$L^2$ Fr\"olicher inequalities

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 $(D_h)^2:=(\overline\partial+\overline\partial^*+h\partial+h\partial^*)^2$ 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.