L² Frölicher inequalities

2025/07/25 by Bei, Francesco, Piovani, Riccardo
#32Q55 #58J10 #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2507.19385

Abstract

We prove a Frölicher inequality between L² Betti and L² 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:=(∂+∂^*+h∂+h∂^*)² for h∈[0,1]. As a by-product, we find a new proof of the classical Frölicher inequality on compact complex manifolds which does not rely at all on spectral sequences.

Citations

• The L² Aeppli-Bott-Chern Hilbert complex
• Von Neumann dimension, Hodge index theorem and geometric applications
• Autour de la cohomologie de Bott-Chern

Related

• L2 to Lp bounds for spectral projectors on thin intervals in Riemannian manifolds
• Amenable covers, volume and L2-Betti numbers of aspherical manifolds
• Von Neumann dimension, Hodge index theorem and geometric applications
• Positivity in the shadow of Hodge index theorem
• Discretization of Riemannian manifolds applied to the Hodge Laplacian