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

Alanís-López, L. ; Artal, E. (Universidad de Zaragoza) ; Bonatti, C. ; Gómez-Mont, X. ; González Villa, M. ; Portilla Cuadrado, P.

Resumen: We study the nilpotent part N' of a pseudo-periodic automorphism h of a real oriented surface with boundary S. We associate a quadratic form Q defined on the first homology group (relative to the boundary) of the surface S. Using the twist formula and techniques from mapping class group theory, we prove that the form Q~ 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 Q~ to the absolute homology group of S 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 S=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 S=F is not even. © 2022 Royal Dutch Mathematical Society (KWG)

Idioma: Inglés

DOI: 10.1016/j.indag.2022.02.007

Año: 2022

Publicado en: INDAGATIONES MATHEMATICAE-NEW SERIES 33, 4 (2022), 816-843
ISSN: 0019-3577

Factor impacto JCR: 0.6 (2022)

Categ. JCR: MATHEMATICS rank: 243 / 329 = 0.739 (2022) - Q3 - T3

Factor impacto CITESCORE: 1.3 - Mathematics (Q3)

Factor impacto SCIMAGO: 0.445 - Mathematics (miscellaneous) (Q2)

Financiación: info:eu-repo/grantAgreement/ES/DGA/E22-20R
Financiación: info:eu-repo/grantAgreement/ES/MICINN/PID2020-114750GB-C31/AEI/10.13039/501100011033

Tipo y forma: Article (PrePrint)

Área (Departamento): Área Geometría y Topología (Dpto. Matemáticas)