[Paper Review] Teichmuller spaces, ergodic theory and global Torelli theorem

Misha Verbitsky|arXiv (Cornell University)|Apr 15, 2014

Geometric and Algebraic Topology28 references6 citations

TL;DR

This paper establishes the ergodicity of the mapping class group action on the Teichmüller space of hyperkähler manifolds, leveraging the Bogomolov-Beauville-Fujiki form and period maps. This ergodicity implies that hyperkähler manifolds are never Kobayashi hyperbolic and leads to new insights into symplectic packing and holomorphic dominance, with applications to global Torelli theorems and moduli theory.

ABSTRACT

A Teichmüller space $Teich$ is a quotient of the space of all complex structures on a given manifold $M$ by the connected components of the group of diffeomorphisms. The mapping class group $Γ$ of $M$ 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 Teichmuller space for a hyperkahler manifold. It turns out that this action is ergodic. We use the ergodicity to show that a hyperkahler manifold is never Kobayashi hyperbolic. This is my ICM submission, with review of some of my work on Teichmuller spaces and moduli; proofs are sketched, new observations and some open problems added.

Motivation & Objective

• To understand the structure of the Teichmüller space and mapping class group action for hyperkähler manifolds.
• To establish the ergodicity of the monodromy group action on the Teichmüller space of hyperkähler manifolds.
• To apply ergodicity to prove that hyperkähler manifolds are not Kobayashi hyperbolic.
• To explore the implications of ergodicity for holomorphic symplectic packing and dominance by complex space.
• To extend the global Torelli theorem to hyperkähler manifolds using period maps and ergodic techniques.

Proposed method

• Define the Teichmüller space as the quotient of the space of Kähler-compatible complex structures on a hyperkähler manifold by the group of isotopies (Diff₀).
• Use the period map to associate each complex structure to its Hodge decomposition in cohomology, invariant under Diff₀.
• Leverage the Bogomolov-Beauville-Fujiki form to analyze the geometry of the Teichmüller space and its action by the mapping class group.
• Apply results from ergodic theory to show that the action of the mapping class group on the Teichmüller space is ergodic.
• Use semicontinuity of symplectic volume and ergodic invariance to study holomorphic symplectic packings and the supremum of radii for symplectic immersions.
• Connect ergodicity to the non-existence of Brody curves and the failure of Brody’s lemma in higher dimensions for dominating maps.

Experimental results

Research questions

• RQ1 Is the action of the mapping class group on the Teichmüller space of a hyperkähler manifold ergodic?
• RQ2 Can ergodicity be used to prove that hyperkähler manifolds are not Kobayashi hyperbolic?
• RQ3 What are the restrictions on holomorphic symplectic packings of hyperkähler manifolds, and are volume bounds the only constraints?
• RQ4 Is every compact hyperkähler manifold dominated by ℂⁿ via a holomorphic symplectomorphism?
• RQ5 How does ergodicity constrain the supremum of radii for symplectic immersions into a hyperkähler manifold?

Key findings

• The mapping class group action on the Teichmüller space of a hyperkähler manifold is ergodic.
• As a consequence, hyperkähler manifolds are not Kobayashi hyperbolic, since ergodic complex structures admit entire curves.
• The Teichmüller space of a hyperkähler manifold is a complex manifold, locally isomorphic to the Kuranishi space, due to the Bogomolov-Tian-Todorov theorem.
• The period map is well-defined on the Teichmüller space and factors through the quotient by Diff₀, preserving Hodge structures.
• The set of possible radii for holomorphic symplectic packings is semicontinuous and constant on ergodic complex structures.
• For K3 surfaces, the supremum of radii for symplectic immersions is finite or infinite depending on the structure, but remains unknown in general.