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

Abstract

Listen

Translate article

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 Q ̃ obtained after killing ker N is positive definite if all the screw numbers associated with certain orbits of annuli are positive. We also prove that the restriction of Q ̃ 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 Q ̃ is computable in terms of the dual resolution or semistable reduction graph, as illustrated with several examples. Numerical invariants associated with Q ̃ 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 Q ̃ to the absolute monodromy of Σ = F is not even.