McDuff, Dusa Lectures on Gromov invariants for symplectic \(4\)-manifolds. (English) Zbl 0881.57037 Hurtubise, Jacques (ed.) et al., Gauge theory and symplectic geometry. Proceedings of the NATO Advanced Study Institute and séminaire de mathématiques supérieures, Montréal, Canada, July 3–14, 1995. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 488, 175-210 (1997). The article under review is an expanded written version of 5 lectures given by the author at the NATO Advanced Study Institute seminar on gauge theory and symplectic geometry in 1995. The main topic of the lectures and article are the results due to C. Taubes on counting \(J\)-holomorphic curves in defining the Gromov invariant Gr\((A).\) After discussing some motivating examples in the first lecture, Lecture 2 is devoted to giving an expository account of the following result due to C. H. Taubes, which later appeared in [J. Am. Math. Soc. 9, No. 3, 845-918 (1996; Zbl 0867.53025)]: Suppose \(J\) is a generic almost-complex structure on the symplectic 4-manifold \(M,\) and \(A\in H_2(M,Z) \) a homology class. (i) Suppose the moduli space \(\mathcal H(A)\) contains a good pair. Then every pair \((\phi,\Sigma)\in\mathcal H(A)\) is good. Moreover: (a) For every component \(\Sigma_i,\phi(\Sigma_i)\) is an embedded curve, disjoint from all other curves \(\phi(\Sigma_j)\). (b) The multiplicity \(m_i\) of \(\phi|\Sigma_i\) is one, unless the genus \(g(\Sigma_i)=1\) and \(\phi(\Sigma_i)\) has zero self-intersection. (c) The moduli space \(\mathcal H(A) \) is 0-dimensional, and finite. (ii) If \((\phi,\Sigma)\in\mathcal H(A)\), then the image \(\phi(\Sigma_i)\) of every \(\Sigma_i\) such that \(\phi(\Sigma_i)\cdot\phi(\Sigma_i)<0\) is an embedded exceptional sphere. However, the multiplicity may be \(> 1\). In this statement \(\mathcal H(A)\) denotes the moduli space of \(J\)-holomorphic curves on the symplectic manifold \((M,\omega)\) with a compatible almost complex structure \(J\) which contains the set \(\Omega_k\) of \(k\) generic points, where \(k=k(A)=(c_1(A)+A\cdot A)/2\). Representatives of elements in this moduli space are pairs \((\phi,\Sigma)\) where \(\Sigma=\coprod\Sigma_i\) is a possibly disconnected Riemann surface, \([\phi_*(\Sigma)]= A\) is a two dimensional integral homology class, and \(\phi\) is a \(J\)-holomorphic map. Lecture 3 concerns the two problems of dealing with multiply covered exceptional curves and understanding how one knows all elements of \(\mathcal H(A)\) have been found. A modified definition \(Gr'(A)\) of the Gromov invariant is proposed to deal with the first problem. Lecture 4 deals with defining a version \(Gr_s(A)\) which just counts spheres. Lecture 5 discusses the calculation of \(Gr(A)\) when \(A\) is represented by tori, as well as giving examples illustrating the difficulties inherent in counting multiply covered tori. We note that this paper was written as much of the work of Taubes on the relation of the Gromov invariant to the Seiberg-Witten invariant was first appearing in preprint form, and the reader should consult the final form of papers by C. H. Taubes [J. Differ. Geom. 44, No. 4, 818-893 (1996); Lect. Notes Pure Appl. Math. 184, 591-601 (1997; Zbl 0873.57017)] in addition to the one cited above.For the entire collection see [Zbl 0861.00016]. Reviewer: T.Lawson (New Orleans) Cited in 4 Documents MSC: 57R57 Applications of global analysis to structures on manifolds 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.) 57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) Keywords:four-manifolds; symplectic; gauge theory; Gromov invariant Citations:Zbl 0867.53025; Zbl 0873.57017 PDFBibTeX XMLCite \textit{D. McDuff}, NATO ASI Ser., Ser. C, Math. Phys. Sci. 488, 175--210 (1997; Zbl 0881.57037) Full Text: arXiv