On the Multipliers at Fixed Points of Quadratic Self-Maps of the Projective Plane with an Invariant Line
Por:
Guillot, Adolfo, Ramirez, Valente
Publicada:
1 ene 2019
Resumen:
This paper deals with holomorphic self-maps of the complex projective plane and the algebraic relations among the eigenvalues of the derivatives at the fixed points. These eigenvalues are constrained by certain index theorems such as the holomorphic Lefschetz fixed-point theorem. A simple dimensional argument suggests there must exist even more algebraic relations that the ones currently known. In this work we analyze the case of quadratic self-maps having an invariant line and obtain all such relations. We also prove that a generic quadratic self-map with an invariant line is completely determined, up to linear equivalence, by the collection of these eigenvalues. Under the natural correspondence between quadratic rational maps of P2 and quadratic homogeneous vector fields on C3, the algebraic relations among multipliers translate to algebraic relations among the Kowalevski exponents of a vector field. As an application of our results, we describe the sets of integers that appear as the Kowalevski exponents of a class of quadratic homogeneous vector fields on C3 having exclusively single-valued solutions. © 2019, Springer-Verlag GmbH Germany, part of Springer Nature.
Filiaciones:
Guillot, Adolfo:
Instituto de Matematicas, Universidad Nacional Autónoma de México, Ciudad Universitaria, Mexico City, 04510, Mexico
Univ Nacl Autonoma Mexico, Inst Matemat, Ciudad Univ, Mexico City 04510, DF, Mexico
Ramirez, Valente:
Institut de Recherche Mathématique de Rennes, Université de Rennes 1, UMR 6625, Rennes, France
Univ Rennes 1, Inst Rech Math Rennes, UMR 6625, Rennes, France
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