Teichmuller spaces, ergodic theory and global Torelli theorem

M. Verbitsky

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Teichmuller spaces, ergodic theory and global Torelli theorem

P. 793–811.

Verbitsky M.

A Teichmüller space Teich is a quotient of the space of all complex structures on a given
manifoldM by the connected components of the group of diffeomorphisms. The mapping class group
Γ ofM is the group of connected components of the diffeomorphism group. The moduli problems can
be understood as statements about the Γ-action on Teich. I will describe the mapping class group and
the Teichmüller space for a hyperkähler manifold. It turns out that this action is ergodic. We use the
ergodicity to show that a hyperkähler manifold is never Kobayashi hyperbolic.

Language: English

Keywords: Torelli theorem Teichmüller space hyperkähler manifold, ergodicity.

In book

Proceedings of the International Congress of Mathematicians, Seoul 2014

Vol. 2. , Seul: Kyung Moon Sa Co. Ltd, 2014.

Lee classes on LCK manifolds with potential

Verbitsky M., Ornea L., Tohoku Mathematical Journal 2024 Vol. 76 No. 1 P. 105 – 125

An LCK manifold is a complex manifold (M,I) equipped with a Hermitian form ω and a closed 1-form θ, called the Lee form, such that dω=θ∧ω. An LCK manifold with potential is an LCK manifold with a positive Kähler potential on its universal cover, such that the deck group multiplies the Kähler potential by a constant. A Lee class of an ...

Added: December 2, 2024

Locality in the Fukaya category of a hyperkähler manifold

Verbitsky M., Solomon J., / Series arXiv "math". 2018.

Let (M,I,J,K,g) be a hyperkahler manifold. Then the complex manifold (M,I) is holomorphic symplectic. We prove that for all real x,y, with x^2+y^2=1 except countably many, any finite energy (xJ+yK)-holomorphic curve with boundary in a collection of I-holomorphic Lagrangians must be constant. By an argument based on the Lojasiewicz inequality, this result holds no matter how the Lagrangians intersect each other. It follows that one ...

Added: December 5, 2018

Ergodic complex structures on hyperkahler manifolds

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

Teichmüller space for hyperkähler and symplectic structures

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

Lectures on universal Teichmüller space

Sergeev A., European Mathematical Society Publishing house, 2014.

This book is based on a lecture course given by the author at the Educational Center of the Steklov Mathematical Institute in 2011. It is designed for a one-semester course for undergraduate students familiar with basic differential geometry and complex and functional analysis. The universal Teichmüller space T is the quotient of the space of quasisymmetric ...

Added: April 9, 2015

Any component of moduli of polarized hyperkähler manifolds is dense in its deformation space

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