Index theorems for couples of holomorphic self-maps

TL;DR

This paper develops index theorems for pairs of holomorphic self-maps on complex manifolds that agree on a hypersurface, using foliation and connection techniques to localize characteristic classes.

Contribution

It introduces a method to define holomorphic foliations and connections on hypersurfaces where two holomorphic maps coincide, enabling new index theorems in complex geometry.

Findings

Defined a holomorphic foliation on the hypersurface minus singularities.

Constructed partial holomorphic connections on vector bundles.

Derived index theorems by localizing characteristic classes.

Abstract

Let $M$ be a $n$-dimensional complex manifold and $f,g:M\to M$ two distinct holomorphic self-maps. Suppose that $f$ and $g$ coincide on a globally irreducible compact hypersurface $S\subset M$. We show that if one of the two maps is a local biholomorphism around $S'=S-\text{Sing}(S)$ and, if needed, $S'$ sits into $M$ in a particular nice way, then it is possible to define a $1$-dimensional holomorphic (possibly singular) foliation on $S'$ and partial holomorphic connections on certain holomorphic vector bundles on $S'$. As a consequence, we are able to localize suitable characteristic classes and thus to get index theorems.