Λ-modules and holomorphic Lie algebroid connections

Abstract

Let X be a complex smooth projective variety, and G a locally free sheaf on X. We show that there is a 1-to-1 correspondence between pairs (Λ, Ξ), where Λ is a sheaf of almost polynomial filtered algebras over X satisfying Simpson's axioms and Ξ : GrΛ → Sym • OX 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 flat L-connections for any holomorphic Lie algebroid L. Particular examples of these are given by generalized holomorphic bundles for any generalized complex structure associated to a holomorphic Poisson manifold.

Representations of Atiyah Algebroids and Logarithmic Connections

“…We shall denote by R the first inclusion, and call its image in L N the isotropy of the Lie algebroid At(N ). Since O Y is abelian as a Lie algebroid, one has the following (see [2], [17]): For example, to reformulate the following propositions using objects in the real category one should use the Lie algebroid At(N ) R ⊲⊳ T 0,1 X as in [11]. Dualizing the sequence (5) one obtains This fact follows from a generalization of Atiyah's theory on the existence of holomorphic connections to the Lie algebroid setting: it is possible to show the following (see [17]):…”

Nonabelian holomorphic Lie algebroid extensions

“…For a comprehensive discussion we refer the reader to [18], see also [3,6,5]. Moreover, the problem of the abelian extensions of holomorphic Lie algebroids was solved in [4] for B = Θ X , and in [28] for a general B (with the notation of equation (5)).…”

Lie algebroid connections, twisted Higgs bundles and motives of moduli spaces

“…The Hodge moduli space M Hod (r, d) is just a particular case of a much more general construction, by Simpson [Sim94], arising from the moduli space M Λ (r, d) of Λ-modules (see Definition 3.8), where Λ is a sheaf of rings of differential operators on X. Now, as proved by Tortella in [Tor11,Tor12], there is an equivalence of categories between a certain subclass of such sheaves (consisting of the split and almost polynomial ones) and algebraic Lie algebroids on X. In turn, such equivalence induces an equivalence of categories between integrable L-connections, where L is a Lie algebroid on X and Λ L -modules, with Λ L the split almost polynomial sheaf of rings of differential operators corresponding to L. This correspondence preserves semistability, hence one can think of M Λ L (r, d) as the moduli space of L-connections of rank r and degree d.…”

“…The two equivalent interpretations of the "same object" -L-connections and Λ L -modules -are actually required in this proof. For instance, in order to prove smoothness, we make use of the deformation theory for integrable L-connections, developed in [Tor12], since deformation theory for Λ-modules is not yet well-understood in the required generality. On the other hand, the proof of the existence of limits of a natural C * -action on the L-Hodge moduli (which is a condition for being semiprojective) is carried out by explicitly using Λ L -modules rather than L-connections.…”

“…In section 2 we give a quick introduction to V -twisted Higgs bundles. In Section 3 we introduce Lie algebroids, sheaves of rings of differential operators, L-connections and Λ-modules and, following [Tor11,Tor12], give a short overview of the equivalence of categories between integrable L-connections and Λ L -modules. This is done over any base variety.…”