10 dec 2020 -- 14:30
DIMaI, Firenze (online)
Seminario di Geometria Differenziale e Analisi Complessa del Dipartimento di Matematica e Informatica "Ulisse Dini" dell'Università di Firenze
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 $X$.