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Invariance of quantum rings under ordinary flops. III: A quantum splitting principle. (English) Zbl 1365.14076

The paper under review is the authors’ ongoing efforts to establish the \(K\)-equivalence conjecture [C.-L. Wang, J. Algebr. Geom. 12, No. 2, 285–306 (2003; Zbl 1080.14510)] for general ordinary flops, which is an important case of the crepant transformation conjecture.
A flop is a birational surgery on a smooth algebraic variety which modifies a small part of the variety called the exceptional locus. The exceptional locus of an ordinary flop is in general a projectivised non-split vector bundles over an arbitrary base. For a simple flop where the base is a point, the \(K\)-equivalence conjecture has been proved in [Y.-P. Lee et al., Ann. Math. (2) 172, No. 1, 243–290 (2010; Zbl 1272.14040)] in the genus zero case, and in [Y. Iwao et al., J. Reine Angew. Math. 663, 67–90 (2012; Zbl 1260.14068)] for all genera.
The background and some recent development on the \(K\)-equivalence relation among birational manifolds has been surveyed in [C.-L. Wang, in: Second international congress of Chinese mathematicians. Proceedings of the congress (ICCM2001), Taipei, Taiwan, December 17–22, 2001. Somerville: International Press. 199–216 (2004; Zbl 1328.14022)].
The current paper is a continuation of Y.-P. Lee et al. [“Invariance of quantum rings under ordinary flops. I: Quantum corrections and reduction to local models”, Preprint, arXiv:1109.5540; “Invariance of quantum rings under ordinary flops. II: A quantum Leray-Hirsch theorem”, Preprint, arXiv:1311.5725] where the authors proved that a general ordinary flop over an smooth base induces an isomorphism of big quantum rings in the genus zero case. In the first two papers, the authors showed that the general case can be reduced to the case of the standard local models for such flops; they then verified the case of split vector bundles for the local models. The major effort of the third paper is to further reduce the non-split case of local models to the split case.
To prove the reduction, the authors observed that after a sequence of blow-ups of the base of a given local model, the vector bundle can be deformed to a split one. Since Gromov-Witten invariants are deformation invariant, the challenge is to analyze the invariants under blow-ups of the base. To relate these invariants of the blow-ups, the authors employed similar ideas as in [D. Maulik and R. Pandharipande, Topology 45, No. 5, 887–918 (2006; Zbl 1112.14065)], and applied the degeneration formula for Gromov-Witten invariants [J. Li, J. Differ. Geom. 60, No. 2, 199–293 (2002; Zbl 1063.14069)] and [A.-M. Li and Y. Ruan, Invent. Math. 145, No. 1, 151–218 (2001; Zbl 1062.53073)].

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
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