Deformations of $(p,q)$-forms and degenerations of the Frölicher spectral sequence

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 $(p,q)$.