Index theorems for holomorphic self-maps

Abstract

We prove several index theorems for holomorphic self-maps having positive-dimensional fixed points set. To do so we show that the fixed points set of a holomorphic self-map has a surprisingly rich structure, expressed by canonically defined meromorphic connections and bundle maps. Finally, we present some applications to holomorphic dynamics

Discrete Holomorphic Local Dynamical Systems

“…The g j 's in (6.5) depend in general on the chosen chart; however, in [ABT1] we proved that setting…”

“…Remark 6.8: It is easy to check that f is tangential if and only if the image of X f is contained in T E. Furthermore, if f is the lifting of a germ f o ∈ End(C n , O) tangent to the identity, then (see [ABT1]) f is tangential if and only if f o is non-dicritical; so in this case tangentiality is generic. Finally, in [A2] we used the term "non degenerate" instead of "tangential".…”

“…Indeed, a not too difficult computation (see [A2], [AT1] and [ABT1]) yields the following Proposition 6.6: Let f ∈ End(M, E) be tangential, and take p ∈ E. If p is not singular, then the stable set of the germ of f at p coincides with E.…”

“…Then in [A2] we proved the following index theorem (see [Br1], [BrT] and [ABT1,2] for multidimensional versions and far reaching generalizations): Theorem 6.7: ([A2], [ABT1]) Let E be a compact Riemann surface inside a 2-dimensional complex manifold M . Take f ∈ End(M, E), and assume that f is tangential to E. Then…”

“…Then in [ABT1] we showed that it is possible to define a holomorphic connection ∇ on N E o by setting…”