#### Vol. 300, No. 1, 2019

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Adiabatic limit and the Frölicher spectral sequence

### Dan Popovici

Vol. 300 (2019), No. 1, 121–158
##### 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 $\partial \stackrel{̄}{\partial }\omega =0$), 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 . 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