Extremal transitions via quantum Serre duality. (English) Zbl 1529.81047
Summary: 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.
MSC:
| 81P68 | Quantum computation |
| 68Q85 | Models and methods for concurrent and distributed computing (process algebras, bisimulation, transition nets, etc.) |
| 32S35 | Mixed Hodge theory of singular varieties (complex-analytic aspects) |
| 14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |
| 35B44 | Blow-up in context of PDEs |
| 32D15 | Continuation of analytic objects in several complex variables |
| 52A05 | Convex sets without dimension restrictions (aspects of convex geometry) |
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