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.

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Mathematical Subject Classification 2010

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