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## Rational Curves on Hyperkähler Manifolds

International Mathematics Research Notices. 2015. Vol. 2015. No. 23. P. 13009-13045.

Let M be an irreducible holomorphically symplectic manifold. We show that all faces of the Kähler cone of M are hyperplanes Hi orthogonal to certain homology classes, called monodromy birationally minimal (MBM) classes. Moreover, the Kähler cone is a connected component of a complement of the positive cone to the union of all Hi. We provide several characterizations of the MBM classes. We show the invariance of MBM property by deformations, as long as the class in question stays of type (1,1). For hyperkähler manifolds with Picard group generated by a negative class z, we prove that ±z is Q-effective if and only if it is an MBM class. We also prove some results toward the Morrison–Kawamata cone conjecture for hyperkähler manifolds.

Publication based on the results of:

Boston : Birkhäuser, 2013

This book features recent developments in a rapidly growing area at the interface of higher-dimensional birational geometry and arithmetic geometry. It focuses on the geometry of spaces of rational curves, with an emphasis on applications to arithmetic questions. Classically, arithmetic is the study of rational or integral solutions of diophantine equations and geometry is the ...

Added: February 14, 2013

Kurnosov N., / Cornell University. Series math "arxiv.org". 2015.

We prove that a generic complex deformation of a generalized Kummer variety contains no complex analytic tori. ...

Added: October 16, 2015

Verbitsky M., Pure and Applied Mathematics Quarterly 2014 Vol. 10 No. 2 P. 325-354

Let S be a smooth rational curve on a complex manifold M. It is called ample if its normal bundle is positive: NS=⨁O(i_k),i_k<0. We assume that M is covered by smooth holomorphic deformations of S. The basic example of such a manifold is a twistor space of a hyperkähler or a 4–dimensional anti-selfdual Riemannian manifold ...

Added: January 23, 2015

Collections of parabolic orbits in homogeneous spaces, homogeneous dynamics and hyperkahler geometry

Amerik E., Verbitsky M., / Cornell University. Series arXiv "math". 2016.

Consider the space M = O(p, q)/O(p) × O(q) of positive p-dimensional subspaces in a pseudo-Euclidean space V of signature (p, q), where p > 0, q > 1 and (p, q) != (1, 2), with integral structure: V = VZ ⊗ R. Let Γ be an arithmetic subgroup in G = O(VZ), and R ...

Added: April 14, 2016

Verbitsky M., Kamenova L., / Cornell University. Series arXiv "math". 2021.

Let M be a hyperkahler manifold of maximal holonomy (that is, an IHS manifold), and let K be its Kahler cone, which is an open, convex subset in the space H1,1(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 ...

Added: November 25, 2021

Tomberg A., Математические заметки 2019 Т. 105 № 6 С. 949-954

...

Added: November 11, 2018

Verbitsky M., Entov M., Selecta Mathematica, New Series 2018 Vol. 24 No. 3 P. 2625-2649

Let M be a closed symplectic manifold of volume V. We say that M admits an unobstructed symplectic packing by balls if any collection of symplectic balls (of possibly different radii) of total volume less than V admits a symplectic embedding to M. In 1994 McDuff and Polterovich proved that symplectic packings of Kahler manifolds ...

Added: September 13, 2018

Amerik E., Campana F., Journal of London Mathematical Society 2017 Vol. 95 No. 1 P. 115-127

We prove that the characteristic foliation F on a nonsingular divisor D in an irreducible projective hyperk¨ahler manifold X cannot be algebraic, unless the leaves of F are rational curves or X is a surface. More generally, we show that if X is an arbitrary projective manifold carrying a holomorphic symplectic 2-form, and D and ...

Added: September 8, 2016

Amerik E., Verbitsky M., Annales Scientifiques de l'Ecole Normale Superieure 2017 Vol. 50 No. 4 P. 973-993

Let M be a simple hyperk¨ahler manifold, that is, a simply connected compact holomorphically symplectic manifold of K¨ahler type with h 2,0 = 1. Assuming b2(M) 6= 5, we prove that the group of holomorphic automorphisms of M acts on the set of faces of its K¨ahler cone with finitely many orbits. This statement is ...

Added: September 8, 2016

Amerik E., Verbitsky M., / Cornell University. Series arXiv "math". 2021.

A parabolic automorphism of a hyperkahler manifold is a holomorphic automorphism acting on H2(M) by a non-semisimple quasi-unipotent linear map. We prove that a parabolic automorphism which preserves a Lagrangian fibration acts on its fibers ergodically. The invariance of a Lagrangian fibration is automatic for manifolds satisfying the hyperkahler SYZ conjecture; this includes all known examples of ...

Added: April 6, 2022

Jardim M., Verbitsky M., Compositio Mathematica 2014 Vol. 150 No. 11 P. 1836-1868

A trisymplectic structure on a complex 2n-manifold is a
three-dimensional space ${\rm\Omega}$ of closed holomorphic forms such
that any element of \Omega has constant rank 2n, n or zero, and
degenerate forms in \Omega belong to a non-degenerate quadric
hypersurface. We show that a trisymplectic manifold is equipped with a
holomorphic 3-web and the Chern connection of this 3-web is
holomorphic, ...

Added: November 28, 2014

Amerik E., Verbitsky M., / Cornell University. Series arXiv "math". 2018.

An MBM class on a hyperkahler manifold M is a second cohomology class such that its orthogonal complement in H^2(M) contains a maximal dimensional face of the boundary of the Kahler cone for some hyperkahler deformation of M. An MBM curve is a rational curve in an MBM class and such that its local deformation ...

Added: December 4, 2018

Kamenova L., Verbitsky M., / Cornell University. Series arXiv "math". 2016.

A projective manifold is algebraically hyperbolic if the degree of any curve is bounded from above by its genus times a constant, which is independent from the curve. This is a property which follows from Kobayashi hyperbolicity. We prove that hyperk¨ahler manifolds are non algebraically hyperbolic when the Picard rank is at least 3, or ...

Added: April 21, 2016

Bogomolov F. A., Kamenova L., Lu S. et al., / Cornell University. Series arXiv "math". 2016.

We define the Kobayashi quotient of a complex variety by identifying points with vanishing Kobayashi pseudodistance between them and show that if a compact complex manifold has an automorphism whose order is infinite, then the fibers of this quotient map are nontrivial. We prove that the Kobayashi quotients associated to ergodic complex structures on a ...

Added: September 6, 2016

Kamenova L., Verbitsky M., Advances in Mathematics 2014 Vol. 260 P. 401-413

A holomorphic Lagrangian fibration on a holomorphically symplectic manifold is a holomorphic map with Lagrangian fibers. It is known that a given compact manifold admits only finitely many holomorphic symplectic structures, up to deformation. We prove that a given compact manifold with $b_2 \geq 7$ admits only finitely many deformation types of holomorphic Lagrangian fibrations. ...

Added: July 11, 2014

Collections of parabolic orbits in homogeneous spaces, homogeneous dynamics and hyperkahler geometry

Amerik E., Verbitsky M., / Cornell University. Series arXiv "math". 2016.

Let M be a hyperk\"ahler manifold with b2(M)≥5. We improve our earlier results on the Morrison-Kawamata cone conjecture by showing that the Beauville-Bogomolov square of the primitive MBM classes (i.e. the classes whose orthogonal hyperplanes bound the K\"ahler cone in the positive cone, or, in other words, the classes of negative extremal rational curves on ...

Added: September 7, 2016

Kurnosov N., / Cornell University. Series math "arxiv.org". 2015.

We prove that b2 is bounded for hyperk¨ahler manifolds with vanishing odd-Betti numbers. The explicit upper boundary is conjectured. Following the method described by Sawon we prove that b2 is bounded in dimension eight and ten in the case of vanishing odd-Betti numbers by 24 and 25 respectively. ...

Added: November 15, 2015

Verbitsky M., Amerik E., / Cornell University. Series arXiv "math". 2019.

We study the exceptional loci of birational (bimeromorphic) contractions of a hyperkähler manifold M. Such a contraction locus is the union of all minimal rational curves in a collection of cohomology classes which are orthogonal to a wall of the Kähler cone. Homology classes which can possibly be orthogonal to a wall of the Kähler cone ...

Added: June 9, 2019

Abasheva A., / Cornell University. Series math "arxiv.org". 2020. No. arXiv:2007.05773.

In this paper we study the geometry of the total space Y of a cotangent bundle to a Kähler manifold N where N is obtained as a Kähler reduction from Cn. Using the hyperkähler reduction we construct a hyperkähler metric on Y and prove that it coincides with the canonical Feix-Kaledin metric. This metric is in general non-complete. We show that the metric completion Y~ of ...

Added: July 21, 2020

Verbitsky M., Acta Mathematica 2015 Vol. 215 No. 276 P. 161-182

Let M be a compact complex manifold. The corresponding Teichm¨uller space Teich is a space of all complex structures on M up to the action of the group Diff0(M) if isotopies. The mapping class group Γ := Diff(M)/ Diff0(M) acts on Teich in a natural way. An ergodic complex structure is the one with a ...

Added: October 27, 2015

Amerik E., Verbitsky M., / Cornell University. Series arXiv "math". 2021.

An MBM locus on a hyperkahler manifold is the union of all deformations of a minimal rational curve with negative self-intersection. MBM loci can be equivalently defined as centers of bimeromorphic contractions. It was shown that the MBM loci on deformation equivalent hyperkahler manifolds are diffeomorphic. We determine the MBM loci on a hyperkahler manifold ...

Added: April 7, 2022

Amerik E., Verbitsky M., Research in the Mathematical Sciences 2016 Vol. 3 No. 7 P. 1-9

Let M be a compact hyperkähler manifold with maximal holonomy (IHS). The group H2(M,ℝ) is equipped with a quadratic form of signature (3,b2−3)(3,b2−3), called Bogomolov–Beauville–Fujiki form. This form restricted to the rational Hodge lattice H1,1(M,ℚ)has signature (1, k). This gives a hyperbolic Riemannian metric on the projectivization H of the positive cone in H1,1(M,ℚ). Torelli ...

Added: August 31, 2016

Verbitsky M., Selecta Mathematica, New Series 2017 Vol. 23 No. 3 P. 2203-2218

The transcendental Hodge lattice of a projective manifold M is the smallest Hodge substructure in pth cohomology which contains all holomorphic p-forms. We prove that the direct sum of all transcendental Hodge lattices has a natural algebraic structure, and compute this algebra explicitly for a hyperkähler manifold. As an application, we obtain a theorem about ...

Added: February 6, 2017

Verbitsky M., Markman E., Mehrotra S., / Cornell University. Series arXiv "math". 2017.

Let S be a K3 surface and M a smooth and projective 2n-dimensional moduli space of stable coherent sheaves on S. Over M x M there exists a rank 2n-2 reflexive hyperholomorphic sheaf E_M, whose fiber over a non-diagonal point (F,G) is Ext^1(F,G). The sheaf E_M can be deformed along some twistor path to a ...

Added: October 10, 2017