[Paper Review] From local Torelli to global Torelli

Kefeng Liu, Yang Shen|arXiv (Cornell University)|Dec 28, 2015

Geometry and complex manifolds34 references7 citations

TL;DR

This paper introduces the concept of strong local Torelli and T-class for polarized manifolds, proving that strong local Torelli implies the global Torelli theorem on Torelli spaces for manifolds in the T-class. By lifting the period map to the Teichmüller space and using level structures, the authors establish global injectivity of the period map, showing that the Torelli space embeds biholomorphically into a bounded pseudoconvex domain in complex space, with applications to moduli spaces with level structures.

ABSTRACT

We introduce the notions of strong local Torelli and T-class for polarized manifolds, and prove that strong local Torelli implies global Torelli theorem on the Torelli spaces for polarized manifolds in the T-class. We discuss many new examples of projective manifolds for which such global Torelli theorem holds. As applications we prove that, in these cases, a canonical completion of the Torelli space is a bounded pseudoconvex domain in complex Euclidean space, and show that generic Torelli implies global Torelli on moduli space with certain level structure.

Motivation & Objective

• To establish a systematic framework for proving the global Torelli theorem beyond isolated cases.
• To resolve the difficulty in proving global Torelli by lifting the period map to the universal cover (Teichmüller space).
• To define and characterize the T-class of polarized manifolds for which global Torelli holds.
• To show that the Torelli space of such manifolds is a bounded pseudoconvex domain in complex Euclidean space.
• To prove global Torelli holds on moduli spaces with level m structure for m ≥ 3.

Proposed method

• Introduce the notion of strong local Torelli as a sufficient condition for global Torelli on Torelli spaces.
• Define the T-class as polarized manifolds admitting a smooth moduli space with level m structure for m ≥ 3.
• Construct the Teichmüller space as the universal cover of the moduli space with level structure.
• Lift the period map from the moduli space to the Teichmüller space, ensuring injectivity under strong local Torelli.
• Use monodromy representations and fundamental group isomorphisms to prove that the period map on the moduli space with level structure is a biholomorphism onto its image.
• Apply results from Hodge theory and complex geometry to identify the image of the period map as a bounded pseudoconvex domain.

Experimental results

Research questions

• RQ1 Under what conditions does strong local Torelli imply the global Torelli theorem for polarized manifolds?
• RQ2 Which classes of polarized manifolds belong to the T-class, and why is this class significant for Torelli-type theorems?
• RQ3 How does the use of level structures and Teichmüller spaces help overcome obstructions in proving global Torelli?
• RQ4 What geometric structure does the Torelli space of a T-class manifold inherit, and how is it related to bounded symmetric domains?
• RQ5 Can the global Torelli theorem be established on moduli spaces with level m structure, and what does this imply for the period map?

Key findings

• Strong local Torelli implies the global Torelli theorem on the Torelli space for manifolds in the T-class.
• The Torelli space of a polarized manifold in the T-class is a bounded pseudoconvex domain in complex Euclidean space.
• The period map from the moduli space with level m structure is a biholomorphic map onto its image when strong local Torelli holds.
• The fundamental group of the moduli space with level structure is isomorphic to the monodromy group Γ.
• The lifted period map to the Teichmüller space is injective, and the image is a domain in the period domain D.
• The global Torelli theorem holds on the moduli space with level m structure for m ≥ 3, extending the result to a broader class of algebraic varieties.