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Let X be a complex smooth projective variety, and G a locally free sheaf on X. We show that there is a one-to-one correspondence between pairs (Λ, Ξ), where Λ is a sheaf of almost polynomial filtered algebras over X satisfying Simpson’s axioms and $\equiv :Gr\Lambda \to Sym{•}_{{𝒪}_X}𝒢$ is an isomorphism, and pairs (L, Σ), where L is a holomorphic Lie algebroid structure on $𝒢$ and Σ is a class in F 1 H 2(L, ℂ), the first Hodge filtration piece of the second cohomology of L. As an application, we construct moduli spaces of semistable...