Deformations of cohomology classes and spectral sequences

Source abstract

We study spectral sequences as an obstruction theory to deforming cohomology classes via the Rees construction. A filtered complex is interpreted as an equivariant family over the affine line, in which higher spectral sequence differentials arise as obstructions to lifting cohomology classes through infinitesimal neighborhoods of the special fibre. This leads to various geometric and module-theoretic degeneration criteria. We give various applications to $\mathrm{Ext}$ spectral sequences for complex Lagrangians and the formality results of Melani-Rubió amongst others. For double complexes, we revisit the Khovanov-Qi decomposition on the Rees side and introduce a two-parameter Rees construction over the affine plane. The higher differentials of the two associated spectral sequences correspond to torsion modules supported on the thickened coordinate axes. We obtain degeneration criteria in terms of these modules and an intrinsic characterisation of the $\partial\bar\partial$-lemma.