Hyperbolic geometry of the ample cone of a hyperkahler manifold
aa r X i v : . [ m a t h . AG ] N ov E. Amerik, M. Verbitsky
Hyperbolic geometry of the ample cone
Hyperbolic geometry of the ample coneof a hyperk¨ahler manifold
Ekaterina Amerik , Misha Verbitsky Abstract
Let M be a compact hyperk¨ahler manifold with maximalholonomy (IHS). The group H ( M, R ) is equipped with aquadratic form of signature (3 , b − H , ( M, Q ), has signature (1 , k ).This gives a hyperbolic Riemannian metric on the projec-tivisation of the positive cone in H , ( M, Q ), denoted by H . Torelli theorem implies that the Hodge monodromygroup Γ acts on H with finite covolume, giving a hyper-bolic orbifold X = H/ Γ. We show that there are finitelymany geodesic hypersurfaces which cut X into finitelymany polyhedral pieces in such a way that each of thesepieces is isometric to a quotient P ( M ′ ) / Aut( M ′ ), where P ( M ′ ) is the projectivization of the ample cone of a bi-rational model M ′ of M , and Aut( M ′ ) the group of itsholomorphic automorphisms. This is used to prove theexistence of nef isotropic line bundles on a hyperk¨ahler bi-rational model of a simple hyperk¨ahler manifold of Picardnumber at least 5, and also illustrates the fact that anIHS manifold has only finitely many birational models upto isomorphism (cf. [MY]). Contents Partially supported by RScF grant, project 14-21-00053, 11.08.14. Partially supported by RScF grant, project 14-21-00053, 11.08.14.
Keywords: hyperk¨ahler manifold, K¨ahler cone, hyperbolic geometry, cusp points – 1 – version 1.0, Nov. 7, 2015 . Amerik, M. Verbitsky
Hyperbolic geometry of the ample cone
Let M be an irreducible holomorphically symplectic manifold, that is, asimply-connected compact K¨ahler manifold with H , ( M ) = C Ω where Ω isnowhere degenerate. In dimension two, such manifolds are K3 surfaces; inhigher dimension 2 n, n > , one knows, up to deformation, two infinite seriesof such manifolds, namely the punctual Hilbert schemes of K3 surfaces andthe generalized Kummer varieties, and two sporadic examples constructedby O’Grady. Though considerable effort has been made to construct otherexamples, none is known at present, and the classification problem for ir-reducible holomorphic symplectic manifolds (IHSM) looks equally out ofreach.One of the main features of an IHSM M is the existence of an inte-gral quadratic form q on the second cohomology H ( M, Z ), the Beauville-Bogomolov-Fujiki form (BBF) form. It generalizes the intersection formon a surface; in particular its signature is (3 , b − H , R ( M ) is (1 , b − { x ∈ H , R ( M ) | x > } thus has two connected components; we call the positive cone Pos( X ) theone which contains the K¨ahler classes. The BBF form is in fact of topolog-ical origin: by a formula due to Fujiki, q ( α ) n is proportional to α n with apositive coefficient depending only on M .To understand better the geometry of an IHSM, it can be useful tofiber it, whenever possible, over a lower-dimensional variety. Note that bya result of Matsushita, the fibers are always Lagrangian (in particular, n -dimensional, where 2 n = dim C M ), and the general fiber is a torus. Suchfibrations are important for the classification-related problems, and one canalso hope to get some interesting geometry from their degenerate fibers (forinstance, use them to construct rational curves on M ).Note that a fibration of M is necessarily given by a linear system | L | where | L | is a nef line bundle with q ( L ) = 0. Conjecturally, any such bundleis semiample, that is, for large m the linear system | L ⊗ m | is base-point-freeand thus gives a desired fibration.It is therefore important to understand which irreducible holomorphicsymplectic varieties carry nef line bundles of square zero. By Meyer’s theo-rem (see for example [Se]), M has an integral (1 , version 1.0, Nov. 7, 2015 . Amerik, M. Verbitsky Hyperbolic geometry of the ample cone soon as the Picard number ρ ( M ) is at least five. By definition, such a classis nef when it is in the closure of the K¨ahler cone Kah( M ) ⊂ Pos( M ). Thequestion is thus to understand whether one can find an isotropic integral(1 , M ) ⊂ Pos( M ) is cut out by the orthogonal hyperplanes to( − , − − ± -effectiveby Riemann-Roch. Let x be an isotropic integral (1 , x Kah( M ), that is, there is a ( − p with h x, p i <
0. Fix anample integral (1 , h . Then the image of x under the reflection in p ⊥ , x ′ = x + h x, p i p , satisfies h x ′ , h i < h x, h i . Therefore the image of x undersuccessive reflexions in such p ’s becomes nef after finitely many steps. Thenon-projective case is even easier, since an isotropic line bundle must thenbe in the kernel of the Neron-Severi lattice and so has zero intersection withevery curve, in particular, it is nef.Trying to apply the same argument to higher-dimensional IHSM we seethat the existence of an isotropic line bundle yields and isotropic element inthe closure of the birational K¨ahler cone BK( M ). By definition, BK( M )is a union of inverse images of the K¨ahler cone on all IHSM birational modelsof M , and its closure is cut out in Pos( X ) by the Beauville-Bogomolovorthogonals to the classes of the prime uniruled exceptional divisors ([Bou1]).One knows that the reflections in those hyperplanes are integral ([M1]); inparticular the divisors have bounded squares and the “reflections argument”above applies with obvious modifications.A priori, the closure of BK( M ) may strictly contain the union of the clo-sures of the inverse images of the K¨ahler cones of all birational models, soan additional argument is required to conclude that there is an isotropic nefclass on some birational model of M . One way to deal with this is explainedin the paper [MZ]: the termination of flops on an IHSM implies that anyelement of the closure of BK( M ) does indeed become nef on some birationalmodel. These observations, though, require the use of rather heavy ma-chinery of the Minimal Model Program (MMP) which are in principle validon all varieties (though the termination of flops itself remains unproven ingeneral).The purpose of the present note is to give another proof of the existenceof nef isotropic classes, which does not rely on the MMP. Instead it relieson the “cone conjecture” which was established in [AV2] using completelydifferent methods, namely ergodic theory and hyperbolic geometry. We findthe hyperbolic geometry picture which appears in our proof particularly ap-– 3 – version 1.0, Nov. 7, 2015 . Amerik, M. Verbitsky Hyperbolic geometry of the ample cone pealing, and believe that it might provide an alternative, perhaps sometimesmore efficient, approach to birational geometry in the particular case of theirreducible holomorphic symplectic manifolds.The main advantage of the present construction is its geometric inter-pretation. The BBF quadratic form, restricted to the rational Hodge lattice H , ( M, Q ), has signature (1 , k ) (unless M is non-algebraic, in which caseour results are tautologies). This gives a hyperbolic Riemannian metricon the projectivisation of the positive cone in H , ( M, Q ), denoted by H .Torelli theorem implies that the group Γ Hdg of Hodge monodromy acts on H with finite covolume, giving a hyperbolic orbifold X = H/ Γ Hdg . UsingSelberg lemma, one easily reduces to the case when X is a manifold. Weprove that X is cut into finitely many polyhedral pieces by finitely manygeodesic hypersurfaces in such a way that each of these pieces is isometricto a quotient Amp( M ′ ) / Aut( M ′ ), where Amp( M ′ ) is the projectivizationof the ample cone of a birational model of M , and Aut( M ′ ) the group ofholomorphic automorphisms.In this interpretation, equivalence classes of birational models are inbijective correspondence with these polyhedral pieces H i , and the isotropicnef line bundles correspond to the cusp points of these H i . Existence of cusppoints is implied by Meyer’s theorem, and finiteness of H i by our results onthe cone conjecture from [AV2] (Section 3). Finally, the geometric finitenessresults from hyperbolic geometry imply the finiteness of the isotropic nefline bundles up to automorphisms. In this section, we recall the definitions and basic properties of hyperk¨ahlermanifolds and MBM classes.
Definition 2.1: A hyperk¨ahler manifold M , that is, a compact K¨ahlerholomorphically symplectic manifold, is called simple (alternatively, ir-reducible holomorphically symplectic (IHSM) ), if π ( M ) = 0 and H , ( M ) = C .This definition is motivated by the following theorem of Bogomolov. Theorem 2.2: ([Bo1]) Any hyperk¨ahler manifold admits a finite coveringwhich is a product of a torus and several simple hyperk¨ahler manifolds.– 4 – version 1.0, Nov. 7, 2015 . Amerik, M. Verbitsky
Hyperbolic geometry of the ample cone
The second cohomology H ( M, Z ) of a simple hyperk¨ahler manifold M carries a primitive integral quadratic form q , called the Bogomolov-Beauville-Fujiki form . It generalizes the intersection product on a K3surface: its signature is (3 , b −
3) on H ( M, R ) and (1 , b −
3) on H , R ( M ).It was first defined in [Bo2] and [Bea], but it is easiest to describe it usingthe Fujiki theorem, proved in [F]. Theorem 2.3: (Fujiki) Let M be a simple hyperk¨ahler manifold, η ∈ H ( M ), and n = dim M . Then R M η n = cq ( η, η ) n , where q is a prim-itive integer quadratic form on H ( M, Z ), and c > Definition 2.4:
Let M be a hyperk¨ahler manifold. The monodromygroup of M is a subgroup of GL ( H ( M, Z )) generated by the monodromytransforms for all Gauss-Manin local systems.It is often enlightening to consider this group in terms of the mappingclass group action. We briefly recall this description.The Teichm¨uller space
Teich is the quotient
Comp ( M ) / Diff ( M ), where Comp ( M ) denotes the space of all complex structures of K¨ahler type on M and Diff ( M ) is the group of isotopies. It follows from a result of Huybrechts(see [H2]) that for an IHSM M , Teich has only finitely many connected com-ponents. Let
Teich M denote the one containing our given complex structure.Consider the subgroup of the mapping class group Diff ( M ) / Diff ( M ) fixing Teich M . Definition 2.5:
The monodromy group
Γ is the image of this subgroupin O ( H ( M, Z ) , q ). The Hodge monodromy group Γ Hdg is the subgroupof Γ preserving the Hodge decomposition.
Theorem 2.6: ([V1], Theorem 3.5) The monodromy group is a finite indexsubgroup in O ( H ( M, Z ) , q ) (and the Hodge monodromy is therefore anarithmetic subgroup of the orthogonal group of the Picard lattice). Definition 2.7:
A cohomology class η ∈ H ( M, R ) is called positive if q ( η, η ) >
0, and negative if q ( η, η ) <
0. The positive cone
Pos( M ) ∈ – 5 – version 1.0, Nov. 7, 2015 . Amerik, M. Verbitsky Hyperbolic geometry of the ample cone H , R ( M ) is that one of the two connected components of the set of positiveclasses on M which contains the K¨ahler classes.Recall e.g. from [M2] that the positive cone is decomposed into theunion of birational K¨ahler chambers , which are monodromy transformsof the birational K¨ahler cone BK( M ). The birational K¨ahler cone is, bydefinition, the union of pullbacks of the K¨ahler cones Kah( M ′ ) where M ′ denote a hyperk¨ahler birational model of M (the “K¨ahler chambers”). The faces of these chambers are supported on the hyperplanes orthogonal tothe classes of prime uniruled divisors of negative square on M .The MBM classes are defined as those classes whose orthogonal hy-perplanes support faces of the K¨ahler chambers.
Definition 2.8:
A negative integral cohomology class z of type (1 ,
1) iscalled monodromy birationally minimal (MBM) if for some isometry γ ∈ O ( H ( M, Z )) belonging to the monodromy group, γ ( z ) ⊥ ⊂ H , R ( M )contains a face of the K¨ahler cone of one of birational models M ′ of M .Geometrically, the MBM classes are characterized among negative inte-gral (1 , M under the identification of H ( M, Q ) with H ( M, Q ) given by the BBF form ([AV1], [AV3], [KLM]).The following theorems summarize the main results about MBM classesfrom [AV1]. Theorem 2.9: ([AV1], Corollary 5.13) An MBM class z ∈ H , ( M ) is alsoMBM on any deformation M ′ of M where z remains of type (1 , Theorem 2.10: ([AV1], Theorem 6.2) The K¨ahler cone of M is a connectedcomponent of Pos( M ) \ ∪ z ∈ S z ⊥ , where S is the set of MBM classes on M .In what follows, we shall also consider the positive cone in the algebraicpart N S ( M ) ⊗ R of H , R ( M ), denoted by Pos Q ( M ). Here and further on, N S ( M ) stands for N´eron-Severi group of M . A face of a convex cone in a vector space V is the intersection of its boundary and ahyperplane which has non-empty interior in the hyperplane. – 6 – version 1.0, Nov. 7, 2015 . Amerik, M. Verbitsky Hyperbolic geometry of the ample cone
Definition 2.11:
The ample chambers are the connected components ofPos Q ( M ) \ ∪ z ∈ S z ⊥ where S is the set of MBM classes on M .One of the ample chambers is, obviously, the ample cone of M , hencethe name.In the same way, one defines birationally ample or movable chambersas the connected components of the complement to the union of orthogonalsto the classes of uniruled divisors and their monodromy transforms, cf. [M2],section 6. These are also described as intersections of the biratonal K¨ahlerchambers with N S ( M ) ⊗ R . Remark 2.12:
Because of the deformation-invariance property of MBMclasses, it is natural to introduce this notion on H ( M, Z ) rather than on(1 , z ∈ H ( M, Z ) an MBM class as soon as it is MBM inthose complex structures where it is of type (1 , The following theorem has been proved in [AV2].
Theorem 2.13: ([AV2]) Suppose that the Picard number ρ ( M ) >
3. Thenthe Hodge monodromy group has only finitely many orbits on the set ofMBM classes of type (1 ,
1) on M .Since the Hodge monodromy group acts by isometries, it follows thatthe primitive MBM classes have bounded square (using the deformationargument, one easily extends this last statement from the case of ρ ( M ) > b ( M ) = 5, but we shall not need this here). In [AV1] we have seenthat this implies some apriori stronger statements on the Hodge monodromyaction. Corollary 2.14:
The Hodge monodromy group has only finitely many orbitson the set of faces of the K¨ahler chambers, as well as on the set of the K¨ahlerchambers themselves.For reader’s convenience, let us briefly sketch the proof (for details, seesections 3 and 6 of [AV1]). It consists in remarking that a face of a chamber isgiven by a flag P s ⊃ P s − ⊃ · · · ⊃ P where P s is the supporting hyperplane– 7 – version 1.0, Nov. 7, 2015 . Amerik, M. Verbitsky Hyperbolic geometry of the ample cone (of dimension s = h , − P s − supports a face of our face, etc., and foreach P i an orientation (“pointing inwards the chamber”) is fixed. Onededuces from the boundedness of the square of primitive MBM classes thatpossible P s − are as well given inside P s by orthogonals to integral vectorsof bounded square, and it follows that the stabilizer of P s in Γ Hdg acts withfinitely many orbits on those vectors; continuing in this way one eventuallygets the statement.By Markman’s version of the Torelli theorem [M2], an element of Γ
Hdg preserving the K¨ahler cone actually comes from an automorphism of M .Thus an immediate consequence is the following K¨ahler version of the Morrison-Kawamata cone conjecture. Corollary 2.15: ([AV2]) Aut( M ) has only finitely many orbits on the setof faces of the K¨ahler cone. Remark 2.16:
As the faces of the ample cone are likewise given by theorthogonals to MBM classes, but in Pos Q ( M ) rather than in Pos( M ), oneconcludes that the same must be true for the ample cone. Definition 3.1: A Kleinian group is a discrete subgroup of isometries ofthe hyperbolic space H n .One way to view H n is as a projectivization of the positive cone P V + ofa quadratic form q of signature (1 , n ) on a real vector space V . The Kleiniangroups are thus discrete subgroups of SO (1 , n ). One calls such a subgroupa lattice if its covolume is finite. Definition 3.2: An arithmetic subgroup of an algebraic group G definedover the integers is a subgroup commensurable with G Z . Remark 3.3:
From Borel and Harish-Chandra theorem (see [BHCh]) itfollows that when q is integral, any arithmetic subgroup of SO (1 , n ) is alattice for n > Definition 3.4: A complete hyperbolic orbifold is a quotient of the– 8 – version 1.0, Nov. 7, 2015 . Amerik, M. Verbitsky Hyperbolic geometry of the ample cone hyperbolic space by a Kleinian group. A complete hyperbolic manifold is a quotient of the hyperbolic space by a Kleinian group acting freely.
Remark 3.5:
One defines a hyperbolic manifold as a manifold of constantnegative bisectional curvature. When complete, such a manifold is uni-formized by the hyperbolic space ([Th]).The following proposition is well-known.
Proposition 3.6:
Any complete hyperbolic orbifold has a finite coveringwhich is a complete hyperbolic manifold (in other words, any Kleinian grouphas a finite index subgroup acting freely).
Proof:
Let Γ be a Kleinian group. Notice first that all stabilizers forthe action of Γ on P V + are finite, since these are identified to discretesubgroups of a compact group SO ( n ). Now by Selberg lemma Γ has a finiteindex subgroup without torsion which must therefore act freely. Remark 3.7: If M is an IHSM, the group of Hodge monodromy Γ Hdg isan arithmetic lattice in SO ( H , ( M, Q )) when rk H , ( M, Q ) >
3. Thehyperbolic manifold P ( H , ( M, Q ) ⊗ Q R ) + / Γ Hdg has finite volume by Boreland Harish-Chandra theorem.
Recall that the rational positive cone
Pos Q ( M ) of a projective hyperk¨ahlermanifold M is one of two connected components of the set of positive vectorsin N S ( M ) ⊗ R .Replacing Γ Hdg by a finite index subgroup if necessary, we may assumethat the quotient P Pos Q ( M ) / Γ Hdg is a complete hyperbolic manifold whichwe shall denote by H .By Borel and Harish-Chandra theorem (see Remark 3.3), H is of finitevolume as soon as the Picard number of M is at least three.Let S = { s i } be the set of MBM classes of type (1 ,
1) on M . Thefollowing is a translation of the Morrison-Kawamata cone conjecture intothe setting of hyperbolic geometry. Theorem 3.8:
The images of the hyperplanes s ⊥ i , s i ∈ S , cut H = P Pos Q ( M ) / Γ Hdg into finitely many pieces. One of those pieces is the image of the ample cone(up to a finite covering, this is the quotient of the ample cone by Aut( M ))– 9 – version 1.0, Nov. 7, 2015 . Amerik, M. Verbitsky Hyperbolic geometry of the ample cone and the others are the images of ample cones of birational models of M .The closure of each one is a hyperbolic manifold with boundary consistingof finitely many geodesic pieces. Proof:
According to Corollary 2.14, up to the action of Γ
Hdg there arefinitely many faces of ample chambers. Each face is a connected componentof the complement to ∪ j = i s ⊥ j in s ⊥ i for some i . It is clear that the imagesof the faces do not intersect hence, being finitely many, cut H into finitelymany pieces which are images of the ample chambers. We have alreadymentioned that an element of Γ Hdg preserving the K¨ahler cone is inducedby an automorphism. Finally, the whole H is covered by the birationalample cone (since the other birational ample chambers are its monodromytransforms) and thus each part of H obtained in this way comes from anample chamber.Let us also mention that the same arguments also prove the followingresult (cf. [MY]). Corollary 3.9:
There are only finitely many non-isomorphic birationalmodels of M . Proof:
Indeed, the K¨ahler (or ample) chambers in the same Γ
Hdg -orbitcorrespond to isomorphic birational models, since one can view the action ofΓ
Hdg as the change of the marking (recall that a marking is a choice of anisometry of H ( M, Z ) with a fixed lattice Λ and that there exists a coarsemoduli space of marked IHSM which in many works (e.g. [H1]) plays thesame role as the Teichm¨uller space in others). Definition 4.1: A horosphere on a hyperbolic space is a sphere which iseverywhere orthogonal to a pencil of geodesics passing through one point atinfinity, and a horoball is a ball bounded by a horosphere.
A cusp point for an n -dimensional hyperbolic manifold H / Γ is a point on the boundary ∂ H such that its stabilizer in Γ contains a free abelian group of rank n − maximal parabolic . For any point p ∈ ∂ H stabilized by Γ ⊂ Γ, and any horosphere S tangent to the boundary in p ,Γ acts on S by isometries. In such a situation, p is a cusp point if and onlyif ( S \ p ) / Γ is compact. – 10 – version 1.0, Nov. 7, 2015 . Amerik, M. Verbitsky Hyperbolic geometry of the ample cone
A cusp point p yields a cusp in the quotient H / Γ, that is, a geometricend of H / Γ of the form B/ Z n − , where B ⊂ H is a horoball tangent to theboundary at p .The following theorem describes the geometry of finite volume completehyperbolic manifolds more precisely. Theorem 4.2: (Thick-thin decomposition)Any n -dimensional complete hyperbolic manifold of finite volume can berepresented as a union of a “thick part”, which is a compact manifold witha boundary, and a “thin part”, which is a finite union of quotients of form B/ Z n − , where B is a horoball tangent to the boundary at a cusp point,and Z n − = St Γ ( B ). Proof:
See [Th, Section 5.10] or [Ka, page 491]).
Theorem 4.3:
Let H / Γ be a hyperbolic manifold, where Γ is an arithmeticsubgroup of SO (1 , n ). Then the cusps of H / Γ are in (1,1)-correspondencewith Z/ Γ, where Z is the set of rational lines l such that l = 0. Proof:
By definition of cusp points, the cusps of H / Γ are in 1 to 1correspondence with Γ-conjugacy classes of maximal parabolic subgroupsof Γ (see [Ka]). Each such subgroup is uniquely determined by the uniquepoint it fixes on the boundary of H .The main result of this paper is the following theorem. Theorem 4.4:
Let M be a hyperk¨ahler manifold with Picard number atleast 5. Then M has a birational model admitting an integral nef (1 , η with η = 0. Moreover each birational model contains only finitely manysuch classes up to automorphism. Proof:
By Meyer’s theorem (see for example [Se]), there exists η ∈ N S ( M ) with η = 0. By Theorem 4.3, the hyperbolic manifold H := P Pos Q ( M ) / Γ Hdg then has cusps, and, being of finite volume, only finitelymany of them. Recall that H is decomposed into finitely many pieces, andeach of those pieces is the image of the ample cone of a birational model of M in Pos Q ( M ). Therefore a lifting of each cusp to the boundary of P Pos Q ( M )gives a BBF-isotropic nef line bundle on a birational model M ′ of M (or,more precisely, the whole line such a bundle generates in N S ( M ) ⊗ R ). Fi-nally, the number of Aut( M ′ )-orbits of such classes is finite, being exactly– 11 – version 1.0, Nov. 7, 2015 . Amerik, M. Verbitsky Hyperbolic geometry of the ample cone the number of cusps in the piece of H corresponding to M ′ : indeed thispiece is just the quotient of the ample cone of M ′ by its stabilizer which isidentified with Aut( M ′ ). Acknowledgements:
We are grateful to S. Cantat, M. Kapovich andV. Gritsenko for interesting discussions and advice.
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Ekaterina AmerikLaboratory of Algebraic Geometry,National Research University HSE,Department of Mathematics, 7 Vavilova Str. Moscow, Russia,
[email protected] , also:
Universit´e Paris-11,Laboratoire de Math´ematiques,Campus d’Orsay, Bˆatiment 425, 91405 Orsay, FranceMisha VerbitskyLaboratory of Algebraic Geometry,National Research University HSE,Department of Mathematics, 7 Vavilova Str. Moscow, Russia, [email protected] , also:
Universit´e Libre de Bruxelles, CP 218,Bd du Triomphe, 1050 Brussels, Belgium – 13 –– 13 –