Index theorems for holomorphic self-maps
By Marco Abate, Filippo Bracci, F. Tovena. 2004
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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.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/dc58c7d73767e23358866e94b660b5608d638cb5
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In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Discrete holomorphic local dynamical systems (2009, background): “The blow-up procedure reduces the study of the dynamics of local holomorphic dynamical systems tangent to the identity to the study of the dynamics of germs f ∈ End(M, E) ,w hereM is a complex n-dimensional manifold, and E ⊂ M is a compact smooth complex hypersurface pointwise fixed by f .I n [A2, B”