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Erschienen in:
2015 | OriginalPaper | Buchkapitel
Atiyah Classes of Lie Algebroids
verfasst von : Francesco Bottacin
Erschienen in: Current Trends in Analysis and Its Applications
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Abstract
Given a smooth morphism of analytic spaces π:X→Y, we introduce the notion of a relative Lie algebroid \((\mathcal {A},\sharp)\) over X. By replacing the relative tangent sheaf \(\mathcal{T}_{X/Y}\) with the Lie algebroid \(\mathcal{A}\), we define the notion of a relative \((\mathcal{A},\sharp)\)-connection on a quasi-coherent \(\mathcal{O}_{X}\)-module \(\mathcal{E}\). Then, we define the \((\mathcal{A},\sharp)\)-Atiyah class of \(\mathcal{E}\) as the obstruction to the existence of a holomorphic \((\mathcal{A},\sharp)\)-connection on \(\mathcal{E}\). Many results of the classical theory of connections can be restated in the more general setting of Lie algebroid connections. As an application we prove the following result.
Let X be a complex manifold and (A,♯) a Lie algebroid over X. For any quasi-coherent sheaf of commutative \(\mathcal{O}_{X}\)-algebras \(\mathcal{F}\), let us write \(\mathfrak{g}_{i} = H^{i-1}(X, A \otimes \mathcal{F})\). The (A,♯)-Atiyah class of A yields maps \(\mathfrak{g}_{i} \otimes\mathfrak{g}_{j} \to\mathfrak{g}_{i+j}\). These maps define a graded Lie algebra structure on the graded vector space \(\mathfrak{g}^{\bullet} = \bigoplus_{i} \mathfrak{g}_{i}\). In a similar way, for any holomorphic vector bundle E over X, let us write \(V_{j} = H^{j-1}(X, E \otimes\mathcal{F})\). Then, for any i and j, the (A,♯)-Atiyah class of E yields a map \(\mathfrak{g}_{i} \otimes V_{j} \to V_{i+j}\), and these maps define a structure of graded module on the graded vector space V
•=⨁
j
V
j
, over the graded Lie algebra \(\mathfrak{g}^{\bullet}\). This generalizes a similar result proved by Kapranov in Compos. Math. 115:71–113, 1999. Similar results have been obtained by Chen, Stiénon and Xu in From Atiyah classes to homotopy Leibniz algebras 2012, by using different techniques.