A survey of Torelli and monodromy results for holomorphic-symplectic varieties

TL;DR

This survey reviews recent advances in Torelli theorems, monodromy, and moduli spaces for holomorphic-symplectic varieties, highlighting key results in Hodge theory, birational geometry, and cone conjectures.

Contribution

It compiles and explains recent progress on Torelli questions, monodromy groups, and moduli space descriptions for holomorphic-symplectic varieties, emphasizing new theoretical insights.

Findings

Hodge theoretic Torelli theorem established

Description of birational Kahler cone as a fundamental domain

Weak version of Morrison's movable cone conjecture proved

Abstract

We survey recent results about the Torelli question for holomorphic-symplectic varieties. Following are the main topics. A Hodge theoretic Torelli theorem. A study of the subgroup W, of the isometry group of the weight 2 Hodge structure, generated by reflection with respect to exceptional divisors. A description of the birational Kahler cone as a fundamental domain for the W-action on the positive cone. A proof of a weak version of Morrison's movable cone conjecture. A description of the moduli spaces of polarized holomorphic symplectic varieties as monodromy quotients of period domains of type IV.