Lefschetz Fixed Point Theorems for Correspondences

Abstract

The classical Lefschetz fixed point theorem states that the number of fixed points, counted with multiplicity ± 1, of a smooth map f from a manifold M to itself can be calculated as the alternating sum $$\sum (-1)^k \operatorname {Tr} f^*|{ }_{H^k(M)}$$ of the trace of the induced homomorphism in cohomology. In 1964, at a conference in Woods Hole, Shimura conjectured a Lefschetz fixed point theorem for a holomorphic map, which Atiyah and Bott proved and generalized into a fixed point theorem for elliptic complexes. However, in Shimura’s recollection, he had conjectured more than the holomorphic Lefschetz fixed point theorem. He said he had made a conjecture for a holomorphic correspondence, but he could not remember the statement. This paper is an exploration of Shimura’s forgotten conjecture, first for a smooth correspondence, then for a holomorphic correspondence in the form of two conjectures and finally in the form of an open problem involving an extension to holomorphic vector bundles over two varieties and the calculation of the trace of a Hecke correspondence.