Localized intersection of currents and the Lefschetz coincidence point theorem

TL;DR

This paper develops a theory of localized intersection of currents and applies it to derive a generalized Lefschetz coincidence point theorem, providing explicit formulas for indices and classes in complex geometric settings.

Contribution

It introduces the notion of Thom class of a current and defines localized intersections, extending classical coincidence theory with explicit residue formulas and a generalized Lefschetz theorem.

Findings

Derived explicit formulas for residues in localized intersections.

Defined global and local coincidence homology classes and indices.

Established a general coincidence point theorem extending Lefschetz's classical result.

Abstract

We introduce the notion of a Thom class of a current and define the localized intersection of currents. In particular we consider the situation where we have a smooth map of manifolds and study localized intersections of the source manifold and currents on the target manifold. We then obtain a residue theorem on the source manifold and give explicit formulas for the residues in some cases. These are applied to the problem of coincidence points of two maps. We define the global and local coincidence homology classes and indices. A representation of the Thom class of the graph as a Cech-de~Rham cocycle immediately gives us an explicit expression of the index at an isolated coincidence point, which in turn gives explicit coincidence classes in some non-isolated components. Combining these, we have a general coincidence point theorem including the one by S. Lefschetz.