UCLA

This thesis has studied global Torelli problems for projective manifolds. In particular, we have focused on projective manifolds of Calabi-Yau type, which is a generalization of Calabi-Yau manifolds.

In collaboration with F. Guan and K. Liu, we have proven the injectivity of the period map on the Teichmuller space of polarized and marked Calabi-Yau type manifolds. Our approach has been to construct the holomorphic affine structure on the Teichmuller space and the Hodge metric completion of the Teichmuller space. As a corollary, we also prove that the Hodge metric completion space of the Teichmuller space for Calabi-Yau type manifolds is a domain of holomorphy and it admits a Kahler-Einstein metric.

More generally, we are interested in global Torelli problems for projective manifolds. We have been working on adopting our techniques to more general projective manifolds, such as Calabi--Yau manifolds, projective hypersurfaces, and projective hyperkahler manifolds.

As direct applications, we will prove some properties for the period map on the moduli space of Calabi-Yau type manifolds; we will also prove a general result about the period map to be biholomorphic from the Hodge metric completion space of the Teichmuller space of Calabi-Yau type manifolds to their period domains, and apply it to the cases of K3 surfaces and cubic fourfolds. Moreover, we will discuss some special cases when the period domain has the same dimension as the Teichmuller space and has a natural affine structure.