Analytic continuations of quantum cohomology

Abstract

Abstract. In the study of local model of ordinary flops, we introduce the Birkhoff factorizaton procedure to produce the generalized mirror map, which is the essential ingredient in performing analytic continuations of quantum cohomology in the Kähler moduli spaces. We survey this procedure in this article and provide an example to illustrate its validity. GW theoryThis article consists of the second author's talk at ICCM 2010 at Beijing under the same title, with more details filled in. The purpose is to introduce the results obtained by the authors in [3] and [4] and to work out a typical example.Let X be a projective manifold over C and M g,n (X, β) be the moduli space of n-pointed stable maps f : (C; p 1 , . . . , p n ) → X from a nodal curve C with arithmetic genus p a (C) = g, [f (C)] = β ∈ NE(X), the Mori cone of effective one cycles.Li-Tian and Behrend-Fantechi had constructed the virtual fundamental classAnd then the (descendent) Gromov-Witten invariants are defined by integration along these cycles: For a i ∈ H(X), The (primary) GW potential is the generating function of all GW invariants without descendent insertions (k i = 0). Let {T i } be a cohomology basis of H = H(X), {T i } be its dual basis, and t = t i T i ∈ H be a general element with t i being the coordinates. Let {q β }'s be the formal (Novikov) variables. Then