Lectures on Gromov invariants for symplectic 4-manifolds

These are notes of lectures given at the NATO Summer School, Montreal 1995. Taubes's recent spectacular work setting up a correspondence between $J$-holomorphic curves in symplectic 4-manifolds and solutions of the Seiberg-Witten equations counts $J$-holomorphic curves in a somewhat new way. The "standard" theory concerns itself with moduli spaces of connected curves, and gives rise to Gromov-Witten invariants. However, Taubes's curves arise as zero sets of sections and so need not be connected. These notes are in the main expository. We first discuss the invariants as Taubes defined them, and then discuss some alternatives, showing, for example, a way of dealing with multiply-covered exceptional spheres. We also calculate some examples, in particular finding the Gromov invariant of the fiber class of an elliptic surface by counting $J$-holomorphic curves, rather than going via Seiberg--Witten theory.