Duality and integrability on contact Fano manifolds
aa r X i v : . [ m a t h . AG ] F e b Duality and integrability on contact Fanomanifolds ∗ Jarosław Buczyński6th December 2018
Abstract
We address the problem of classification of contact Fano manifolds. It isconjectured that every such manifold is necessarily homogeneous. We provethat the Killing form, the Lie algebra grading and parts of the Lie bracketcan be read from geometry of an arbitrary contact manifold. Minimalrational curves on contact manifolds (or contact lines) and their chains arethe essential ingredients for our constructions. author’s e-mail: [email protected] keywords: complex contact manifold, Fano variety, minimal rational curves, adjoint variety,Killing form, Lie bracket, Lie algebra grading, complex contact manifold;
AMS Mathematical Subject Classification 2010:
Primary: 14M17; Secondary: 53C26,14M20, 14J45;
In this article we are interested in the classification of contact Fano manifolds. Wereview the relevant definitions in §2. So far the only known examples of contactFano manifolds are obtained as follows. For a simple Lie group G consider itsadjoint action on P ( g ) , where g is the Lie algebra of G . This action has a uniqueclosed 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 thecoadjoint action of G on P ( g ∗ ) and X is isomorphic to the unique closed orbit in P ( g ∗ ) .In order to study the non-homogeneous contact manifolds (potentially non-existent) it is natural to assume Pic X ≃ Z and further that X is not isomorphic ∗ Dedicated in memory of Marcin Hauzer.
1o 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 homogeneousstructure on adjoint varieties: the Killing form B on g , the Lie algebra grading g = g − ⊕ g − ⊕ g ⊕ g ⊕ g (see [LM02, §6.1] and references therein) and a partof the Lie bracket on g . Understanding the underlying geometry allows us todefine the appropriate generalisations of these notions on arbitrary contact Fanomanifolds.An essential building block for our constructions is the notion of a contactline (or simply line ) on X . These contact lines were studied by Kebekus [Keb01],[Keb05] and Wiśniewski [Wiś00]. Also they are an instance of minimal rationalcurves, which are studied extensively. The geometry of contact lines was the ori-ginal motivation to study Legendrian subvarieties in projective space (see [Buc08]for an overview and many details). We briefly review the subject of lines on con-tact Fano manifolds in §3.1.The key ingedient is the construction of a family of divisors D x parametrisedby points x ∈ X (see §3.3). These divisors are swept by pairs of intersectingcontact 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 intersectingcontact lines. The idea to study these loci comes from Wiśniewski [Wiś00] wherehe observed, that (under an additional minor assumption) these loci containsome non-trivial divisorial components and he studied the intersection numbersof certain curves on X with the divisorial components. Here we prove all thecomponents of D x are divisorial and draw conclusions from that observation goinginto a different direction than those of [Wiś00]. Theorem 1.1.
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 thepairs of intersecting contact lines, one of which passes through x ∈ X is a of purecodimension and thus D x determines a divisor on X . Let h D i ⊂ H ( O ( D x )) be the linear system spanned by these divisors. Let φ : X → P h D i ∗ be the mapdetermined by the linear system h D i and let ψ : X → P h D i 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 h D i ,which determines an isomorphism P h D i ∗ ≃ P h D i making the following dia-gram commutative: P h D i ∗≃ (cid:15) (cid:15) X φ gggggggggggggg ψ + + WWWWWWWWWWWWWW P h D i . 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 h D i = 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 h D i , which we denote h D i − , h D i − , h D i , h D i and h D i (see §5 for details). Theorem 1.2. 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 choiceof a maximal torus of G and a choice of order of roots of g , such that h D i i = g i for every i ∈ {− , − , , , } , where g = g − ⊕ g − ⊕ g ⊕ g ⊕ g is the Liealgebra 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 . Werecover this bracket restricted to D for general contact manifolds: Theorem 1.3.
For X and D x and in Theorem 1.1, let D ⊂ X × X be the divisorconsisting of pairs ( x, z ) ∈ X × X , such that z ∈ D x . There exists a rational map [ · , · ] : D X , such that [ x, z ] = y , where y is an intersection point of a pair ofcontact lines that join x and z . In particular, this intersection point y and thepair of lines are unique for general pair ( x, z ) ∈ D . Moreover, if X is the adjointvariety of a simple Lie group G , then [ · , · ] 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 con-tinue by studying certain types of loci swept by those lines and calculate theirdimensions. In particular we prove there Theorem 3.6, which is a part of res-ults summarised in Theorem 1.1. We also study the tangent bundle to D x as asubspace of T X .In §4 we study the duality of maps φ and ψ introduced in Theorem 1.1 togetherwith the consequences of this duality. This section is culminated with the proofof Theorem 1.1.In §5 we generalise the Lie algebra grading to arbitrary contact manifolds andprove Theorem 1.2.In §6 we prove that certain lines are integrable with respect to a specialdistribution on D x and we apply this to prove Theorem 1.3.3 cknowledgements The author is supported by Marie Curie International Outgoing Fellowship.The author would like to thank the following people for many enlighteningdiscussions: Jun-Muk Hwang, Stefan Kebekus, Joseph M. Landsberg, JarosławWiśniewski and Fyodor L. Zak. Comments of Laurent Manivel helped to improvethe presentation in the paper.
Throughout the paper all our projectivisations P are naive. This means, if V isa vector space, then P V = ( V \ / C ∗ , and similarly for vector bundles.A complex manifold X of dimension n + 1 is contact if there exists a vectorsubbundle F ⊂ T X of rank n fitting into an exact sequence: → F → T X θ → L → such that the derivative d θ ∈ H ( V F ∗ ⊗ L ) of the twisted form θ ∈ H ( T ∗ X ⊗ L ) is nowhere degenerate. In particular, d θ x is a symplectic form on the fibre ofcontact distribution F x . See [Buc08, §E.3 and Chapter C] and references thereinfor an overview of the subject.A projective manifold X is Fano, if its anticanonical divisor K X ∗ = V dim X T X is ample.If X is a projective contact manifold, then by Theorem of Kebekus, Peter-nell, Sommese and Wiśniewski [KPSW00] combined with a result by Demailly[Dem02], X is either a projectivisation of a cotangent bundle to a smooth pro-jective manifold or X is a contact Fano manifold, with Pic X ≃ Z . In the secondcase, since K X ≃ ( L ∗ ) ⊗ ( n +1) , by [KO73], either X ≃ P n +1 or Pic X = Z · [ L ] . Herewe are interested in the case X P n +1 . Thus our assumption spelled out belowonly exclude some well understood cases (the projectivised cotangent bundlesand the projective space) and they agree with the assumptions of Theorems 1.1,1.2 and 1.3. 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 contactdistribution on X . We also assume dim X = 2 n + 1 .From Theorem of Ye [Ye94] it follows that n ≥ .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, notof types A or C (i.e. not isomorphic to SL n nor Sp n nor their discrete quotients).Further g is the Lie algebra of G . Thus g is one of so n (types B and D ), or oneof the exceptional Lie algebras g , f , e , e or e .4he contact structure on P n − = P ( C n ) is determined by a symplectic form ω on C n . The precise relation between the contact and symplectic structures isdecribed for instance in [Buc08, §E.1] (see also [LeB95, Ex. 2.1]). In particular,for all x ∈ X , the projectivisation of a fibre of the contact distribution P F x comeswith a natural contact structure.Let M be a projective contact manifold (in our case M = X with X as inNotation 2.1 or M = P n − ). A subvariety Z ⊂ M is Legendrian , if for all smoothpoints z ∈ Z the tangent space T z Z is contained in the contact distribution of M and Z is of pure dimension (dim M − .Recall from [Har95, Lecture 20] or [Mum99, III.§3,§4] the notion of tangentcone . For a subvariety Z ⊂ X , and a point x ∈ Z let τ x Z ⊂ T x X be thetangent cone of Z at x . In this article we will only need the following elementaryproperties 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 ⊂ Z ⊂ X , then τ x Z ⊂ τ x Z . • If Z is smooth at x , then τ x Z = T x Z .Since τ x Z is a cone, let P τ x Z ⊂ P T x be the corresponding projective variety. 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 curveson X , as in [Kol96, II.2.11]. Let Lines ( X ) ⊂ RatCurves n ( X ) be the subschemeparametrising lines. Then every component of Lines ( X ) is a minimal componentof X in the sense of [HM04]. We fix H 6 = ∅ a union of some irreducible componentsof Lines ( X ) .By a slight abuse of notation, from now on we say l is a (contact) line if andonly if l ∈ H . For simplicity, the reader may choose to restrict his attentionto one of the extreme cases: either to the case H = Lines ( X ) (and thus beconsistent 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 consistentwith [Keb01, Keb05]). In general it is expected that Lines ( X ) (with X as inNotation 2.1) is irreducible and all the cases are the same.5 .1 Legendrian varieties swept by lines We denote by C x ⊂ X the locus of contact lines through x ∈ X . Let C x := P τ x C x ⊂ P ( T X ) . Note that with our assumptions both C x and C x are closedsubsets 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 irre-ducible 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 ≃ ˜ C x ⊂ P ( F x ⊕ C ) , in such a way that lines through x are mapped bijectively ontothe generators of the cone and restriction of L to C x via this isomorph-ism is identified with the restriction of O P ( F x ⊕ C ) (1) to ˜ C x . In particular alllines through x are smooth and two different lines intersecting at x will notintersect 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]. Explicitstatements 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 thehomogeneous 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 isnever used in the proof, with the exception of the argument for the irreducibilityof C x — see however Remark 3.2. Thus C x is smooth and C x is isomorphic to thecone over C x as claimed. Each irreducible component C x is non-degenerate on P F x by [Keb01, Thm 4.4] — again the statement is only for C x , not for its com-ponents, 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 alinear subspace with polarised by O (1) . Thus the other results of this theoremgive alternate (but more complicated) proof of that generalised non-degeneracy. (cid:3) Remark . Note that (assuming H is irreducible) Kebekus [Keb05] also statedthat C x and C x are irreducible for general x . However it was observed by Kebekushimself together with the author that there is a gap in the proof. This gapis on page 234 in Step 2 of proof of Proposition 3.2 where Kebekus claims toconstruct “a well defined family of cycles” parametrised by a divisor D . Thisis not necessarily a well defined family of cycles: Condition (3.10.4) in [Kol96,6I.3.10] is not necessarily satisfied if D is not normal and there seem to beno reason to expect that D is normal. As a consequence the map Φ : D → Chow( X ) is not necessarily regular at non-normal points of D and it mightcontract some curves.Let us define: C ⊂ X × XC := { ( 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 := C × X C so that: C ⊂ X × X × XC := { ( x, y, z ) | y ∈ C x , z ∈ C y } . Finally, for x ∈ X we also define C x : C x ⊂ X × X ≃ { x } × X × XC x := { ( y, z ) | y ∈ C x , z ∈ C y } , with the scheme structure of the fibre of C under the projection on the first co-ordinate. 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 , C x , C are projective subschemes, they are all of puredimension, and their dimensions are: • dim C = 3 n + 1 . • dim C x = 2 n . • dim C = 4 n + 1 . (cid:3) For subvarieties Y , Y ⊂ P N recall that their join Y ∗ Y is the closure of the locusof lines between points y ∈ Y and y ∈ Y . Note that the expected dimensionof Y ∗ Y is dim Y + dim Y + 1 . We are only concerned with two special cases:either Y and Y are disjoint or Y = Y .7 emma 3.4. If Y , Y ⊂ P N are two disjoint subvarieties of dimensions k − and N − k respectively, then their join Y ∗ Y fills out the ambient space, i.e. thisjoin is of expected dimension. Proof.
Let p ∈ P N be a general point and consider the projection π : P N P N − away from p . Let Z i = π ( Y i ) for i = 1 , . Since p is general, dim Z i = dim Y i and thus Z ∩ Z is non-empty. Let q ∈ Z ∩ Z be any point. The preimage π − ( q ) is a line in P N intersecting both Y and Y and passing through p . (cid:3) Recall, that the special case of join is when Y = Y = Y and σ ( Y ) := Y ∗ Y is the secant variety of Y . Proposition 3.5. • Let Y ⊂ P n − be an irreducible linearly non-degenerateLegendrian variety. Then σ ( Y ) = P n − . • Let Y , Y ⊂ P n − be two disjoint Legendrian subvarieties. Then Y ∗ Y = P n − . Proof. If Y is irreducible, then this is proved in the course of proof of Prop. 17(2)in [LM07].If Y and Y are disjoint, then the result follows from Lemma 3.4. (cid:3) Following the idea of Wiśniewski [Wiś00] we introduce the locus of broken lines(or reducible conics, or chains of 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 ofprojective variety C x ⊂ X × X under a proper map, which is the projection ontothe last coordinate. By analogy to the case of lines consider also: D ⊂ X × XD := { ( x, z ) | ∃ y ∈ C x s.t. z ∈ C y } , i.e. D is the projection of C onto first and third coordinates. Thus again D isa closed subset of the product. Set theoretically D x is the fibre over x of (eitherof) the projection D → X and if we consider D as a reduced scheme, then wecan assign to D x the scheme structure of the fibre.It follows immediately from the above discussion and Proposition 3.3, thatevery component of D x has dimension at most n and every component of D has dimension at most n + 1 . In fact the equality holds.8 heorem 3.6. Let x ∈ X be any point. Then the locus D x is of pure codimen-sion . Proof.
Assume first that x ∈ X is a general point. Recall, that C x ⊂ X × X has two projections: C x π / / / / π (cid:15) (cid:15) (cid:15) (cid:15) D x C x Fix ( D x ) • to be an irreducible component of D x . Then ( D x ) • is dominated bysome component ( C x ) • of C x . Dimension of ( C x ) • is equal to n by Proposi-tion 3.3.For y ∈ C x the fiber π − ( y ) ⊂ C x is equal to { y } × C y . In particular, byTheorem 3.1(ii) the fibers of π have constant dimension n . Thus ( C x ) • is mappedonto an irreducible component ( C x ) • of C x . Finally, let C ′ be an irreduciblecomponent of the preimage π − ( x ) which is contained in ( C x ) • . Note that C ′ canbe identified with an irreducible component of C x , because π − ( x ) = { x } × C x .We claim that the projectivised tangent cone P τ x ( D x ) • contains the join oftwo tangent cones ( P τ x C ′ ) ∗ ( P τ x ( C x ) • ) ⊂ P F x ⊂ P T x X. The proof of the claim is a baby version of [HK05, Thm 3.11]. There howeverHwang and Kebekus assume C x is irreducible and thus their results do not nec-cessarily apply directly here. Let l 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 isenough to show there exists a surface S ⊂ D x containing l and l which is smoothat x , since in such a case T x S ⊂ τ x D x and P T x S is the line between P T x l and P T x l .We obtain S by varying l . 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 iswell defined on an open subset containg all lines through x .By generality of our choices, l is a smooth point of H l and ξ is submersiveat l . In the neighbourhood of l choose a curve A ⊂ H l smooth at l for which ξ | A is submersive at l . Then the locus in X of lines which are in A sweeps asurface S ⊂ X , which is smooth at x , contains l , and contains an open subsetof l around x . Thus the claim is proved and: ( P τ x C ′ ) ∗ ( P τ x ( C x ) • ) ⊂ P τ x ( D x ) • (3.7)Now we claim that F x ⊂ τ x D x . For this purpose we separate two cases.9n the first case C ′ = ( C x ) • . Then P τ x C ′ is non-degenerate by Theorem 3.1and thus ( P τ x C ′ ) ∗ ( P τ x ( C x ) • ) = σ ( P τ 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 bygenerality of x and by Theorem 3.1, the two tangent cones ( P τ x C ′ ) and ( P τ x ( C x ) • ) are disjoint. Thus again ( P τ x C ′ ) ∗ ( P τ 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 dimensionat least n . The dimension can only jump up at special points when one hasa fibration, thus also at special points every component of D x has dimension atleast n . Earlier we observed that dim D x ≤ n , thus the theorem is proved. (cid:3) Proposition 3.8. If X is the adjoint variety of G , and x ∈ X , then D x is thehyperplane 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 isdetermined 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 asin the statment of proposition.So let ∆ be a P -invariant divisor linearly equivalent to L k for some k ≥ . Alsolet ρ ∆ be a section of L k which determines ∆ . The module of sections H ( L k ) isan irreducible G -module by Borel-Weil theorem (see [Ser95]), with some highestweight ω . Since the Lie algebra p of P contains all positive root spaces, by [FH91,Prop. 14.13] there is a unique -dimensional p -invariant submodule of H ( L k ) , itis the highest weight space H ( L k ) ω . So ρ ∆ ∈ H ( L k ) ω and ∆ is unique.The hyperplane section of X ⊂ P ( g ) perpendicular to x via the Killing formis a divisor in | L | , and it is P -invariant, and so are its multiples in | L k | . So bythe uniqueness ∆ must be equal to k times this hyperplane section. Thus ∆ isreduced if and only k = 1 and so D x is the hyperplane section. (cid: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 − ⊕ O ln − ⊕ O l F | l ≃ O l (2) ⊕ O l (1) n − ⊕ O ln − ⊕ O l ( − T l ≃ O l (2) and for general z ∈ l : T C z | l \{ z } ≃ O l (2) ⊕ O l (1) n − . If x ∈ X is a general point and y ∈ C x is a general point of any of theirreducible 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 isswept by deformations l t of l = l such that l t intersects C x . By the standarddeformation theory argument taking derivative of l t by t at a point z ∈ l , weobtain that: T z D x ⊃ (cid:8) s ( z ) ∈ T z X | ∃ s ∈ H ( T X | l ) s.t. s ( y ) ∈ T y C x (cid:9) (3.9)Moreover, at a general point z we have equality in (3.9). If we mod out T X | l bythe rank n positive bundle ( T X | l ) > := O l (2) ⊕ O l (1) n − , then we are left witha trivial bundle O ln +1 . Thus, since by Theorem 3.6 the dimension of T z D x = 2 n for general z ∈ l , the vector space T y C x must be transversal to ( T X | l ) > at y . Inparticular, if z = y , then dimension of the right hand side in (3.9) is n 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 pointof any of the irreducible components of C x and l be any line through y . Thenthere exists a subbundle Γ ⊂ T X | l such that: Γ = O l (2) ⊕ O l (1) n − ⊕ O ln ,Γ ∩ F | l = O l (2) ⊕ O l (1) n − ⊕ O ln − = ( F | l ) ≥ and if z ∈ l is a smooth point of D x , then T z D x = Γ z . (cid:3) An effective divisor ∆ on X is an element of divisor group (and thus a positiveintegral combination of codimension subvarieties of X ) and also a point in theprojective space P ( H O X (∆)) or a hyperplane in P ( H O X (∆) ∗ ) . In this sectionwe will constantly interchange these three interpretations of ∆ . In order to avoidconfusion we will write: 11 ∆ div to mean the divisor on X ; • ∆ P to mean the point in P ( H O X (∆)) or in a fixed linear subsystem. • ∆ P ⊥ to mean the hyperplane in P ( H O X (∆) ∗ ) or in dual of the fixed sub-system.In §3.3 we have defined D ⊂ X × X , which we now view as a family of divisorson X parametrised by X . Since the Picard group of X is discrete and X is smoothand connected, it follows that all the divisors D x are linearly equivalent. Thuslet E ≃ L ⊗ k be the line bundle O X ( D x ) . Consider the following vector space h D i ⊂ H ( E ) : h D i := span { s x : x ∈ X } where s x is a section of E vanishing on D x .Hence P h D i is the linear system spanned by all the D x .Further, consider the map φ : X → P h D i ∗ determined by the linear system h D i , i.e. mapping point x ∈ X to the hyperplanein P h D i consisting of all divisors containing x . Remark . 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 notcontract 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 adjointvariety. Proof. If X is the adjoint variety of G , and x ∈ X , then D x is the hyperplanesection 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 adjointvariety. (cid:3) In algebraic geometry it is standard to consider maps determined by linear sys-tems (such as φ defined above). However in our situation, we also have anothermap determined by the family of divisors D . Namely: ψ : X → P h D i x D x P .
12o let
S ⊂ O X ⊗ h D i ∗ ≃ X × h D i ∗ be the pullback under φ of the universalhyperplane bundle, i.e. the corank 1 subbundle such that the fibre of S over x is D x P ⊥ ⊂ h D i ∗ . We note that P ( S ) is both a projective space bundle on X andalso it is a divisor on X × P h D i ∗ . Also D = (id X × φ ) ∗ P ( S ) as divisors.We can also consider the line bundle dual to the cokernel of S → O X ⊗ h D i ∗ ,i.e. the subbundle S ⊥ ⊂ O X ⊗ h D i . This line subbundle determines section X → X × P h D i , where x ( x, D x P ) . So ψ is the composition of the section andthe projection: X → X × P h D i → P h D i . Every map to a projective space is determined by some linear system. Weclaim the ψ is determined by h D i , precisely the system that defines φ and thusthat there is a natural linear isomorphism between P h D i and P h D i ∗ . Proposition 4.3.
We have ψ ∗ O P h D i (1) ≃ E and the linear system cut out byhyperplanes ψ ∗ H (cid:0) O P h D i (1) (cid:1) := { ψ ∗ s : s ∈ h D i ∗ } ⊂ H ( E ) is equal to h D i . Proof.
For fixed x ∈ X let φ ( x ) ⊥ ⊂ P h D i be the hyperplane dual to φ ( x ) ∈ P h D i ∗ . To prove the proposition it is enough to prove ψ ∗ ( φ ( x ) ⊥ ) = D xdiv . (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 moreof gymnastics, which translates the equivalences above into local equations. Thedetails are below.The pull back of φ ( x ) ⊥ by the projection X × P h D i → P h D i is just X × φ ( x ) ⊥ .Then the pull-back of the product by the section X → X × P h D i associated to S ⊥ is just the subscheme of X defined by (cid:8) y ∈ X | ( S ⊥ ) y ⊂ φ ( x ) ⊥ (cid:9) (locally, this isjust a single equation: the spanning section of S ⊥ satisfies the defining equationof φ ( x ) ⊥ ). But this is clearly equal to the dual equation { y | P ( S y ) ∋ φ ( x ) } . Ifwe let ρ x be the section ρ x : X → X × Xρ x ( y ) := ( y, x ) ψ ∗ ( φ ( x ) ⊥ ) = ρ x ∗ ◦ (id X × φ ) ∗ ( P ( S )) = ρ x ∗ ( D ) = D xdiv as claimed. (cid:3) Thus we have a canonical linear isomorphism f : P h D i ∗ → P h D i giving riseto the following commutative diagram: P h D i ∗≃ (cid:15) (cid:15) X φ gggggggggggggg ψ + + WWWWWWWWWWWWWW P h D i . (4.5)We will denote the underlying vector space isomorphism h D i ∗ → h D i (which isunique up to scalar) with the same letter f . The choice of f combined with thecanonical pairing h D i × h D i ∗ → C , determines a non-degenerate bilinear form B : h D i × h D i → 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 h D i = H ( L ) ≃ g and B is (up to scalar) the Killing form on g . Proof.
Follows immediately from Proposition 3.8 and Equation 4.6. (cid:3)
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). (cid:3) 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. roof. Consider two linear maps h D i → h D i ∗ : α ( v ) := B ( v, · ) and β ( v ) := B ( · , v ) . If v = φ ( x ) for some x ∈ X , then ker (cid:0) α ( v ) (cid:1) = span (cid:16) ker (cid:0) α ( v ) (cid:1) ∩ φ ( X ) (cid:17) = span (cid:0) φ ( D x ) (cid:1) 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 ≡ ± . Therefore ± α ( φ ( x )) = β ( φ ( x )) and by linearity this extends to ± α = β so B is eithersymmetric or skew-symmetric as stated in the proposition. (cid:3) Example . If X is one of the adjoint varieties, then B is symmetric (becausethe Killing form is symmetric). Remark . Consider P n +1 with a contact structure arising from a symplecticform ω on C n +2 . Recall, that this homogeneous contact Fano manifold doesnot satisfy our assumptions, namely, its Picard group is not generated by theequivalence class of L — in this case L ≃ O P n +1 (2) . However, Wiśniewski in[Wiś00] considers also this generalised situation and defines D x to be the divisorswept by contact conics (i.e. curves C with degree of L | C = 2 ) tangent to thecontact distribution F . Then for the projective space D x is just the hyperplaneperpendicular to x with respect to ω . And thus in this case h D i = H ( O P n +1 (1)) and the bilinear form B defined from such family of divisors would be proportionalto ω , hence skew-symmetric. Proof of Theorem 1.1. D x is a divisor by Theorem 3.6. φ is regular byRemark 4.1. ψ is regular by (4.5). The non-degenerate bilinear form B is con-structed in §4.1. It is either symmetric or skew-symmetric by Proposition 4.9. Inthe adjoint case B is the Killing form by Proposition 4.7. (cid:3) Corollary 4.12. If B is symmetric, then ψ ( X ) ⊂ P h D i is contained in thequadric B ( v, v ) = 0 . orollary 4.13. If x ∈ X , then ψ ( C x ) is contained in a linear subspace ofdimension at most j dim h D i k . Proof. If y, z ∈ C x , then z ∈ D y , so B ( ψ ( y ) , ψ ( z )) = 0 . Therefore span( ψ ( C x )) isan isotropic linear subspace, which cannot have dimension bigger than j dim h D i k . (cid:3) Suppose X ⊂ P g is the adjoint variety of G . Assume further that a maximaltorus and an order of roots in g has been chosen, then g has a natural grading(see [LM02, §6.1]): g = g − ⊕ g − ⊕ g ⊕ g ⊕ g where:(i) g ⊕ g ⊕ g is the parabolic subalgebra p of X .(ii) g is the maximal reductive subalgebra of p .(iii) for all i ∈ {− , − , , , } the vector space g i is a g -module.(iv) g is the -dimensional highest root space,(v) g − is the -dimensional lowest root space.(vi) The restriction of the Killing form to each g ⊕ g − , g ⊕ g − and g 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 / ∈ {− , − , , , } ).In fact the grading is determined by g − and g together with the geometryof X only. So let X be as in Notation 2.1 and let x and w be two general pointsof X . Define the following subspaces of h D i : • h D i to be the -dimensional subspace ψ ( x ) ; • h D i − to be the -dimensional subspace ψ ( w ) ; • h D i to be the linear span of affine cone of ψ ( C x ∩ D w ) ; • h D i − to be the linear span of affine cone of ψ ( C w ∩ D x ) ;16 h D i to be the vector subspace of h D i , 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 1.2.
First note that the classes of the 1-dimensional linearsubspaces g and g − are both in X (as points in P g ). Moreover, they are a pairof general points, because the action of the parabolic subgroup P < G preserves g and moves freely g − . This is because ˆ T [ g − ] X = [ g − , g ] = [ g − , p ] .So fix x = [ g ] and w = [ g − ] . We claim the linear span of C x (respectively C w ) is just g ⊕ g (respectively g − ⊕ g − ). To see that, the lines on X through x are in the intersection of X and the projectivised tangent space P ( ˆ T 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 P g through x and y intersects X with multiplicity at least , but X is cut out by quadrics (see for instance [Pro07, §10.6.6]), so this line must becontained in X . Also C x is non-degenerate in P ( ˆ F x ) ⊂ P ( ˆ T x X ) . However ˆ F x is a p -invariant hyperplane in P ( ˆ T x X ) and the unique p -invariant hyperplane in ˆ T x X = [ g , g ] = [ g − , g ] ⊕ g ⊕ g is ˆ F x = [ g − ⊕ g ⊕ g ⊕ g , g ] = g ⊕ g . Further we have seen in Proposition 3.8 that D x (respectively D w ) is theintersection of P ( g ⊥ B ) = P ( g ⊕ g ⊕ g ⊕ g − ) and X (respectively P ( g − ⊕ g − ⊕ g ⊕ g ) and X ). Equivalently, f ( D x P ⊥ ) = P ( g ⊕ g ⊕ g ⊕ g − ) . Thus: C x ∩ D w = C x ∩ f ( D w P ⊥ ) = C x ∩ P ( g − ⊕ g − ⊕ g ⊕ g ) = C x ∩ P ( g ) .C x ∩ P ( g ) is non-degenerate in P ( g ) , thus h D i = g and analogously h D i − = g − .It remains to prove h D i = g . P h D i = \ y ∈ C x ∪ C w f ( D z P ⊥ )= ( C x ∪ C w ) ⊥ B = P ( g ⊕ g ⊕ g − ⊕ g − ) ⊥ B = P ( g ) . (cid:3) We also note the following lemma in the homogeneous case:17 emma 5.1. If X is the adjoint variety of G , then X ∩ P ( g ) ⊂ C x where x is the point of projective space corresponding to g . Proof.
Suppose y ∈ X ∩ P g and let l ⊂ P g be the line through x and y . Notethat l ⊂ P ( g ⊕ g ) Since g ⊕ g ⊂ [ g , g ] = ˆ T x X , hence l ∩ X has multiplicity atleast at x . Thus l ∩ X has degree at least and since X is cut out by quadrics, l is contained in X . (cid:3) Definition 6.1.
A subvariety ∆ ⊂ X is F -cointegrable if T x ∆ ∩ F x ⊂ F x is acoisotropic subspace for general point x of each irreducible component of ∆ .Note that this is equivalent to the definition given in [Buc08, §E.4] — thisfollows from the local description of the symplectic form on the symplectisationof the contact manifold (see [Buc08, (C.15)]).Clearly, every codimension subvariety of X is F -cointegrable.Assume ∆ ⊂ X is a subvariety of pure dimension, which is F -cointegrable andlet ∆ be the locus where T x ∆ ∩ F x ⊂ F x is a coisotropic subspace of dimension dim ∆ − . We define the ∆ -integrable distribution ∆ ⊥ to be the distributiondefined over ∆ by: ∆ ⊥ x := ( T x ∆ ∩ F x ) ⊥ d θ ⊂ F x We say an irreducible subvariety A ⊂ X is ∆ -integral if A ⊂ ∆ , A ∩ ∆ = ∅ , and T A ⊂ ∆ ⊥ over the smooth points of A ∩ ∆ . Lemma 6.2.
Let A and A be two irreducible ∆ -integral subvarieties. Assume dim A = dim A = codim X ∆ . Then either A = A or A ∩ A ⊂ ∆ \ ∆ . (cid:3) 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 isunique. roof. 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 ) ≥ and for general z ∈ l we have ( T z D x ∩ F z ) ⊥ d θz ⊂ F z is the O (2) part, i.e. the parttangent 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 wheretwo different lines through u do not share the tangent direction and do not meetin any other point. Note that since x is a general point, x ∈ U and thus eachirreducible component of C x and D x intersects U . Thus generality of z impliesthat 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. Inthe 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 pointand therefore: C x ∩ C z = { y } . (cid:3) As a consequence of part (iii) of the theorem the surjective map π : C → D is birational. Thus consider the inverse rational map D C and composeit with the projection on the middle coordinate π : C → X . We define thecomposition to be the bracket map : [ · , · ] D : D C π → X. In this setting, for ( x, z ) ∈ D , one has [ x, z ] D = y = C x ∩ C z , whenever theintersection is just one point. Theorem 6.4. If X is the adjoint variety of G , then the bracket map definedabove agrees with the Lie bracket on g , in the following sense: Let ξ, ζ ∈ g andset η := [ ξ, ζ ] (the Lie bracket on g ). Denote by x , y and z the projective classesin P g of ξ , η and ζ respectively. If x ∈ D z and η = 0 , then the bracket mapsatisfies [ x, z ] D = y . Proof.
It is enough to prove the statement for a general pair ( x, z ) ∈ D . Supposefurther w ∈ C z is a general point. Then the pair ( x, w ) ∈ X × X is a general pair.Thus by Proposition 1.2, we may assume ξ ∈ g and ζ ∈ g − . The restriction ofthe Lie bracket to [ ξ, g − ] determines an isomorphism g − → g of g -modules.In particular the minimal orbit X ∩ P g − is mapped onto X ∩ P g under thisisomorphism. In particular y ∈ X ∩ P g ⊂ C x (see Lemma 5.1). Analogously y ∈ C z , so y ∈ C x ∩ C z . (cid:3) eferences [Bea98] Arnaud Beauville. Fano contact manifolds and nilpotent orbits. Com-ment. Math. Helv. , 73(4):566–583, 1998.[Buc08] Jarosław Buczyński.
Algebraic Legendrian Varieties . PhD thesis,Institute of Mathematics, Warsaw University, 2008. To appear inDissertationes Mathematicae; arXiv:0805.3848v2.[Dem02] Jean-Pierre Demailly. On the Frobenius integrability of certain holo-morphic p -forms. In Complex geometry (Göttingen, 2000) , pages 93–98. Springer, Berlin, 2002.[FH91] William Fulton and Joe Harris.
Representation theory , volume 129 of
Graduate Texts in Mathematics . Springer-Verlag, New York, 1991. Afirst course, Readings in Mathematics.[Har95] Joe Harris.
Algebraic geometry , volume 133 of
Graduate Texts inMathematics . Springer-Verlag, New York, 1995. A first course, Cor-rected reprint of the 1992 original.[HK05] Jun-Muk Hwang and Stefan Kebekus. Geometry of chains of minimalrational curves.
J. Reine Angew. Math. , 584:173–194, 2005.[HM99] Jun-Muk Hwang and Ngaiming Mok. Varieties of minimal rationaltangents on uniruled projective manifolds. In
Several complex vari-ables (Berkeley, CA, 1995–1996) , volume 37 of
Math. Sci. Res. Inst.Publ. , pages 351–389. Cambridge Univ. Press, Cambridge, 1999.[HM04] Jun-Muk Hwang and Ngaiming Mok. Birationality of the tangentmap for minimal rational curves.
Asian J. Math. , 8(1):51–63, 2004.[Keb01] Stefan Kebekus. Lines on contact manifolds.
J. Reine Angew. Math. ,539:167–177, 2001.[Keb05] Stefan Kebekus. Lines on complex contact manifolds. II.
Compos.Math. , 141(1):227–252, 2005.[KO73] Shoshichi Kobayashi and Takushiro Ochiai. Characterizations of com-plex projective spaces and hyperquadrics.
J. Math. Kyoto Univ. ,13:31–47, 1973.[Kol96] János Kollár.
Rational curves on algebraic varieties , volume 32 of
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Seriesof Modern Surveys in Mathematics [Results in Mathematics and Re-lated Areas. 3rd Series. A Series of Modern Surveys in Mathematics] .Springer-Verlag, Berlin, 1996.20KPSW00] Stefan Kebekus, Thomas Peternell, Andrew J. Sommese, andJarosław A. Wiśniewski. Projective contact manifolds.
Invent. Math. ,142(1):1–15, 2000.[LeB95] Claude LeBrun. Fano manifolds, contact structures, and quaternionicgeometry.
Internat. J. Math. , 6(3):419–437, 1995.[LM02] Joseph M. Landsberg and Laurent Manivel. Construction and classi-fication of complex simple Lie algebras via projective geometry.
Se-lecta Math. (N.S.) , 8(1):137–159, 2002.[LM07] Joseph M. Landsberg and Laurent Manivel. Legendrian varieties.
Asian J. Math. , 11(3):341–359, 2007.[Mum99] David Mumford.
The red book of varieties and schemes , volume 1358of
Lecture Notes in Mathematics . Springer-Verlag, Berlin, expandededition, 1999. Includes the Michigan lectures (1974) on curves andtheir Jacobians, With contributions by Enrico Arbarello.[Pro07] Claudio Procesi.
Lie groups . Universitext. Springer, New York, 2007.An approach through invariants and representations.[Ser95] Jean-Pierre Serre. Représentations linéaires et espaces homogèneskählériens des groupes de Lie compacts (d’après Armand Borel etAndré Weil). In
Séminaire Bourbaki, Vol. 2