Advances in Theoretical and Mathematical Physics

Volume 20 (2016)

Number 2

Applications of affine structures to Calabi–Yau moduli spaces

Pages: 313 – 349



Kefeng Liu (School of Mathematics, Capital Normal University, Beijing, China; and Department of Mathematics, University of California at Los Angeles)

Yang Shen (Center of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang, China)


In this paper, we review our recent results and the methods of proofs in [15], in which it is proved that the Hodge metric completion of the moduli space of polarized and marked Calabi–Yau manifolds, i.e. the Torelli space, is a complex affine manifold. As applications it is proved that the period map from the Torelli space and the extended period map from its completion space, both are injective into the period domain, and that the period map from the moduli space of polarized Calabi–Yau manifolds with level $m$ structure is also injective.

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Published 16 August 2016