Extremal transitions via quantum Serre duality

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 $${\widetilde{Z}} \rightarrow {\overline{Z}}$$ . 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.