A Recursive Formula for Log Gromov-Witten Invariants

Given a smooth variety X containing a smooth divisor Y, the relative Gromov-Witten invariants of (X,Y) are defined as certain counts of algebraic curves in X with specified orders of tangency to Y. Their intrinsic interest aside, they are an important part of any Gromov-Witten theorist’s toolkit, thanks to their role in the celebrated “degeneration formula.” In recent years these invariants have been significantly generalised, using techniques in logarithmic geometry. The resulting “log Gromov-Witten invariants” are defined for a large class of targets, and in particular give a rigorous definition of relative invariants for (X,D) where D is a normal crossings divisor. Besides being more general, these numbers are  intimately related to constructions in Mirror Symmetry, via the Gross-Siebert program. In this talk, we will describe a recursive formula for computing the invariants of (X,D) in genus zero. The result relies on a comparison theorem which expresses the log Gromov-Witten invariants as classical (i.e. non log-geometric) objects.