Invariance of Quantum Rings under Ordinary Flops

Details Invariance of Quantum Rings under Ordinary Flops Invariance of Quantum Rings under Ordinary Flops

2011-09-26

arxiv.org/abs/1109.5540v2

Mathematics

Algebraic Geometry

70 pages

Scientific paper

For ordinary flops over a smooth base, we determine the defect of the cup product under the canonical correspondence and show that it is corrected by the small quantum product attached to the extremal ray. If the flop is of splitting type, the big quantum cohomology ring is also shown to be invariant after an analytic continuation in the K\"ahler moduli space. Viewed from the context of the K-equivalence (crepant transformation) conjecture, there are two new features of our results. First, there is no semipositivity assumption on the varieties. Second, the local structure of the exceptional loci can not be deformed to any explicit (e.g. toric) geometry and the analytic continuation is nontrivial. This excludes the possibility of an ad hoc comparison by explicit computation of both sides. To achieve that, we have to clear a few technical hurdles. One technical breakthrough is a quantum Leray--Hirsch theorem for the local models (a certain toric bundle) which extends the quantum D modules of Dubrovin connection on the base by a Picard--Fuchs system of the toric fibers. Nonsplit flops as well as further applications of the quantum Leray--Hirsch theorem will be discussed in subsequent papers.

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