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Compactification of the moduli of polarized abelian varieties and mirror symmetry


Author: Yuecheng Zhu
Journal: Trans. Amer. Math. Soc. 370 (2018), 1693-1758
MSC (2010): Primary 14K10
DOI: https://doi.org/10.1090/tran/7008
Published electronically: October 16, 2017
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Abstract: We show that Martin Olsson's compactification of moduli space of polarized abelian varieties can be interpreted in terms of KSBA stable pairs. We find that any degenerating family of polarized abelic sheme over a local normal base is equipped with a canonical set of divisors $ S(K_2)$. Choosing any divisor $ \Theta $ from the set $ S(K_2)$, we get a KSBA stable pair. Then the limit in the moduli space of KSBA pairs $ \overline {\mathscr {AP}}_{g,d}$ agrees with the canonical degeneration given by Martin Olsson's compactification. Moreover, we give an alternative construction of the compactification by using mirror symmetry. We construct a toroidal compactification $ \overline {\mathscr {A}}_{g,\delta }^m$ that is isomorphic to Olsson's compactification over characteristic zero. The collection of fans needed for a toroidal compactification is obtained from the Mori fans of the minimal models of the mirror families.


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Additional Information

Yuecheng Zhu
Affiliation: Department of Mathematics, the University of Texas at Austin, 2515 Speedway Stop C1200, Austin, Texas 78712
Email: yuechengzhu@math.utexas.edu

DOI: https://doi.org/10.1090/tran/7008
Received by editor(s): June 3, 2015
Received by editor(s) in revised form: May 27, 2016
Published electronically: October 16, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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