Extremal transitions via quantum Serre duality

Abstract

Two varieties $Z$ and $\tilde{Z}$ 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 $\tilde{Z}\to \overline{Z}$. 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 $\tilde{Z}$, 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.

53 pages, comments welcome