Pure and Applied Mathematics Quarterly

Volume 18 (2022)

Number 1

Special Issue in Honor of Bernie Shiffman

Guest Editors: Yuan Yuan, Christopher Sogge, and Steven Morris Zelditch

On Calabi–Yau fractional complete intersections

Pages: 317 – 342

DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n1.a10


Tsung-Ju Lee (Center of Mathematical Sciences and Applications, Harvard University, Cambridge, Massachusetts, U.S.A.)

Bong H. Lian (Brandeis University, Waltham, Massachusetts, U.S.A.)

Shing-Tung Yau (Department of Mathematics,, Harvard University, Cambridge, Massachusetts, U.S.A.; and Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)


In this article, we study mirror symmetry for pairs of singular Calabi–Yau varieties which are double covers of toric manifolds. Their period integrals can be seen as certain ‘fractional’ analogues of those of ordinary complete intersections. This new structure can then be used to solve their Riemann–Hilbert problems. The latter can then be used to answer definitively questions about mirror symmetry for this class of Calabi–Yau varieties.


Calabi–Yau, mirror symmetry, fractional complete intersections

2010 Mathematics Subject Classification

14D07, 32G20

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Received 24 June 2020

Accepted 29 September 2020

Published 10 February 2022