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## Construction of automorphisms of hyperkahler manifolds

Cornell University
,
2016.

Amerik E., Verbitsky M.

Let M be an irreducible holomorphic symplectic (hyperk¨ahler) manifold. If b2(M) >= 5, we construct a deformation M′ of M which admits a symplectic automorphism of infinite order. This automorphism is hyperbolic, that is, its action on the space of real (1, 1)-classes is hyperbolic. If b2(M) >= 14, similarly, we construct a deformation which admits a parabolic automorphism.

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., 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

Amerik E., Verbitsky M., 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 ...

Added: October 28, 2015

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

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

Entov M., Verbitsky M., Journal of Topology and Analysis 2016 Vol. 08 No. 4 P. 589–626

Let M be a closed symplectic manifold of volume V. We say that the symplectic packings of M by balls are unobstructed if any collection of disjoint 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 ...

Added: December 3, 2015

Голубин Р. В., Емельянова Е. Н., Ильина И. С. et al., Н. Новгород: Издательство Нижегородского государственного университета им. Н.И. Лобачевского, 2015

The publication is dedicated to Nikolai Ivanovich Lobachevsky (1792-1856) — a great Russian mathematician, creator of non-Euclidean geometry, a native of Nizhny Novgorod. The basis of the illustrated album is provided by copies of archival materials relating to the biographies of N.I. Lobachevsky and his contemporaries. The publication is timed to celebrate the centennial (2016) ...

Added: March 13, 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

Kamenova L., Verbitsky M., New York Journal of Mathematics 2017 Vol. 23 P. 489–495

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 hyperkähler manifolds are not algebraically hyperbolic when the Picard rank is at least 3, or ...

Added: April 10, 2017

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

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., 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

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

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

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

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

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

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

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

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

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

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

Kalinin N., Shkolnikov M., Journal of Knot Theory and Its Ramifications 2016 Vol. 25 No. 12 P. 1–12

We study perturbations of the maximal stable state in a sandpile model on the set of faces of the heptagonal tiling on the hyperbolic plane. An explicit description for relaxations of such states is given. ...

Added: October 10, 2017

Прасолов В. В., Schwarzman O., МЦНМО, 2014

Книга, адресованная студентам физико-математических специальностей написана на основе лекций, прочитанных авторами в Независимом
московском университете.
В первой части изложены основы теории алгебраических кривых, рассматриваемых как римановы поверхности. Здесь преобладают сравнительно элементарные алгебраические и геометрические методы. Новинкой для
учебной литературы такого уровня является обсуждение связи алгебраических кривых с теорией Галуа. Впервые на русском языке изложены теоремы
Ритта о композициях многочленов ...

Added: March 18, 2015