Comparing Absolute and relative Gromov--Witten invariants

Abstract

This note compares the usual (absolute) Gromov-Witten invariants of a symplectic manifold with the invariants that count the curves relative to a (symplectic) divisor D. We give explicit examples where these invariants differ even though it seems at first that they should agree, for example when counting genus zero curves in a class \be such that \be\cdot D=0. The main tool is the decomposition formula in the form developed by A. Li--Ruan.