Let $M$ be a complex manifold and $S\subset M$ a (possibly singular) subvariety of $M$. Let $f\colon M\to M$ be a holomorphic map such that $f$ restricted to $S$ is the identity. We show that one can associate to $f$ a holomorphic section $X_f$ of a sheaf related to the embedding of $S$ in $M$ and that such a section reads the dynamical behavior of $f$ along $S$. In particular we prove that under generic hypotheses the canonical section $X_f$ induces a holomorphic action in the sense of Bott on the normal bundle of (the regular part of) $S$ in $M$ and this allows to obtain for holomorphic self-maps with non- isolated fixed points index theorems similar to Camacho-Sad, Baum-Bott and variation index theorems for holomorphic foliations. Finally we apply our index theorems to obtain information about topology and dynamics of holomorphic self-maps of surfaces with a compact curve of fixed points.

Abate, M., Bracci, F., Tovena, F. (2004). Index theorems for holomorphic self-maps. ANNALS OF MATHEMATICS, 159(2), 819-864.

### Index theorems for holomorphic self-maps

#### Abstract

Let $M$ be a complex manifold and $S\subset M$ a (possibly singular) subvariety of $M$. Let $f\colon M\to M$ be a holomorphic map such that $f$ restricted to $S$ is the identity. We show that one can associate to $f$ a holomorphic section $X_f$ of a sheaf related to the embedding of $S$ in $M$ and that such a section reads the dynamical behavior of $f$ along $S$. In particular we prove that under generic hypotheses the canonical section $X_f$ induces a holomorphic action in the sense of Bott on the normal bundle of (the regular part of) $S$ in $M$ and this allows to obtain for holomorphic self-maps with non- isolated fixed points index theorems similar to Camacho-Sad, Baum-Bott and variation index theorems for holomorphic foliations. Finally we apply our index theorems to obtain information about topology and dynamics of holomorphic self-maps of surfaces with a compact curve of fixed points.
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Rilevanza internazionale
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Sì, ma tipo non specificato
Settore MAT/03 - Geometria
English
Con Impact Factor ISI
VECTOR-FIELDS; FIXED-POINTS; VARIETIES; RESIDUES; IDENTITY; TANGENT
Abate, M., Bracci, F., Tovena, F. (2004). Index theorems for holomorphic self-maps. ANNALS OF MATHEMATICS, 159(2), 819-864.
Abate, M; Bracci, F; Tovena, F
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/31008