### ?

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

Cornell University
,
2021.

Verbitsky M., Kamenova L.

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 the space of signature (1,n) is a disconnected union of two convex cones. The "positive cone" is the component which contains the Kahler cone. We say that the Kahler cone is "round" if it is equal to the positive cone. The manifolds with round Kahler cones have unique bimeromorphic model and correspond to Hausdorff points in the corresponding Teichmuller space. We prove thay any maximal holonomy hyperkahler manifold with b2>4 has a deformation with round Kahler 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 hyperkahler manifolds admit a deformation with two transversal Lagrangian fibrations, and the Kobayashi metric vanishes unless the Picard rank is maximal.

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., / 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 a generic complex deformation of a generalized Kummer variety contains no complex analytic tori. ...

Added: October 16, 2015

Soldatenkov A. O., Verbitsky M., Journal of Geometry and Physics 2014

Let (M,I,J,K) be a hyperkahler manifold, and Z⊂(M,I) a complex subvariety in (M,I). We say that Z is trianalytic if it is complex analytic with respect to J and K, and absolutely trianalytic if it is trianalytic with respect to any hyperk\"ahler triple of complex structures (M,I,J′,K′) containing I. For a generic complex structure I ...

Added: December 26, 2014

Katzarkov L. V., Abouzaid M., Auroux D., Publications Mathématiques de l'IHÉS 2016 Vol. 123 No. 1 P. 199–282

https://link.springer.com/article/10.1007/s10240-016-0081-9 ...

Added: October 23, 2017

Kamenova L., Verbitsky M., European Journal of Mathematics 2021

Let M be a holomorphic symplectic Kähler manifold equipped with a Lagrangian fibration 𝜋π with compact fibers. The base of this manifold is equipped with a special Kähler structure, that is, a Kähler structure (I,g,ω)and a symplectic flat connection ∇∇ such that the metric g is locally the Hessian of a function. We prove that any Lagrangian subvariety Z⊂M which intersects smooth fibers of 𝜋π and ...

Added: September 3, 2021

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

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., / 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

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

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

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

Let p: M -> B be a Lagrangian fibration on a hyperkahler manifold of maximal holonomy (also known as IHS), and H the generator of the Picard group of B. We prove that the pullback p∗(H) is a primitive class on M. ...

Added: April 10, 2017

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

Kamenova L., Lu S., Verbitsky M., Journal of London Mathematical Society 2014 Vol. 90 No. 2 P. 436–450

The Kobayashi pseudometric on a complex manifold is the maximal pseudometric such that any holomorphic map from the Poincaré disk to the manifold is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi–Yau manifolds. Using ergodicity of complex structures, we prove this for all hyperkähler manifold with b_2\geqslant 7 that admits a deformation with ...

Added: September 19, 2014

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

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

Kurnosov N., Bogomolov F. A., / Cornell University. Series arXiv "math". 2018.

In this paper we study the Lagrangian fibrations for projective irreducible symplectic fourfolds and exclude the case of non-smooth base. Our method could be extended to the higher-dimensional cases. ...

Added: December 2, 2018

Verbitsky M., Kamenova L., / Cornell University. Series arXiv "math". 2019.

Let M be a holomorphic symplectic Kähler manifold equipped with a Lagrangian fibration π with compact fibers. The base of this manifold is equipped with a special Kähler structure, that is, a Kähler structure (I,g,ω) and a symplectic flat connection ∇ such that the metric g is locally the Hessian of a function. We prove that any Lagrangian subvariety Z⊂M which intersects smooth fibers of π and smoothly projects ...

Added: June 9, 2019

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

Kotelnikova M. V., Aistov A., Вестник Нижегородского университета им. Н.И. Лобачевского. Серия: Социальные науки 2019 Т. 55 № 3 С. 183–189

The article describes a method that allows to improve the content of disciplines of the mathematical cycle by dividing them into invariant (general) and variable parts. The invariants were identified for such disciplines as «Linear algebra», «Mathematical analysis», «Probability theory and mathematical statistics» delivered to Bachelors program students of economics at several universities. Based on ...

Added: January 28, 2020

Borzykh D., ЛЕНАНД, 2021

Книга представляет собой экспресс-курс по теории вероятностей в контексте начального курса эконометрики. В курсе в максимально доступной форме изложен тот минимум, который необходим для осознанного изучения начального курса эконометрики. Данная книга может не только помочь ликвидировать пробелы в знаниях по теории вероятностей, но и позволить в первом приближении выучить предмет «с нуля». При этом, благодаря доступности изложения и небольшому объему книги, ...

Added: February 20, 2021

В. Л. Попов, Математические заметки 2017 Т. 102 № 1 С. 72–80

Мы доказываем, что аффинно-треугольные подгруппы являются борелевскими подгруппами групп Кремоны. ...

Added: May 3, 2017

Красноярск: ИВМ СО РАН, 2013

Труды Пятой Международной конференции «Системный анализ и информационные технологии» САИТ-2013 (19–25 сентября 2013 г., г.Красноярск, Россия): ...

Added: November 18, 2013