Coherent sheaves, Chern character, and RRG (Zoom)

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.

• 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.
• 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.