Deformations of $(p,q)$-forms and degenerations of the Frölicher spectral sequence
Xueyuan Wan, Wei Xia
TL;DR
This work investigates how the degeneration behavior of the Frölicher spectral sequence controls the deformation invariance of Hodge-type data for complex manifolds. It introduces a power-series deformation framework for (p,q)-forms, leveraging r-filtered forms and canonical deformations to derive unobstructedness criteria under vanishing sums of differentials, namely $⊕_{r≥1} d_r^{p,q}=0$ and $⊕_{r≥1,i≥1} d_r^{p-i,q+i}=0$, which imply invariance of certain Hodge-number combinations and persistence of vanishing across nearby fibers. A second main result addresses deformation stability of $E_r^{p,q}=E_∞^{p,q}$, providing conditions under which higher-page degenerations are open under deformation and yielding corollaries for the case $r=2$. The paper complements the theory with explicit examples at degenerations $E_2$ and $E_3$, including the Iwasawa manifold and a nilmanifold with $\mathfrak{g}\cong \mathfrak{h}_{15}$, illustrating how canonical deformations can be computed and how the new criteria apply in concrete settings.
Abstract
It is well-known that Hodge numbers are invariant under deformations of complex structures if the Frölicher spectral sequence of the central fiber degenerates at the first page (i.e. $E_1=E_\infty$). As a result, the deformations of $(p,q)$-forms are unobstructed for all $(p,q)$ if $E_1=E_\infty$. We refine this classical result by showing that for any fixed $(p,q)$ the deformations of $(p,q)$-forms are unobstructed if the differentials $d_r^{p,q}$ in the Frölicher spectral sequence satisfy \[ \bigoplus_{r\geq 1}d_r^{p,q}=0\quad\text{and}\quad \bigoplus_{ r\geq i\geq 1 } d_r^{p-i,q+i}=0. \] Moreover, the deformation stability of the degeneration property for Frölicher spectral sequences in the first page and higher pages is also studied. In particular, we have found suitable conditions to ensure the deformation stability of $E_r^{p,q}=E_\infty^{p,q}$ ($r\geq1$) for fixed $(p,q)$.