On a quadratic form associated with a surface automorphism and its applications to Singularity Theory

Abstract

We study the nilpotent part N′ of a pseudo-periodic automorphism h of a real oriented surface with boundary Σ. We associate a quadratic form Q defined on the first homology group (relative to the boundary) of the surface Σ. Using the twist formula and techniques from mapping class group theory, we prove that the form Q̃ obtained after killing kerN is positive definite if all the screw numbers associated with certain orbits of annuli are positive. We also prove that the restriction of Q̃ to the absolute homology group of Σ is even whenever the quotient of the Nielsen–Thurston graph under the action of the automorphism is a tree. The case of monodromy automorphisms of Milnor fibers Σ=F of germs of curves on normal surface singularities is discussed in detail, and the aforementioned results are specialized to such situation. This form Q is determined by the Seifert form but can be much more easily computed. Moreover, the form Q̃ is computable in terms of the dual resolution or semistable reduction graph, as illustrated with several examples. Numerical invariants associated with Q̃ are able to distinguish plane curve singularities with different topological types but same spectral pairs. Finally, we discuss a generic linear germ defined on a superisolated surface. In this case the plumbing graph is not a tree and the restriction of Q̃ to the absolute monodromy of Σ=F is not even.