$\Lambda$-modules and holomorphic Lie algebroid connections

TL;DR

This paper establishes a correspondence between algebraic structures called sheaves of filtered algebras and geometric structures known as holomorphic Lie algebroids, and constructs moduli spaces of flat connections related to these structures.

Contribution

It introduces a novel correspondence between filtered algebra sheaves satisfying Simpson's axioms and holomorphic Lie algebroids with cohomology classes, expanding the understanding of their interplay.

Findings

Established a 1-to-1 correspondence between algebraic and geometric structures.

Constructed moduli spaces of semistable flat Lie algebroid connections.

Applied results to generalized holomorphic bundles on Poisson manifolds.

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.