# Open Torelli problems

I just finished studying the proof of the Torelli Theorem for K3 surfaces made by Daniel Huybrechts (following the approach of Misha Verbitsky).

This theorem states that two K3 surfaces $X$ and $Y$ are isomorphic if and only if there is an isometry $\phi:H^2(X,\mathbb{Z})\rightarrow H^2(Y,\mathbb{Z})$ (with respect to the intersection form) which respects the Hodge decomposition.

Huybrechts (and Verbitsky before him) proved this theorem using the period map defined from the moduli space of K3 surfaces to the period domain. This approach is particularly interesting because it can be extended to the case of Hyperkahler manifolds. For this type of manifolds however there can be no strong result as for K3s, as Namikawa proved in this article.

The Torelli problem consists in knowing when the Hodge structure of a manifold determine the manifold.

Question: Do you know other classes of manifolds for which the Torelli problem is an open problem?

• I think you should add some sort of "minimality" condition for your varieties $X$. Obviously the Torelli theorem can fail for a blowing up, and you can "essentially" transform many non-existence conjectures for polarized varieties (e.g., as in Hwang-Mok rigidity) into infinitesimal Torelli conjectures via blowing up. Speaking of which, you should probably also specify whether you are interested in infinitesimal Torelli, local Torelli, global Torelli, generic Torelli, etc. – Jason Starr Aug 28 '13 at 22:03
• yes, you're right: i'm interested in local and global Torelli. – fabio alves Aug 28 '13 at 22:07