Another proof of the persistence of Serre symmetry in the Frölicher spectral sequence

Source abstract

Abstract Serre’s duality theorem implies a symmetry between the Hodge numbers, h p , q = h n − p , n − q , on a compact complex n –manifold. Equivalently, the first page of the associated Frölicher spectral sequence satisfies \dim E_1^{p,q} = \dim E_1^{n - p,n - q} for all p , q . Adapting an argument of Chern, Hirzebruch, and Serre [3] in an obvious way, in this short note we observe that this “Serre symmetry” \dim E_k^{p,q} = \dim E_k^{n - p,n - q} holds on all subsequent pages of the spectral sequence as well. The argument shows that an analogous statement holds for the Frölicher spectral sequence of an almost complex structure on a nilpotent real Lie group as considered by Cirici and Wilson in [4].