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On a quadratic form associated with a surface automorphism and its applications to singularity theory. (English) Zbl 1495.14003

The authors study the nilpotent part of a pseudo-periodic automorphism \(h\) of a real oriented surface \(\Sigma\) with boundary \(\partial \Sigma\). Let \(e\) be the least common multiple of the orders of \(h\) restricted to each periodic piece. Consider the operator \(N\colon H_1(\Sigma,\partial \Sigma;\mathbb Z)\to H_1(\Sigma;\mathbb Z)\) defined by \(N([\gamma])=[h^e(\gamma)-\gamma]\). The quadratic form \(Q(v,w)=\langle Nv,w\rangle\) descends to a quadratic form \(\widetilde Q\) on \(H_1(\Sigma,\partial \Sigma;\mathbb Z)/\ker N\). A formula for \(Q\) is given, involving the so called screw numbers measuring the amount of rotation in the collection of annuli in the Nielsen-Thurston decomposition.
The bilinear form \(\widetilde Q\) is positive definite if all the screw numbers associated to orbits of annuli whose core curves are non-nullhomotopic are positive. This condition is satisfied if \(h\) is the geometric monodromy of a reduced function \(f \colon (X, 0) \to (\mathbb C, 0) \) on a normal surface singularity. The screw numbers can then be computed in terms of the embedded resolution.
Explicit computations and examples are given. Numerical invariants associated to \(\widetilde Q\) are able to distinguish the examples of pairs of plane curve singularities with different topological type but same spectral pairs, due to R. Schrauwen et al. [Proc. Symp. Pure Math. 53, 305–328 (1991; Zbl 0749.14003)]. The form \(\widetilde Q\) is also computed for the two infinite families of reducible plane singularities that are not topologically equivalent but have the same Seifert form given by P. Du Bois and F. Michel [J. Algebr. Geom. 3, No. 1, 1–38 (1994; Zbl 0810.32005)]. In that case \(\widetilde Q\) is always defined on an abelian group of rank 4; while this form is a weaker invariant than the Seifert form it is also much easier to compute.

MSC:

14B05 Singularities in algebraic geometry
32S25 Complex surface and hypersurface singularities
14J17 Singularities of surfaces or higher-dimensional varieties
32S05 Local complex singularities
14J50 Automorphisms of surfaces and higher-dimensional varieties
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