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Birational Calabi-Yau manifolds have the same small quantum products. (English) Zbl 1436.14092

V. V. Batyrev [Lond. Math. Soc. Lect. Note Ser. 264, 1–11 (1999; Zbl 0955.14028)] showed that birationally equivalent Calabi-Yau (CY) manifolds have equal Betti numbers, and it follows from work of several authors that their integral cohomology groups are isomorphic. While the cup product structures may differ, A.-M. Li and Y. Ruan [Invent. Math. 145, No. 1, 151–218 (2001; Zbl 1062.53073)] proved that small quantum cohomology (QC) rings are isomorphic for CY \(3\)-folds. Even big QC rings were proven isomorphic, but only under ordinary flops.
In this paper the author proves the isomorphism of small QC rings under a change of Novikov ring. Unlike previous proofs, that rely on a degeneration argument and the quantum Leray-Hirsch theorem, this one uses Hamiltonian Floer cohomology, which is known to be isomorphic to small QC. The proof works in greater generality and “explains” why the result is true, but, as a tradeoff, the isomorphism is not explicit. As a corollary, the author gives a new proof of Batyrev’s result.
The idea of proof is inspired by Borman and Sheridan. The Hamiltonians on the two manifolds are chosen constant outside of a “large” compact set \(K\) that sits inside their isomorphic Zariski dense affine open subsets, and the Kähler forms are modified to agree near \(K\). Ignoring \(1\)-periodic orbits outside of \(K\) makes the Floer groups “identical”, but, alas, they are no longer isomorphic to the small QC groups. The fix is to use sequences of Hamiltonians that tend to infinity outside of \(K\) to define “symplectic cohomology” that restores the isomorphism.

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
53D40 Symplectic aspects of Floer homology and cohomology
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