Semiregularity maps and deformations of modules over Lie algebroids
Ruggero Bandiera, Emma Lepri, Marco Manetti
TL;DR
This work generalizes semiregularity maps from vector bundles to locally free modules over Lie algebroids by introducing the $(\mathcal{L},\mathcal{A})$-Atiyah class $\operatorname{At}_{\mathcal{L}/\mathcal{A}}(\mathcal{E})$ and establishing a DG-Lie framework that governs infinitesimal deformations via the Tot functor applied to $\Omega^*(\mathcal{A})\otimes\operatorname{End}(\mathcal{E})$. It constructs semiregularity maps $\tau_k$ from $\mathbb{H}^2(\mathcal{A};\operatorname{End}(\mathcal{E}))$ to $\mathbb{H}^{2+k}(\mathcal{A};\bigwedge^k(\mathcal{L}/\mathcal{A})^\vee)$ and proves that, under a suitable Leray-degeneracy condition for the Lie pair, these maps kill obstructions to deformations. The paper introduces reduced and simplicial Atiyah structures to manage extensions of $\mathcal{A}$-connections to $\mathcal{L}$-connections, and uses Thom--Whitney totalisation to build abelian targets for obstruction vanishing via $L_\infty$ morphisms. Consequently, the main result asserts that the obstructions to deforming a locally free $\mathcal{A}$-module vanish after applying the composite semiregularity maps, provided the Leray filtration behaves well; this yields powerful obstruction-vanishing criteria in the Lie algebroid setting with potential applications to relative and Poisson-geometric contexts.
Abstract
We determine a DG-Lie algebra controlling deformations of a locally free module over a Lie algebroid $\mathcal{A}$. Moreover, for every flat inclusion of Lie algebroids $\mathcal{A}\subset \mathcal{L}$ we introduce semiregularity maps and prove that they annihilate obstructions, provided that the Leray spectral sequence of the pair $(\mathcal{L},\mathcal{A})$ degenerates at $E_1$.