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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?

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    $\begingroup$ 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. $\endgroup$ – Jason Starr Aug 28 '13 at 22:03
  • $\begingroup$ yes, you're right: i'm interested in local and global Torelli. $\endgroup$ – fabio alves Aug 28 '13 at 22:07
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There are versions of the Global Torelli Problem for Calabi-Yau threefolds which are open. Infinitesimal Torelli is true for CY3s, i.e. (families of) projective 3-folds with trivial canonical bundle and no holomorphic 1-forms, as proved already by Griffiths. The strongest possible form of global Torelli fails, as proved in the late 90s in my thesis and early papers, eg. "Calabi-Yau threefolds with a curve of singularities and counterexamples to the Torelli problem", Int. J. Math. 11 constructs explicit families of CY3s in which the polarized Hodge structure up to isomorphism does not determine the varieties up to isomorphism. But there are outstanding questions. Todorov, Yau and collaborators have studied a variant where the period map maps from the Teichmuller space to the period domain (without taking any discrete group quotient); this map could still be injective. There are several papers on the arXiv claiming results of this nature, such as http://xxx.lanl.gov/abs/1112.1163 and http://arxiv.org/abs/1205.4207. One can also ask whether in the simply-connected case, the variety up to birational equivalence could be recovered from the polarized Hodge structure; I don't know any counterexample to this. As a special case, one can ask Global Torelli for Calabi-Yau threefolds of Picard number one; I am not sure we know the result for any specific family, apart from the generic Torelli for the family of quintics in Voisin's "A generic Torelli theorem for the quintic threefold", in New trends in algebraic geometry (Warwick, 1996).

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  • $\begingroup$ "I am not sure we know the result for any specific family, apart from the generic Torelli for the family of qunitics in Voisin's ..." We also know generic Torelli also for many complete intersections in weighted projective spaces, cf. Donagi, Saito, etc. For instance, this applies to octic double solids. These complete intersections "usually" have Picard number one. $\endgroup$ – Jason Starr Aug 29 '13 at 14:53
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    $\begingroup$ The generic Torelli holds for the mirror quintic (due to Usui), which has 1-dim complex moduli space, instead of 1-dim Kahler moduli space. $\endgroup$ – Atsushi Kanazawa Aug 29 '13 at 15:23
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    $\begingroup$ The Calabi-Yau case is excluded in most "generic Torelli for hypersurfaces" type papers. This is most explicitly visible in the original Donagi papers, but if you check through the conditions of the later papers (Donagi-Tu, Cox-Green, etc) then the CY case is never covered. This is true for Saito's paper also if you read the conditions in the actual paper (e.g. the double octic is also excluded) - the MathSciNet review has a mistake in the numerical conditions. There is a technical reason: the symmetrizer lemma never applies in the CY case. Voisin has to work a lot harder to prove the result. $\endgroup$ – Balazs Aug 29 '13 at 20:00
  • $\begingroup$ What about octic double solids? If we include the data of the involution, then we can recover the Hodge structure of the octic hypersurface, and thus the octic hypersurface by the generic Torelli. Of course maybe we should not include the data of the involution. $\endgroup$ – Jason Starr Aug 29 '13 at 22:15
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Calabi-Yaus? I think A. Todorov and his coauthors were working on this, but I believe the general case is open.

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