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arXiv · 2006.09907

Extremal transitions via quantum Serre duality

Rongxiao Mi·Mark Shoemaker
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Abstract

Two varieties $Z$ and $\widetilde Z$ are said to be related by extremal transition if there exists a degeneration from $Z$ to a singular variety $\overline Z$ and a crepant resolution $\widetilde Z \to \overline Z$. In this paper we compare the genus-zero Gromov--Witten theory of toric hypersurfaces related by extremal transitions arising from toric blow-up. We show that the quantum $D$-module of $\widetilde Z$, after analytic continuation and restriction of a parameter, recovers the quantum $D$-module of $Z$. The proof provides a geometric explanation for both the analytic continuation and restriction parameter appearing in the theorem.

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Rongxiao Mi, Mark Shoemaker. 2020-06-17. Extremal transitions via quantum Serre duality. https://arxiv.org/abs/2006.09907

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