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

Cornell University
,
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 ⊂ VZ a Γ-invariant set of vectors with negative square. Denote by R ⊥ the set of all positive p-planes W ⊂ V such that the orthogonal complement W⊥ contains r ∈ R. We prove that either R ⊥ is dense in M or Γ acts on R with finitely many orbits. This is used to prove that the squares of primitive classes giving the rational boundary of the K¨ahler cone (i.e. the classes of “negative” minimal rational curves) on a hyperkahler manifold X are bounded by a number which depends only on the deformation class of X. We also state and prove the density of orbits in a more general situation when M is the space of maximal compact subgroups in a simple real Lie group.

Verbitsky M., Duke Mathematical Journal 2013 Vol. 162 No. 15 (2013) P. 2929-2986

A mapping class group of an oriented manifold is a quotient of its diffeomorphism group by the isotopies. We compute a mapping class group of a hyperkähler manifold $M$, showing that it is commensurable to an arithmetic lattice in SO(3, b_2-3). A Teichmüller space of $M$ is a space of complex structures on $M$ up ...

Added: December 10, 2013

Amerik E., Verbitsky M., Journal of Geometry and Physics 2015 Vol. 97 P. 44-50

Let S be an infinite-dimensional manifold of all symplectic, or hyperkähler, structures on a compact manifold M, and Diff0 the connected component of its diffeomorphism group. The quotient S/Diff0 is called the Teichmüller space of symplectic (or hyperkähler) structures on M. MBM classes on a hyperkähler manifold M are cohomology classes which can be represented ...

Added: September 8, 2015

Ananʼin S., Verbitsky M., Journal de Mathématiques Pures et Appliquées 2014 Vol. 101 No. 2 P. 188-197

Let M be a compact hyperkähler manifold, and W the coarse moduli of complex deformations of M. Every positive integer class v in H^2(M) defines a divisor Dv in W consisting of all algebraic manifolds polarized by v. We prove that every connected component of this divisor is dense in W. ...

Added: January 28, 2015

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

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., 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

Polishchuk A., Lekili Y., Journal fuer die reine und angewandte Mathematik 2019 Vol. 2019 No. 755 P. 151-189

We show that a certain moduli space of minimal A∞-structures coincides with the modular compactification ℳ_{1,n}(n−1)of ℳ_{1,n} constructed by Smyth in [26]. In addition, we describe these moduli spaces and the universal curves over them by explicit equations, prove that they are normal and Gorenstein, show that their Picard groups have no torsion and that they have rational ...

Added: May 10, 2020

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., 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

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

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

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., / 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

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

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

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

Gritsenko V., Ванг Х., Математический сборник 2019 Т. 210 № 12 С. 43-66

Задача о построении антисимметричных парамодульных форм канонического веса 3 была поставлена в 1996 году. Любая параболическая форма этого типа определяет каноническую диф- ференциальную форму на любой гладкой компактификации про- странство модулей куммеровых поверхностей, отвечающих (1,t)- поляризованным абелевым поверхностям. В этой статье мы строим первое бесконечное семейство антисимметричных парамодулярных форм веса 3 как автоморфные произведения Борчердса, чьи пер- вые коэффициенты Фурье–Якоби ...

Added: October 29, 2019

Moduli of symplectic instanton vector bundles of higher rank on projective space $\mathbb{P^3}$. II.

Tikhomirov A. S., Bruzzo U., Markushevich D., / Max Planck Institute for Mathematics. Series MPIM "MPIM". 2014. No. 2014-22.

Symplectic instanton vector bundles on the projective space $\mathbb{P^3}$ are a natural generalization of mathematical instantons of rank 2. We study the moduli space $I_{n,r}$ of rank-$2r$ symplectic instanton vector bundles on $\mathbb{P^3}$ with $r\ge2$ and second Chern class $n\ge r+1,\ n-r \equiv 1(\mod2)$. We introduce the notion of tame symplectic instantons by excluding a ...

Added: October 19, 2014

Natanzon S. M., Pratoussevitch A., Russian Mathematical Surveys 2016 Vol. 71 No. 2 P. 382-384

In this paper, we present all higher spinor structures on Klein surfaces. We present also topological invariants that describe the connected components of moduli of Klein surfaces with higher spinor structure. Each connected component is represented as a cell factorable by a discrete group . ...

Added: March 25, 2016

Kochetkov Y., Фундаментальная и прикладная математика 2014 Т. 19 № 1 С. 45-63

Мы рассматриваем открытое пространство модулей $\mathcal{M}_{2,1}$ комплексных кривых рода 2 с одной отмеченной точкой. На языке хордовых диаграмм мы описываем клеточную структуру пространства $\mathcal{M}_{2,1}$ и структуру примыкания клеток. Это позволяет нам построить матрицы граничных операторов и найти числа Бетти пространства $\mathcal{M}_{2,1}$ над Q. ...

Added: November 11, 2014

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

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

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

Feigin B. L., Finkelberg M. V., Rybnikov L. G. et al., Selecta Mathematica, New Series 2011 Vol. 17 No. 3 P. 573-607

Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GLn. We construct the action of the Yangian of sln in the cohomology of Laumon spaces by certain natural correspondences. We construct the action of the affine Yangian (two-parametric deformation of the universal ...

Added: October 9, 2012