$\Lambda$-modules and holomorphic Lie algebroid connections

Abstract

Let $X$ be a complex smooth projective variety, and $\mathcal{G}$ a locally free sheaf on $X$. We show that there is a 1-to-1 correspondence between pairs $(\Lambda,\Xi)$, where $\Lambda$ is a sheaf of almost polynomial filtered algebras over $X$ satisfying Simpson's axioms and $\Xi: \Gr\Lambda \rightarrow \Sym^\bullet_{\corO_X} \mathcal{G}$ is an isomorphism, and pairs $(\mathcal{L},\Sigma)$, where $\mathcal{L}$ is a holomorphic Lie algebroid structure on $\mathcal{G}$ and $\Sigma$ is a class in $F^1H^2(\mathcal{L},\C)$, the first Hodge filtration piece of the second cohomology of $\bella$. As an application, we construct moduli spaces of semistable flat $\mathcal{L}$-connections for any holomorphic Lie algebroid $\mathcal{L}$. Particular examples of these are given by generalized holomorphic bundles for any generalized complex structure associated to a holomorphic Poisson manifold.