On the spectral theory and analytic K-homology of complex spaces

Abstract: Let $(X, h)$ be a compact and irreducible Hermitian complex space. In the last thirty years, motivated among other things by the Cheeger-Goresky-MacPherson conjecture and the Riemann-Roch theorem of Baum-Fulton-MacPherson, the $L^2$- theory of the Hodge-de Rham operator $d + d^t$, the Hodge-Dolbeault operator $\overline\partial + \overline\partial^t$ and the associated Laplacians on $(X, h)$ has been the subject of many investigations. In the first part of this talk we will report about some recent results concerning the existence of self-adjoint extensions of the Hodge- Kodaira Laplacian with entirely discrete spectrum. Then in the second part we will describe some applications to the K-homology of X. In particular assuming $\dim(sing(X)) = 0$ we will show how the operator $\overline\partial+\overline\partial^t$ induces an analytic K-homology class in $K^{an}_{0}(X)$ and we will give a geometric interpretation of this class in terms of a resolution of $X$.

Mathematics

Audience: researchers in the topic

Geometry Seminar - University of Florence

Series comments: If you are interested in attending, please send a message to daniele.angella@unifi.it or francesco.pediconi@unifi.it.