Event description
Abstract: The Frölicher inequalities, established by Frölicher in 1955, state that all the k-Betti numbers are bounded from above by the sum of the (p,q)-Hodge numbers for p+q=k. In this talk we will sketch how to prove new generalised Frölicher inequalities between L^2 Betti numbers and L^2 Hodge numbers on normal coverings of compact complex manifolds. The new proof uses analytical tools to replace the algebraic ones in the original proof by Frölicher. In particular, we replace spectral sequences with the use of spectral projectors of a family of self-adjoint operators joining the Hodge-de Rham Laplacian to the Dolbeault Laplacian. Thanks to this new proof, we are able to construct an injection between de Rham cohomology and Dolbeault cohomology on compact complex manifolds. Finally, we will give a brief overview of some open problems which might be approached with similar techniques.