Adiabatic limit and the Frölicher spectral sequence

Dan Popovici

Vol. 300 (2019), No. 1, 121–158

DOI: 10.2140/pjm.2019.300.121

Abstract

Motivated by our conjecture of an earlier work predicting the degeneration at the second page of the Frölicher spectral sequence of any compact complex manifold supporting an SKT metric $\omega$ (i.e., such that $\partial \bar{\partial }\omega =0$), we prove degeneration at $E_2$ whenever the manifold admits a Hermitian metric whose torsion operator $\tau$ and its adjoint vanish on ${\Delta }^{\prime \prime }$-harmonic forms of positive degrees up to ${\text{ dim}}_{ℂ}X$. Besides the pseudodifferential Laplacian inducing a Hodge theory for $E_2$ that we constructed in earlier work and Demailly’s Bochner–Kodaira–Nakano formula for Hermitian metrics, a key ingredient is a general formula for the dimensions of the vector spaces featuring in the Frölicher spectral sequence in terms of the asymptotics, as a positive constant $h$ decreases to zero, of the small eigenvalues of a rescaled Laplacian ${\Delta }_h$, introduced here in the present form, that we adapt to the context of a complex structure from the well-known construction of the adiabatic limit and from the analogous result for Riemannian foliations of Álvarez López and Kordyukov.

Keywords

Frölicher spectral sequence, Hodge theory, Hermitian metrics, elliptic differential and pseudodifferential operators, spectral theory

Mathematical Subject Classification 2010

Primary: 14C30, 14F40, 32W05, 53C55

Milestones

Received: 10 October 2017

Revised: 15 July 2018

Accepted: 13 August 2018

Published: 20 July 2019