Celebratio Mathematica — McDuff

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D. McDuff: “Lectures on Gromov invariants for symplectic 4-manifolds,” pp. 175–210 in Gauge theory and symplectic geometry (Montreal, 3–14 July 1995). Edited by J. Hurtubise and F. Lalonde. NATO ASI Series C: Mathematical and Physical Sciences 488. Kluwer (Dordrecht), 1997. Based on notes taken by Wladyslav Lorek. MR 1461573 Zbl 0881.57037 incollection

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: see, for example, [McDuff and Salamon 1994; Ruan and Tian 1995, 1996]. 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.

Wladyslav Lorek	Related
François Lalonde	Related
Jacques Hurtubise	Related