An inverse image functor for Lie algebroids.
The author generalizes to Lie algebroids many results from the theory of \({\mathcal D}\)-modules. Let \(X\) be a complex analytic manifold, \({\mathcal O}_X\) the sheaf of holomorphic functions on \(X\), and \(\Theta_X\) the \({\mathcal O}_X\)-module of holomorphic vector fields on \(X\). A complex Lie algebroid over \(X\) is a pair \(({\mathcal L}_X,\omega)\) when \({\mathcal L}_X\) is a locally-free \({\mathcal O}_X\)-module of finite constant rank and a sheaf of \(\mathbb{C}\)-Lie algebras and \(\omega:{\mathcal L}_X\to \Theta_X\) an \({\mathcal O}_X\)-linear morphism of sheaves of \(\mathbb{C}\)-Lie algebras such that a natural compatibility condition holds. An important example are the Poisson manifolds. If \(X\) is a Poisson manifold, \((X,\Omega^1_X)\) is a Lie algebroid. The sheaf \({\mathcal D}({\mathcal L}_X)\) of generalized differential operators is constructed in a similar way to the sheaf \({\mathcal D}_X({\mathcal O}_X)\) of differential operators on \(X\) (and when \({\mathcal L}_X= \Theta_X\) and \(\omega=\) identity \({\mathcal D}({\mathcal L}_X)\) is just \({\mathcal D}_X({\mathcal O}_X)\); when \(X\) reduces to a point and \({\mathcal L}_X\) is a Lie algebra \({\mathcal D}({\mathcal L}_X)\) is the enveloping algebra of \({\mathcal L}_X\)). The author constructs a Koszul resolution of \({\mathcal O}_X\) considered as a left \({\mathcal D}({\mathcal L}_X)\) module, and an equivalence of categories between complex of left \({\mathcal D}({\mathcal L})\)-modules and complex of right \({\mathcal D}({\mathcal L}_X)\)-modules in a natural way. The definition of Lie algebroid morphisms given by the author in [\textit{S. Chemla}, Manusc. Math. 87, No. 2, 199--223 (1995; Zbl 0999.17009) and Math. Z. 232, No. 2, 367--388 (1999; Zbl 0933.32015)] coincide with that given previously by \textit{R. Almeida} and \textit{A. Kumpera} [An. Acad. Brasil, Cienc. 53, 247--250 (1981; Zbl 0484.58033)]. If \(\Phi= (f,F)\) is a Lie algebraic morphism from \(({\mathcal L}_X, \omega_X)\) to \(({\mathcal L}_Y, \omega_Y)\), then the direct image function is defined by \(\Phi_!({\mathcal M}^\circ)= Rf_!({\mathcal M}^\circ\bigotimes^L_{D({\mathcal L}_X)}{\mathcal D}_{{\mathcal L}_X\to{\mathcal L}_Y})\) where \({\mathcal M}^\circ\) is an object of \(D^{\mathcal C}({\mathcal D}({\mathcal L}_X)^{\text{op}})\), and \({\mathcal D}_{{\mathcal L}_X\to{\mathcal L}}\) extends the transfer bimodule in the \({\mathcal D}\)-module theory. The author easily extends Schneider's proof (the case \({\mathcal D}\)-modules) to show that if \({\mathcal M}^\bullet\) is in \(D^{\mathcal C}_{\text{good}}({\mathcal D}({\mathcal L}_X)^{\text{op}})\) (``good'' in the sense of Kashiwara) and that if \(f\) is proper on \(\text{Supp\,}{\mathcal M}\) then \(\Phi_!({\mathcal M})\) is in \(D^{\mathcal C}_{\text{good}}({\mathcal D}({\mathcal L}_Y)^{\text{op}})\). If \(\phi= (f, F)\) is a Lie algebroid morphism, if \({\mathcal M}^\bullet\in D^{\mathcal C}_{\text{good}}({\mathcal D}){\mathcal L}_X)^{\text{op}})\) and \(f\) is proper in the support of \({\mathcal M}^circ\) then there is a functional isomorphism from \(\Phi_*\Delta{\mathcal L}_X({\mathcal M}^\circ)\) to \(\Delta_{{\mathcal L}_Y}\Phi_!({\mathcal M}^{\mathcal C})\) in \(D^{\mathcal C}_{\text{good}}({\mathcal D}({\mathcal L}_Y)^{\text{op}})\). This generalizes a result of \textit{J. P. Schneider} [Thèse, Université de Liège, 1986). The inverse image is defined as an object in \(D^{\mathcal C}({\mathcal D}({\mathcal L}_Y))\) by \(\Phi^{-1}({\mathcal R}^\bullet)= {\mathcal D}_{{\mathcal L}_X\to {\mathcal L}_Y}\otimes^L_{f^{-1}{\mathcal D}({\mathcal L}_Y)} f^{-1}{\mathcal R}^\bullet\) for \({\mathcal R}^\bullet\) in \(D^{\mathcal C}({\mathcal D}({\mathcal L}_Y))\). With a suitable notion of noncharacteristic with respect to a morphism \(\phi\), the autor proves that if \({\mathcal R}^\bullet\in D^{\mathcal C}_{\text{coh}}({\mathcal D}({\mathcal L}_Y))\) is not characteristic with respect to \(\phi\), then \(\phi^{-1}({\mathcal R}^\bullet)\) is in \(D^{\mathcal C}_{\text{coh}}({\mathcal D}({\mathcal L}_X))\). The author constructs a functional arrow \(D_{{\mathcal L}_X}\phi^{-1}({\mathcal R}^\bullet)\to \phi^{-1}D{\mathcal L}_Y({\mathcal R}^\bullet)\) for any \({\mathcal R}^\bullet\in D^b({\mathcal D}({\mathcal L}_Y))\) such that \(\phi^{-1}({\mathcal R}^\bullet)\) is in \(D^{\mathcal C}_{\text{coh}}({\mathcal D}({\mathcal L}_X))\). If \({\mathcal R}^\bullet\in D^{\mathcal C}_{\text{coh}}({\mathcal D}({\mathcal L}_Y))\) is noncharacteristic with respect to a Lie algebroid morphism then the above functional morphism is an isomorphism. This generalizes a theorem of \textit{M. Sato}, \textit{T. Kawa} and \textit{M. Kashiwara} [Lect. Notes Math. 287, 263--529 (1973; Zbl 0277.46039)], and the proof of this in the case of \({\mathcal D}\)-modules is different from that of S-K-K. The author derives from these results two adjunction formulas for modules over Lie algebras which generalize analogous results of [\textit{M. Kashiwara} and \textit{P. Shapira}, Mém. Soc. Math. Fr., Nouv. Sér. 64, III (1996; Zbl 0881.58060)]. The author gives also two applications; the first concerns finite-dimensional Lie algebras, and the second analytic Poisson manifolds.
scientific article; zbMATH DE number 2005902
scientific article