Analytic continuations of quantum cohomology (Q2900289)

This article provides a survey of the authors' work on the relationship between birational geometry and Gromov-Witten theory. For an ordinary flop \(f: X \relbar \to X'\), the graph closure induces an isomorphism of the cohomology groups \(H^*(X) \to H^*(X')\), but this isomorphism does not generally preserve the ring structure. Based on a conjecture of Y. Ruan and C.-L. Wang, one expects an isomorphism of rings for quantum cohomology after analytic continuation of the Kähler parameters, and this has been established in various cases. The authors summarize the four main steps in their approach to proving invariance of the big quantum cohomology ring, and they include quantum corrections, divisorial reconstruction, Birkoff factorization and the generalized mirror transformation, and analytic continuation. The last two sections of the paper are devoted to a worked example which helps to illustrate the main ideas at work.NEWLINENEWLINEFor the entire collection see [Zbl 1235.00045].