2020/11/24 by Liliana Alanís-López, Enrique Artal Bartolo, Alanís-López, Lilia +10
Mathematics · #32S05 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2011.12332
openalex publication_date 2020/11/24 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
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. 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.