Flops, Type III contractions and Gromov-Witten invariants on Calabi-Yau threefolds

Abstract

We investigate Gromov-Witten invariants associated to exceptional classes for primitive birational contractions on a Calabi-Yau threefold X. It was observed in a previous paper that these invariants are locally defined, in that they can be calculated from knowledge of an open neighbourhood of the exceptional locus of the contraction; in this paper, we make this explicit. For Type I contractions (i.e. only finitely many exceptional curves), a method is given for calculating the Gromov-Witten invariants, and these in turn yield explicit expressions for the changes in the cubic form $D^3$ and the linear form $D.c_2$ under the corresponding flop. For Type III contractions (when a divisor E is contracted to a smooth curve C of singularities), there are only two relevant Gromov-Witten numbers n(1) and n(2). Here n(2) is the number (suitably defined) of simple pseudo-holomorphic rational curves representing the class of a fibre of E over C, and n(1) the number of simple curves representing half this class. Explicit formulae for n(1) and n(2) are given (n(1) in terms of the singular fibres of E over C and n(2)=2g(C)-2). An easy proof of these formulae is provided when g(C)>0. The main part of the paper then gives a proof valid in general (including the case g(C)=0).