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Donaldson-Floer invariants and singularities. (English) Zbl 0936.57016

Lanteri, A. (ed.) et al., Geometry of complex projective varieties. Proceedings of the conference, Cetraro, Italy, May 28-June 2, 1990. Rende: Mediterranean Press, Semin. Conf. 9, 203-217 (1993).
This note is a slightly expanded abstract of my lecture delivered at the Cetraro Conference “Geometry of Complex Projective Varieties”. In this talk I reported on a joint work with Wolfgang Ebeling which in the meantime appeared under the title “Donaldson invariants, monodromy groups and singularities” [Int. J. Math. 1, No. 3, 233-250 (1990; Zbl 0783.57006)].
In his seminal paper S. K. Donaldson [Topology 29, No. 3, 257-315 (1990; Zbl 0715.57007)] has introduced a series of invariants for certain closed, differentiable 4-manifolds. These invariants, which are polynomials on the second homology group of the manifold with values in the integers, are very powerful tools for distinguishing between distinct \(C^\infty\)-structures on a fixed topological 4-manifold. In the integrable case, i.e. for manifolds underlying algebraic surfaces, techniques from algebraic geometry can be used to calculate Donaldson’s polynomials (in good cases) [S. K. Donaldson, loc. cit.; R. Friedman and J. W. Morgan, Topology Appl. 32, No. 2, 135-139 (1989; Zbl 0694.14013); the author with A. Van de Ven, Several complex variables, VI: Complex manifolds, Encycl. Math. Sci. 69, 197-249 (1990); Russian translation in Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 69, 222-277 (1991; Zbl 0752.14013)].
Forthcoming work of Donaldson will generalize this to the relative case, and will lead to invariants for certain 4-manifolds with boundary a \(\mathbb{Z}\)-homology sphere. Again these invariants are polynomials on the second homology group, but this time take their values in the Floer-homology [A. Floer, Commun. Math. Phys. 118, No. 2, 215-240 (1988; Zbl 0684.53027)] of the boundary.
In his H. Weyl lectures M. Atiyah [Proc. Symp. Pure Math. 48, 285-299 (1988; Zbl 0667.57018)] announced this work and posed the problem to find a method to compute the relative invariants. In our paper [Ebeling and Okonek, loc. cit.] we contribute to this problem again in an essentially algebraic situation. We describe the structure of the Donaldson-Floer invariants for Milnor fibers of certain isolated surface singularities and obtain – by using a natural compactification in the weighted homogeneous case – strong non-triviality results.
For the entire collection see [Zbl 0930.00040].

MSC:

57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
14J17 Singularities of surfaces or higher-dimensional varieties
57R55 Differentiable structures in differential topology
57R57 Applications of global analysis to structures on manifolds
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