Motivated by our conjecture of an earlier work predicting the degeneration at the second page of the Fr��licher spectral sequence of any compact complex manifold supporting an SKT metric $��$ (i.e. such that $\partial\bar\partial��=0$), we prove degeneration at $E_2$ whenever the manifold admits a Hermitian metric whose torsion operator $��$ and its adjoint vanish on $��''$-harmonic forms of positive degrees up to $\mbox{dim}_\C X$. Besides the pseudo-differential 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 Fr��licher spectral sequence in terms of the asymptotics, as a positive constant $h$ decreases to zero, of the small eigenvalues of a rescaled Laplacian $��_h$, 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 ��lvarez L��pez and Kordyukov.

32 pages