.10] is not necessarily satisfied if D^{0} is not normal and there seem to be no reason to expect that D^{0} is normal. As a consequence the map Φ: D^{0} → Chow(X) is not necessarily regular at non-normal points of D^{0} and it might contract some curves. Let us define:

C^{2} ⊂ X × X
C^{2} := {(x, y) | y ∈ C_{x}} ,

i.e. this is the locus of those pairs (x, y), which are both on the same contact line. Again this locus is a closed subset of X × X. Analogously, define:

C^{3} := C^{2} ×_{X} C^{2}

so that:

C^{3} ⊂ X × X × X
C^{3} := {(x, y, z) | y ∈ C_{x}, z ∈ C_{y}} .

Finally, for x ∈ X we also define C_{x}^{2}:

C_{x}^{2} ⊂ X × X ≃ {x} × X × X
C_{x}^{2} := {(y, z) | y ∈ C_{x}, z ∈ C_{y}} ,

with the scheme structure of the fibre of C^{3} under the projection on the first coordinate. Since for all x ∈ X all irreducible components of C_{x} are of dimension n (see Theorem 3.1) we conclude:

Proposition 3.3. All C^{2}, C_{x}^{2}, C^{3} are projective subschemes, they are all of pure dimension, and their dimensions are:

• dim C^{2} = 3n + 1.

• dim C_{x}^{2} = 2n.

• dim C^{3} = 4n + 1.

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3.2 Joins and secants of Legendrian subvarieties

For subvarieties Y_{1}, Y_{2} ⊂ ℙ^{N} recall that their join Y_{1} ∗ Y_{2} is the closure of the locus of lines between points y_{1} ∈ Y_{1} and y_{2} ∈ Y_{2}. Note that the expected dimension of Y_{1} ∗ Y_{2} is dim Y_{1} + dim Y_{2} + 1. We are only concerned with two special cases: either Y_{1} and Y_{2} are disjoint or Y_{1} = Y_{2}.

Lemma 3.4. If Y_{1}, Y_{2} ⊂ ℙ^{N} are two disjoint subvarieties of dimensions k − 1 and N −k respectively, then their join Y_{1}∗Y_{2} fills out the ambient space, i.e. this join is of expected dimension.

Proof. Let p ∈ ℙ^{N} be a general point and consider the projection π : ℙ^{N} −−→ ℙ^{N−1} away from p. Let Z_{i} = π(Y_{i}) for i = 1, 2. Since p is general, dim Z_{i} = dim Y_{i} and thus Z_{1} ∩ Z_{2} is non-empty. Let q ∈ Z_{1} ∩ Z_{2} be any point. The preimage π^{−1}(q) is a line in ℙ^{N} intersecting both Y_{1} and Y_{2} and passing through p.

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Recall, that the special case of join is when Y = Y_{1} = Y_{2} and σ_{2}(Y) := Y ∗ Y is the secant variety of Y.

Proposition 3.5. • Let Y ⊂ ℙ^{2n−1} be an irreducible linearly nondegenerate Legendrian variety. Then σ_{2}(Y) = ℙ^{2n−1}.

• Let Y_{1}, Y_{2} ⊂ ℙ^{2n−1} be two disjoint Legendrian subvarieties. Then Y_{1}∗Y_{2} = ℙ^{2n−1}.

Proof. If Y is irreducible, then this is proved in the course of proof of Prop. 17(2) in [LM07]. If Y_{1} and Y_{2} are disjoint, then the result follows from Lemma 3.4.

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3.3 Divisors swept by broken lines

Following the idea of Wiśniewski [Wiś00] we introduce the locus of broken lines (or reducible conics, or chains of 2 lines) through x:

D_{x} := ⋃_{y∈C_{x}} C_{y}.

Note that D_{x} is a closed subset of X as it can be interpreted as the image of projective variety C_{x}^{2} ⊂ X × X under a proper map, which is the projection onto the last coordinate. By analogy to the case of lines consider also:

D^{2} ⊂ X × X
D^{2} := {(x, z) | ∃_{y∈C_{x}} s.t. z ∈ C_{y}} ,

i.e. D^{2} is the projection of C^{3} onto first and third coordinates. Thus again D^{2} is a closed subset of the product. Set theoretically D_{x} is the fibre over x of (either of) the projection D^{2} → X and if we consider D^{2} as a reduced scheme, then we can assign to D_{x} the scheme structure of the fibre. It follows immediately from the above discussion and Proposition 3.3, that every component of D_{x} has dimension at most 2n and every component of D^{2} has dimension at most 4n + 1. In fact the equality holds.

Theorem 3.6. Let x ∈ X be any point. Then the locus D_{x} is of pure codimension 1.

Proof. Assume first that x ∈ X is a general point. Recall, that C_{x}^{2} ⊂ X × X has two projections:

\xymatrix{C_{x}^{2} \ar@{->>}[r]^{π_{2}} \ar@{->>}[d]^{π_{1}} & D_{x} \\ C_{x}}

Fix (D_{x})^{•} to be an irreducible component of D_{x}. Then (D_{x})^{•} is dominated by some component (C_{x}^{2})^{•} of C_{x}^{2}. Dimension of (C_{x}^{2})^{•} is equal to 2n by Proposition 3.3. For y ∈ C_{x} the fiber π_{1}^{−1}(y) ⊂ C_{x}^{2} is equal to {y} × C_{y}. In particular, by Theorem 3.1(ii) the fibers of π_{1} have constant dimension n. Thus (C_{x}^{2})^{•} is mapped onto an irreducible component (C_{x})^{•} of C_{x}. Finally, let C′ be an irreducible component of the preimage π_{1}^{−1}(x) which is contained in (C_{x}^{2})^{•}. Note that C′ can be identified with an irreducible component of C_{x}, because π_{1}^{−1}(x) = {x} × C_{x}. We claim that the projectivised tangent cone ℙτ_{x}(D_{x})^{•} contains the join of two tangent cones

(ℙτ_{x}C′) ∗ (ℙτ_{x}(C_{x})^{•}) ⊂ ℙF_{x} ⊂ ℙT_{x}X.

The proof of the claim is a baby version of [HK05, Thm 3.11]. There however Hwang and Kebekus assume C_{x} is irreducible and thus their results do not neccessarily apply directly here. Let l_{0} be a general line through x contained in C′ and let l be a general line through x contained in (C_{x})^{•}. To prove the claim it is enough to show there exists a surface S ⊂ D_{x} containing l_{0} and l which is smooth at x, since in such a case T_{x}S ⊂ τ_{x}D_{x} and ℙT_{x}S is the line between ℙT_{x}l and ℙT_{x}l_{0}.

We obtain S by varying l_{0}. Consider H_{l} ⊂ H the parameter space for lines on X, which intersect l. By Theorem 3.1(iii) the space H_{l} comes with a projection ξ: H_{l} −−→ l, which maps l′ ∈ H_{l} to the intersection point of l and l′, and which is well defined on an open subset containg all lines through x. By generality of our choices, l_{0} is a smooth point of H_{l} and ξ is submersive at l_{0}. In the neighbourhood of l_{0} choose a curve A ⊂ H_{l} smooth at l_{0} for which ξ|_{A} is submersive at l_{0}. Then the locus in X of lines which are in A sweeps a surface S ⊂ X, which is smooth at x, contains l_{0}, and contains an open subset of l around x. Thus the claim is proved and:

(ℙτ_{x}C′) ∗ (ℙτ_{x}(C_{x})^{•}) ⊂ ℙτ_{x}(D_{x})^{•}  (3.7)

Now we claim that F_{x} ⊂ τ_{x}D_{x}. For this purpose we separate two cases. In the first case C′ = (C_{x})^{•}. Then ℙτ_{x}C′ is non-degenerate by Theorem 3.1 and thus

(ℙτ_{x}C′) ∗ (ℙτ_{x}(C_{x})^{•}) = σ_{2}(ℙτ_{x}C′) = ℙ(F_{x})

by Proposition 3.5. Combining with (3.7) we obtain the claim. In the second case C′ and (C_{x})^{•} are different components of C_{x}. Then by generality of x and by Theorem 3.1, the two tangent cones (ℙτ_{x}C′) and (ℙτ_{x}(C_{x})^{•}) are disjoint. Thus again

(ℙτ_{x}C′) ∗ (ℙτ_{x}(C_{x})^{•}) = ℙ(F_{x})

by Proposition 3.5. Combining with (3.7) we obtain the claim. Thus in any case for a general x ∈ X, every component of D_{x} has dimension at least 2n. The dimension can only jump up at special points when one has a fibration, thus also at special points every component of D_{x} has dimension at least 2n. Earlier we observed that dim D_{x} ≤ 2n, thus the theorem is proved.

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Proposition 3.8. If X is the adjoint variety of G, and x ∈ X, then D_{x} is the hyperplane section of X ⊂ ℙ(𝔤) perpendicular to x via the Killing form.

Proof. Let X = G/P, where P is the parabolic subgroup preserving x. Notice, that D_{x} must be reduced (because D is reduced and D_{x} is a general fibre of D). Also D_{x} is P -invariant, because the set of lines is G invariant and D_{x} is determined by x and the geometry of lines on X. We claim, there is a unique P -invariant reduced divisor on X, and thus it must be the hyperplane section as in the statment of proposition. So let Δ be a P -invariant divisor linearly equivalent to L^{k} for some k ≥ 0. Also let ρ_{Δ} be a section of L^{k} which determines Δ. The module of sections H^{0}(L^{k}) is an irreducible G-module by Borel-Weil theorem (see [Ser95]), with some highest weight ω. Since the Lie algebra 𝔭 of P contains all positive root spaces, by [FH91, Prop. 14.13] there is a unique 1-dimensional 𝔭-invariant submodule of H^{0}(L^{k}), it is the highest weight space H^{0}(L^{k})_{ω}. So ρ_{Δ} ∈ H^{0}(L^{k})_{ω} and Δ is unique.

The hyperplane section of X ⊂ ℙ(𝔤) perpendicular to x via the Killing form is a divisor in |L|, and it is P -invariant, and so are its multiples in |L^{k}|. So by the uniqueness Δ must be equal to k times this hyperplane section. Thus Δ is reduced if and only k = 1 and so D_{x} is the hyperplane section.

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3.4 Tangent bundles restricted to lines

Let l be a line through a general point y ∈ X. Recall from [Keb05, Fact 2.3] that:

T X|_{l} ≃ O_{l}(2) ⊕ O_{l}(1)^{n−1} ⊕ O_{l}^{n−1} ⊕ O_{l}^{2}
F |_{l} ≃ O_{l}(2) ⊕ O_{l}(1)^{n−1} ⊕ O_{l}^{n−1} ⊕ O_{l}(−1)
T l ≃ O_{l}(2)

and for general z ∈ l:

T C_{z}|_{l\{z}} ≃ O_{l}(2) ⊕ O_{l}(1)^{n−1}.

If x ∈ X is a general point and y ∈ C_{x} is a general point of any of the irreducible components of C_{x} and l is a line through y, then we want to express T D_{x}|_{l} in terms of those splittings. In a neighbourhood of l the divisor D_{x} is swept by deformations l_{t} of l = l_{0} such that l_{t} intersects C_{x}. By the standard deformation theory argument taking derivative of l_{t} by t at a point z ∈ l, we obtain that:

T_{z}D_{x} ⊃ {s(z) ∈ T_{z}X | ∃s ∈ H^{0}(T X|_{l}) s.t. s(y) ∈ T_{y}C_{x}}  (3.9)

Moreover, at a general point z we have equality in (3.9). If we mod out T X|_{l} by the rank n positive bundle (T X|_{l})^{>0} := O_{l}(2) ⊕ O_{l}(1)^{n−1}, then we are left with a trivial bundle O_{l}^{n+1}. Thus, since by Theorem 3.6 the dimension of T_{z}D_{x} = 2n for general z ∈ l, the vector space T_{y}C_{x} must be transversal to (T X|_{l})^{>0} at y. In particular, if z ≠ y, then dimension of the right hand side in (3.9) is 2n and thus (3.9) is an equality for each point z ∈ l, such that z is a smooth point of D_{x}. We conclude:

Proposition 3.10. Let x ∈ X be a general point and y ∈ C_{x} be a general point of any of the irreducible components of C_{x} and l be any line through y. Then there exists a subbundle Γ ⊂ T X|_{l} such that:

Γ = O_{l}(2) ⊕ O_{l}(1)^{n−1} ⊕ O_{l}^{n},
Γ ∩ F |_{l} = O_{l}(2) ⊕ O_{l}(1)^{n−1} ⊕ O_{l}^{n−1} = (F |_{l})^{≥0}

and if z ∈ l is a smooth point of D_{x}, then T_{z}D_{x} = Γ_{z}.

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4 Duality

An effective divisor Δ on X is an element of divisor group (and thus a positive integral combination of codimension 1 subvarieties of X) and also a point in the projective space ℙ(H^{0}O_{X}(Δ)) or a hyperplane in ℙ(H^{0}O_{X}(Δ)^{∗}). In this section we will constantly interchange these three interpretations of Δ. In order to avoid confusion we will write:

• Δ^{div} to mean the divisor on X;

• Δ^{ℙ} to mean the point in ℙ(H^{0}O_{X}(Δ)) or in a fixed linear subsystem.

• Δ^{ℙ⊥} to mean the hyperplane in ℙ(H^{0}O_{X}(Δ)^{∗}) or in dual of the fixed subsystem.

In §3.3 we have defined D ⊂ X × X, which we now view as a family of divisors on X parametrised by X. Since the Picard group of X is discrete and X is smooth and connected, it follows that all the divisors D_{x} are linearly equivalent. Thus let E ≃ L^{⊗k} be the line bundle O_{X}(D_{x}). Consider the following vector space ⟨D⟩ ⊂ H^{0}(E):

⟨D⟩ := span {s_{x} : x ∈ X} where s_{x} is a section of E vanishing on D_{x}.

Hence ℙ⟨D⟩ is the linear system spanned by all the D_{x}. Further, consider the map

ϕ : X → ℙ⟨D⟩^{∗}

determined by the linear system ⟨D⟩, i.e. mapping point x ∈ X to the hyperplane in ℙ⟨D⟩ consisting of all divisors containing x.

Remark 4.1. Note that ϕ is regular, since for every x ∈ X there exists w ∈ X, such that x ∉ D_{w} (or equivalently, w ∉ D_{x}).

Since E is ample, it must intersect every curve in X and hence ϕ does not contract any curve. Therefore ϕ is finite to one.

Proposition 4.2. If X is an adjoint variety, then k = 1, i.e. E ≃ L. If k = 1 and the automorphism group of X is reductive, then X is isomorphic to an adjoint variety.

Proof. If X is the adjoint variety of G, and x ∈ X, then D_{x} is the hyperplane section of X ⊂ ℙ(𝔤) by Proposition 3.8. If k = 1 and the automorphism group of X is reductive, since ϕ is finite to one, we can apply Beauville Theorem [Bea98]. Thus X is isomorphic to an adjoint variety.

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4.1 Dual map

In algebraic geometry it is standard to consider maps determined by linear systems (such as ϕ defined above). However in our situation, we also have another map determined by the family of divisors D. Namely:

ψ: X → ℙ⟨D⟩
x ↦ D_{x}^{ℙ}.

So let S ⊂ O_{X} ⊗ ⟨D⟩^{∗} ≃ X × ⟨D⟩^{∗} be the pullback under ϕ of the universal hyperplane bundle, i.e. the corank 1 subbundle such that the fibre of S over x is D_{x}^{ℙ⊥} ⊂ ⟨D⟩^{∗}. We note that ℙ(S) is both a projective space bundle on X and also it is a divisor on X × ℙ⟨D⟩^{∗}. Also D = (id_{X} ×ϕ)^{∗}ℙ(S) as divisors. We can also consider the line bundle dual to the cokernel of S → O_{X} ⊗ ⟨D⟩^{∗}, i.e. the subbundle S^{⊥} ⊂ O_{X} ⊗ ⟨D⟩. This line subbundle determines section X → X × ℙ⟨D⟩, where x ↦ (x, D_{x}^{ℙ}). So ψ is the composition of the section and the projection:

X → X × ℙ⟨D⟩ → ℙ⟨D⟩.

Every map to a projective space is determined by some linear system. We claim the ψ is determined by ⟨D⟩, precisely the system that defines ϕ and thus that there is a natural linear isomorphism between ℙ⟨D⟩ and ℙ⟨D⟩^{∗}.

Proposition 4.3. We have ψ^{∗}O_{ℙ⟨D⟩}(1) ≃ E and the linear system cut out by hyperplanes

ψ^{∗}H^{0} (O_{ℙ⟨D⟩}(1)) := {ψ^{∗}s : s ∈ ⟨D⟩^{∗}} ⊂ H^{0}(E)

is equal to ⟨D⟩.

Proof. For fixed x ∈ X let ϕ(x)^{⊥} ⊂ ℙ⟨D⟩ be the hyperplane dual to ϕ(x) ∈ ℙ⟨D⟩^{∗}. To prove the proposition it is enough to prove

ψ^{∗}(ϕ(x)^{⊥}) = D_{x}^{div}.  (4.4)

Since we have the following symmetry property of D:

x ∈ D_{y} ⟺ y ∈ D_{x},

the set theoretic version of (4.4) follows easily:

y ∈ ψ^{∗}(ϕ(x)^{⊥}) ⟺ ψ(y) ∈ ϕ(x)^{⊥} ⟺ D_{y}^{ℙ⊥} ∋ ϕ(x) ⟺ D_{y} ∋ x.

However, in order to prove the equality of divisors in (4.4) we must do a bit more of gymnastics, which translates the equivalences above into local equations. The details are below. The pull back of ϕ(x)^{⊥} by the projection X × ℙ⟨D⟩ → ℙ⟨D⟩ is just X × ϕ(x)^{⊥}. Then the pull-back of the product by the section X → X × ℙ⟨D⟩ associated to S^{⊥} is just the subscheme of X defined by {y ∈ X | (S^{⊥})_{y} ⊂ ϕ(x)^{⊥}} (locally, this is just a single equation: the spanning section of S^{⊥} satisfies the defining equation of ϕ(x)^{⊥}). But this is clearly equal to the dual equation {y | ℙ(S_{y}) ∋ ϕ(x)}. If we let ρ_{x} be the section

ρ_{x}: X → X × X
ρ_{x}(y) := (y, x)

then we have:

ψ^{∗}(ϕ(x)^{⊥}) = ρ_{x}^{∗} ∘ (id_{X} ×ϕ)^{∗}(ℙ(S)) = ρ_{x}^{∗}(D) = D_{x}^{div}

as claimed.

□

Thus we have a canonical linear isomorphism f: ℙ⟨D⟩^{∗} → ℙ⟨D⟩ giving rise to the following commutative diagram:

\xymatrix{ & ℙ⟨D⟩^{∗} \ar[dd]^{≃} \\ X \ar[ur]^{ϕ} \ar[dr]^{ψ} \\  & ℙ⟨D⟩.}  (4.5)

We will denote the underlying vector space isomorphism ⟨D⟩^{∗} → ⟨D⟩ (which is unique up to scalar) with the same letter f. The choice of f combined with the canonical pairing ⟨D⟩ × ⟨D⟩^{∗} → ℂ, determines a non-degenerate bilinear form B : ⟨D⟩ × ⟨D⟩ → ℂ, with the following property:

B(ϕ(x), ϕ(y)) = 0 ⟺ (x, y) ∈ D ⟺ x ∈ D_{y} ⟺ y ∈ D_{x}.  (4.6)

Proposition 4.7. If X is the adjoint variety of G, then ⟨D⟩ = H^{0}(L) ≃ 𝔤 and B is (up to scalar) the Killing form on 𝔤.

Proof. Follows immediately from Proposition 3.8 and Equation 4.6.

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Corollary 4.8. ϕ(x) = ϕ(y) if and only if D_{x} = D_{y}.

Proof. It is immediate from the definition of ψ and from Diagram (4.5).

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4.2 Symmetry

Note that B has the property that for x ∈ X,

B(ϕ(x), ϕ(x)) = 0

(because x ∈ D_{x}).

Proposition 4.9. The bilinear form B is either symmetric or skew-symmetric.

Proof. Consider two linear maps ⟨D⟩ → ⟨D⟩^{∗}:

α(v) := B(v, ·) and β(v) := B(·, v).

If v = ϕ(x) for some x ∈ X, then

ker(α(v)) = span(ker(α(v)) ∩ ϕ(X)) = span(ϕ(D_{x}))

and analogously ker(β(v)) = span(ϕ(D_{x})). So ker(α(v)) = ker(β(v)) and hence α(v) and β(v) are proportional. Therefore there exists a function λ : X → ℂ such that:

λ(x)α(ϕ(x)) = β(ϕ(x)).

So for every x, y ∈ X we have:

B(ϕ(x), ϕ(y)) = λ(x)B(ϕ(y), ϕ(x)) = λ(x)λ(y)B(ϕ(x), ϕ(y))

and hence:

∀(x, y) ∈ X × X \ D λ(x)λ(y) = 1.

Taking three different points we see that λ is constant and λ ≡ ±1. Therefore ±α(ϕ(x)) = β(ϕ(x)) and by linearity this extends to ±α = β so B is either symmetric or skew-symmetric as stated in the proposition.

□

Example 4.10. If X is one of the adjoint varieties, then B is symmetric (because the Killing form is symmetric).

Remark 4.11. Consider ℙ^{2n+1} with a contact structure arising from a symplectic form ω on ℂ^{2n+2}. Recall, that this homogeneous contact Fano manifold does not satisfy our assumptions, namely, its Picard group is not generated by the equivalence class of L — in this case L ≃ O_{ℙ^{2n+1}}(2). However, Wiśniewski in [Wiś00] considers also this generalised situation and defines D_{x} to be the divisor swept by contact conics (i.e. curves C with degree of L|_{C} = 2) tangent to the contact distribution F. Then for the projective space D_{x} is just the hyperplane perpendicular to x with respect to ω. And thus in this case ⟨D⟩ = H^{0} (O_{ℙ^{2n+1}}(1)) and the bilinear form B defined from such family of divisors would be proportional to ω, hence skew-symmetric.

Proof of Theorem 1.3. D_{x} is a divisor by Theorem 3.6. ϕ is regular by Remark 4.1. ψ is regular by (4.5). The non-degenerate bilinear form B is constructed in §4.1. It is either symmetric or skew-symmetric by Proposition 4.9. In the adjoint case B is the Killing form by Proposition 4.7.

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836 Jarosław Buczyński Corollary 4.12. If B is symmetric, then ψ(X) ⊂ ℙ⟨D⟩ is contained in the quadric B(v, v) = 0. Corollary 4.13. If x ∈ X, then ψ(C_{x}) is contained in a linear subspace of dimension at most ⌊\frac{dim ⟨D⟩}{2}⌋.

Proof. If y, z ∈ C_{x}, then z ∈ D_{y}, so B(ψ(y), ψ(z)) = 0. Therefore span(ψ(C_{x})) is an isotropic linear subspace, which cannot have dimension bigger than ⌊\frac{dim ⟨D⟩}{2}⌋.

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5 Grading Suppose X ⊂ ℙ𝔤 is the adjoint variety of G. Assume further that a maximal torus and an order of roots in 𝔤 has been chosen, then 𝔤 has a natural grading (see [LM02, §6.1]):

𝔤 = 𝔤_{−2} ⊕ 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1} ⊕ 𝔤_{2}

where:

(i) 𝔤_{0} ⊕ 𝔤_{1} ⊕ 𝔤_{2} is the parabolic subalgebra 𝔭 of X. (ii) 𝔤_{0} is the maximal reductive subalgebra of 𝔭. (iii) for all i ∈ {−2, −1, 0, 1, 2} the vector space 𝔤_{i} is a 𝔤_{0}-module. (iv) 𝔤_{2} is the 1-dimensional highest root space,

(v) 𝔤_{−2} is the 1-dimensional lowest root space. (vi) The restriction of the Killing form to each 𝔤_{2} ⊕ 𝔤_{−2}, 𝔤_{1} ⊕ 𝔤_{−1} and 𝔤_{0}

is non-degenerate, and the Killing form B(𝔤_{i}, 𝔤_{j}) is identically zero for i ≠ −j.

(vii) The Lie bracket on 𝔤 respects the grading, [𝔤_{i}, 𝔤_{j}] ⊂ 𝔤_{i+j} (where 𝔤_{k} = 0

for k ∉ {−2, −1, 0, 1, 2}). In fact the grading is determined by 𝔤_{−2} and 𝔤_{2} together with the geometry of X only. So let X be as in Notation 2.1 and let x and w be two general points of X. Define the following subspaces of ⟨D⟩:

• ⟨D⟩_{2} to be the 1-dimensional subspace ψ(x);

• ⟨D⟩_{−2} to be the 1-dimensional subspace ψ(w);

• ⟨D⟩_{1} to be the linear span of affine cone of ψ(C_{x} ∩ D_{w});

• ⟨D⟩_{−1} to be the linear span of affine cone of ψ(C_{w} ∩ D_{x});

Documenta Mathematica 15 (2010) 821–841

• ⟨D⟩_{0} to be the vector subspace of ⟨D⟩, whose projectivisation is:

⋂_{y∈C_{x}∪C_{w}} f(D_{y}^{ℙ⊥})

In the homogeneous case this is precisely the grading of 𝔤.

Proof of Theorem 1.4. First note that the classes of the 1-dimensional linear subspaces 𝔤_{2} and 𝔤_{−2} are both in X (as points in ℙ𝔤). Moreover, they are a pair of general points, because the action of the parabolic subgroup P < G preserves 𝔤_{2} and moves freely 𝔤_{−2}. This is because T̂_{[𝔤_{−2}]}X = [𝔤_{−2}, 𝔤] = [𝔤_{−2}, 𝔭]. So fix x = [𝔤_{2}] and w = [𝔤_{−2}]. We claim the linear span of C_{x} (respectively C_{w}) is just 𝔤_{2} ⊕ 𝔤_{1} (respectively 𝔤_{−2} ⊕ 𝔤_{−1}). To see that, we observe the lines on X through x are contained in the intersection of X and the projective tangent space ℙ(T̂_{x}X) ⊂ ℙ(𝔤). In fact this intersection is equal to C_{x}: if y ≠ x is a point of the intersection, then the line in ℙ𝔤 through x and y intersects X with multiplicity at least 3, but X is cut out by quadrics (see for instance [Pro07, §10.6.6]), so this line must be contained in X. Also C_{x} is non-degenerate in ℙ(F̂_{x}) ⊂ ℙ(T̂_{x}X). However F̂_{x} is a 𝔭-invariant hyperplane in ℙ(T̂_{x}X) and the unique 𝔭-invariant hyperplane in

T̂_{x}X = [𝔤, 𝔤_{2}] = [𝔤_{−2}, 𝔤_{2}] ⊕ 𝔤_{1} ⊕ 𝔤_{2}

is

F̂_{x} = [𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1} ⊕ 𝔤_{2}, 𝔤_{2}] = 𝔤_{1} ⊕ 𝔤_{2}.

Further we have seen in Proposition 3.8 that D_{x} (respectively D_{w}) is the intersection of ℙ(𝔤_{2}^{⊥_{B}}) = ℙ(𝔤_{2} ⊕ 𝔤_{1} ⊕ 𝔤_{0} ⊕ 𝔤_{−1}) and X (respectively ℙ(𝔤_{−2} ⊕ 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1}) and X). Equivalently, f(D_{x}^{ℙ⊥}) = ℙ(𝔤_{2} ⊕ 𝔤_{1} ⊕ 𝔤_{0} ⊕ 𝔤_{−1}). Thus:

C_{x} ∩ D_{w} = C_{x} ∩ f(D_{w}^{ℙ⊥}) = C_{x} ∩ ℙ(𝔤_{−2} ⊕ 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1}) = C_{x} ∩ ℙ(𝔤_{1}).

C_{x}∩ℙ(𝔤_{1}) is non-degenerate in ℙ(𝔤_{1}), thus ⟨D⟩_{1} = 𝔤_{1} and analogously ⟨D⟩_{−1} = 𝔤_{−1}. It remains to prove ⟨D⟩_{0} = 𝔤_{0}.

ℙ⟨D⟩_{0} = ⋂_{y∈C_{x}∪C_{w}} f(D_{z}^{ℙ⊥})
= (C_{x} ∪ C_{w})^{⊥_{B}}
= ℙ(𝔤_{2} ⊕ 𝔤_{1} ⊕ 𝔤_{−1} ⊕ 𝔤_{−2})^{⊥_{B}}
= ℙ(𝔤_{0}).

□

We also note the following lemma in the homogeneous case:

Lemma 5.1. If X is the adjoint variety of G, then

X ∩ ℙ(𝔤_{1}) ⊂ C_{x}

where x is the point of projective space corresponding to 𝔤_{2}.

Proof. Suppose y ∈ X ∩ℙ𝔤_{1} and let l ⊂ ℙ𝔤 be the line through x and y. Note that l ⊂ ℙ(𝔤_{1} ⊕ 𝔤_{2}). Since 𝔤_{1} ⊕ 𝔤_{2} ⊂ [𝔤, 𝔤_{2}] = T̂_{x}X, hence l ∩ X has multiplicity at least 2 at x. Thus l ∩ X has degree at least 3 and since X is cut out by quadrics, l is contained in X.

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6 Cointegrable subvarieties

Definition 6.1. A subvariety Δ ⊂ X is F -cointegrable if T_{x}Δ ∩ F_{x} ⊂ F_{x} is a coisotropic subspace for general point x of each irreducible component of Δ.

Note that this is equivalent to the definition given in [Bucz09, §E.4] — this follows from the local description of the symplectic form on the symplectisation of the contact manifold (see [Bucz09, (C.15)]). Clearly, every codimension 1 subvariety of X is F -cointegrable. Assume Δ ⊂ X is a subvariety of pure dimension, which is F -cointegrable and let Δ_{0} be the locus where T_{x}Δ∩F_{x} ⊂ F_{x} is a coisotropic subspace of dimension dim Δ − 1. We define the Δ-integrable distribution Δ^{⊥} to be the distribution defined over Δ_{0} by:

Δ^{⊥}_{x} := (T_{x}Δ ∩ F_{x})^{⊥_{dθ}} ⊂ F_{x}

We say an irreducible subvariety A ⊂ X is Δ-integral if A ⊂ Δ, A ∩ Δ_{0} ≠ ∅, and T A ⊂ Δ^{⊥} over the smooth points of A ∩ Δ_{0}.

Lemma 6.2. Let A_{1} and A_{2} be two irreducible Δ-integral subvarieties. Assume dim A_{1} = dim A_{2} = codim_{X} Δ. Then either A_{1} = A_{2} or A_{1} ∩ A_{2} ⊂ Δ \ Δ_{0}.

□

Theorem 6.3. Consider a general point x ∈ X. Then:

(i) D_{x} (as reduced, but possibly not irreducible subvariety of X) is F -cointe-grable.

(ii) For general y in any of the irreducible components of C_{x} all lines through y are D_{x}-integral.

(iii) For general z in any of the irreducible components of D_{x} the intersection C_{x} ∩ C_{z} is a unique point and the chain of two lines connecting x to z is unique.

Proof. Part (i) is immediate, since D_{x} is a divisor, by Theorem 3.6. To prove part (ii) let l be a line through y. Then by Proposition 3.10:

T_{z}D_{x} ∩ F_{z} = (F |_{l})^{≥0}

and for general z ∈ l we have (T_{z}D_{x} ∩ F_{z})^{⊥_{dθz}} ⊂ F_{z} is the O(2) part, i.e. the part tangent to l. So l is D_{x}-integral as claimed. To prove (iii), let U ⊂ X be an open dense subset of points u ∈ X where two different lines through u do not share the tangent direction and do not meet in any other point. Note that since x is a general point, x ∈ U and thus each irreducible component of C_{x} and D_{x} intersects U. Thus generality of z implies that z ∈ U and thus each irreducible component of C_{z} and D_{z} intersects U. Also C_{x} ∩ C_{z} intersects U. So fix y ∈ C_{x} ∩ C_{z} ∩ U. By (ii) and Lemma 6.2 the line l_{z} through z which intersects C_{x} is unique. In the same way let l_{x} be the unique line through x intersecting C_{z}. Thus

C_{x} ∩ C_{z} = l_{x} ∩ l_{z}.

In particular, y ∈ l_{x} ∩ l_{z}. But since y ∈ U the intersection l_{x} ∩ l_{z} is just one point and therefore:

C_{x} ∩ C_{z} = {y} .
□

As a consequence of part (iii) of the theorem the surjective map π_{13} : C^{3} → D is birational. Thus consider the inverse rational map D −−→ C^{3} and compose it with the projection on the middle coordinate π_{2} : C^{3} → X. We define the composition to be the bracket map:

[·, ·]^{D}: D −−→ C^{3} →^{π_{2}} X.

In this setting, for (x, z) ∈ D, one has [x, z]^{D} = y = C_{x} ∩ C_{z}, whenever the intersection is just one point.

Theorem 6.4. If X is the adjoint variety of G, then the bracket map defined above agrees with the Lie bracket on 𝔤, in the following sense: Let ξ, ζ ∈ 𝔤 and set η := [ξ, ζ] (the Lie bracket on 𝔤). Denote by x, y and z the projective classes in ℙ𝔤 of ξ, η and ζ respectively. If x ∈ D_{z} and η ≠ 0, then the bracket map satisfies [x, z]^{D} = y.

Proof. It is enough to prove the statement for a general pair (x, z) ∈ D. Suppose further w ∈ C_{z} is a general point. Then the pair (x, w) ∈ X × X is a general pair. Thus by Theorem 1.4, we may assume ξ ∈ 𝔤_{2} and ζ ∈ 𝔤_{−1}. The restriction of the Lie bracket to [ξ, 𝔤_{−1}] determines an isomorphism 𝔤_{−1} → 𝔤_{1} of 𝔤_{0}-modules. In particular the minimal orbit X ∩ ℙ𝔤_{−1} is mapped onto X ∩ ℙ𝔤_{1} under this isomorphism. In particular y ∈ X ∩ ℙ𝔤_{1} ⊂ C_{x} (see Lemma 5.1). Analogously y ∈ C_{z}, so y ∈ C_{x} ∩ C_{z}.

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Documenta Mathematica 15 (2010)