$L^2$ Frölicher inequalities

Abstract

We prove a Frölicher inequality between $L^2$ Betti and $L^2$ Hodge numbers on normal coverings of compact complex manifolds. This is achieved by building an injection using suitable spectral projectors associated to the self-adjoint operators $(D_h{)}^2:=(\overline{\partial }+{\overline{\partial }}^{\ast }+h\partial +h{\partial }^{\ast }{)}^2$ for $h\in [0,1]$. With similar techniques, we show that the positivity of the spectrum of the Dolbeault Laplacian implies the positivity of the spectrum of the Hodge Laplacian; moreover, if equality holds in the $L^2$ Frölicher inequality, then we can replace "positivity of the spectrum" with "spectral gap at 0" in the previous statement. As a by-product, in the case of compact complex manifolds, we find a new proof of the classical Frölicher inequality which does not rely at all on spectral sequences and build an explicit injection from de Rham to Dolbeault cohomology.

20 pages, new results added in section 4, comments are welcome!