Geometry Seminar

DATE2017-04-26 15:10-16:00


SPEAKER杜武亮 教授(Tufts University

TITLEThe Lefschetz Fixed Point Theorem for Correspondences

ABSTRACT The classical Lefschetz fixed point theorem states that the number of fixed points, counted with multiplicity $\pm 1$, of a smooth map $f$ from a manifold $M$ to itself can be calculated as the alternating sum $\sum (-1)^q {\mathrm tr} f^*|_{H^q(M;\mathbb Q)}$ 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 what it said nor did he have any notes. This talk is an exploration of Shimura's forgotten conjecture, first a proof for a smooth correspondence, and then a conjectural statement for a holomorphic correspondence.