Hyperbolic geometry of the ample cone of a hyperkähler manifold

Amerik, Ekaterina; Verbitsky, Misha

doi:10.1186/s40687-016-0059-8

Cited by 6 publications

(3 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The shape of the Kähler cone of a hyperkähler manifold is more or less understood by now (see [AV3]). However, finding examples of manifolds with prescribed shape of their Kähler cone is a complicated task.…”

Section: Introductionmentioning

confidence: 99%

“…As shown in [AV3] (the result is essentially due to E. Markman,[Mar2]), the positive cone Pos(M, I) of a hyperkähler manifold is cut into pieces by hyperplanes orthogonal to the MBM classes which lie in H 1,1 (M, I), and each of the connected components of this complement can be realized as a Kähler cone of a certain hyperkähler birational model of (M, I). In other words, the Kähler cone is a connected component of the set Pos • (M, I) := Pos(M, I)…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Roundness of the ample cone and existence of double Lagrangian fibrations on hyperkahler manifolds

Kamenova¹,

Verbitsky²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

Let M be a hyperkähler manifold of maximal holonomy (that is, an IHS manifold), and let K be its Kähler cone, which is an open, convex subset in the space H 1,1 (M, R) of real (1,1)-forms. This space is equipped with a canonical bilinear symmetric form of signature (1, n) obtained as a restriction of the Bogomolov-Beauville-Fujiki form. The set of vectors of positive square in the space of signature (1, n) is a disconnected union of two convex cones. The "positive cone" is the component which contains the Kähler cone. We say that the Kähler cone is "round" if it is equal to the positive cone. The manifolds with round Kähler cones have unique bimeromorphic model and correspond to Hausdorff points in the corresponding Teichmüller space. We prove thay any maximal holonomy hyperkähler manifold with b245 has a deformation with round Kähler cone and the Picard lattice of signature (1,1), admitting two non-collinear integer isotropic classes. This is used to show that all known examples of hyperkähler manifolds admit a deformation with two transversal Lagrangian fibrations, and their Kobayashi metric vanishes, unless the Picard rank is maximal.

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Roundness of the ample cone and existence of double Lagrangian fibrations on hyperkahler manifolds

Kamenova¹,

Verbitsky²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…In other words, the set of ample line bundles on X 0 of fixed degree is finite up to the action of the group Aut(X 0 ) of automorphisms of X 0 . As the cone conjecture for hyperkähler manifolds has recently been established in great generality in [AV14,AV15], see also [MY15] for a proof for the two standard series, the theorem can also be seen as a consequence of the cone conjecture. In fact, the full conjecture is not needed to conclude the above result from the global Torelli theorem, a shortcut is outlined in Section 1.4.…”

mentioning

confidence: 99%

Finiteness of polarized K3 surfaces and hyperkähler manifolds

Huybrechts¹

2019

Annales Henri Lebesgue

View full text Add to dashboard Cite

In the moduli space of polarized varieties (X, L) the same unpolarized variety X can occur more than once. However, for K3 surfaces, compact hyperkähler manifolds, and abelian varieties the 'orbit' of X, i.e. the subset {(Xi, Li) | Xi ≃ X}, is known to be finite, which may be viewed as a consequence of the Kawamata-Morrison cone conjecture. In this note we provide a proof of this finiteness not relying on the cone conjecture and, in fact, not even on the global Torelli theorem. Instead, it uses the geometry of the moduli space of polarized varieties to conclude the finiteness by means of Baily-Borel type arguments. We also address related questions concerning finiteness in twistor families associated with polarized K3 surfaces of CM type.The paper studies the connection between moduli spaces of polarized varieties, on the one hand, and the shape of the ample cone on a fixed variety, on the other hand. To illustrate our point of departure, let us revue a few well-known results. 0.1. The classical Torelli theorem shows that two complex smooth projective curves C and C ′ are isomorphic if and only if their polarized Hodge structures are isomorphic, i.e. there exists a Hodge isometry H 1 (C, Z) ≃ H 1 (C, Z), or, equivalently, if their principally polarized Jacobians J(C) ≃ J(C ′ ) are isomorphic. Dropping the compatibility with the polarizations, so only requiring isomorphisms of unpolarized Hodge structures or unpolarized abelian varieties, the geometric relation between C and C ′ becomes less clear. In moduli theoretic terms, one may wonder about the geometric nature of the quotient map M g / / M g / ∼ . Here, M g denotes the moduli space of genus g curves and C ∼ C ′ if and only if J(C) ≃ J(C ′ ) unpolarized.Similarly, two polarized K3 surfaces (S, L) and (S ′ , L ′ ) are isomorphic if and only if there exists a Hodge isometry H 2 (S, Z) ≃ H 2 (S ′ , Z) that maps L to L ′ . Dropping the latter condition has a clear geometric meaning and corresponds to considering isomorphisms between unpolarized surfaces S and S ′ . Therefore, dividing out by the resulting equivalence relation yields a map M d / / M d / ∼ from the moduli space of polarized K3 surfaces (S, L) of degree d to the space of isomorphism classes of K3 surfaces that merely admit a polarization of this degree. Considering only isomorphisms of Hodge structures without any further compatibilities leads to the analogue of the aforementioned question for curves. At this time, there is no clear picture of what the existence of an unpolarized isomorphism of Hodge structures could mean for the geometry of the two K3 surfaces, but finiteness has recently been established in [Ef17].The author is supported by the SFB/TR 45 'Periods, Moduli Spaces and Arithmetic of Algebraic Varieties' of the DFG (German Research Foundation) and the Hausdorff Center for Mathematics.1 2 D. HUYBRECHTS Other types of varieties, like abelian varieties, Calabi-Yau or hyperkähler varieties, can be discussed from the same perspective. 0.2. Let us move to the cone side. For a K3 surface S, t...

show abstract

Finiteness of Klein actions and real structures on compact hyperkähler manifolds

Cattaneo

2019

Math. Ann.

View full text Add to dashboard Cite

One central problem in real algebraic geometry is to classify the real structures of a given complex manifold. We address this problem for compact hyperkähler manifolds by showing that any such manifold admits only finitely many real structures up to equivalence. We actually prove more generally that there are only finitely many, up to conjugacy, faithful finite group actions by holomorphic or anti-holomorphic automorphisms (the socalled Klein actions). In other words, the automorphism group and the Klein automorphism group of a compact hyperkähler manifold contain only finitely many conjugacy classes of finite subgroups. We furthermore answer a question of Oguiso by showing that the automorphism group of a compact hyperkähler manifold is finitely presented.

show abstract

Hyperbolic geometry of the ample cone of a hyperkähler manifold

Cited by 6 publications

References 18 publications

Roundness of the ample cone and existence of double Lagrangian fibrations on hyperkahler manifolds

Roundness of the ample cone and existence of double Lagrangian fibrations on hyperkahler manifolds

Finiteness of polarized K3 surfaces and hyperkähler manifolds

Finiteness of Klein actions and real structures on compact hyperkähler manifolds

Contact Info

Product

Resources

About