Geometric Analysis and Topology Seminar

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Coherent sheaves, Chern character, and RRG

Speaker: Jean-Michel Bismut, Université Paris-Saclay

Location: Online

Videoconference link: https://nyu.zoom.us/j/95759976381

Date: Friday, October 14, 2022, 11 a.m.

Synopsis:

Let X be a compact complex manifold. On X, one can consider holomorphic vector bundles, and also coherent sheaves. When X is projective, the corresponding Grothendieck groups coincide.

When X is non-projective, a result of Voisin shows that in general, coherent sheaves may not have finite locally free resolutions.

In our talk, we will focus on two results.

1. The construction of a Chern character for coherent sheaves with values in Bott-Chern cohomology, which strictly refines on de Rham cohomology. This will be done using a fundamental construction of Block.
2. The proof of a Riemann-Roch-Grothendieck formula for direct images of coherent sheaves. It relies in particular on the theory of the hypoelliptic Laplacian.

Our results refine on earlier work by Levy, Toledo-Tong, and Grivaux.

This is joint work with Shu SHEN and Zhaoting WEI, available in https://arxiv.org/abs/2102.08129.