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The Cauchy-Kowalevski theorem for \(\mathcal E\)-modules. (English) Zbl 0906.35004
The author proves a version of the Cauchy-Kowalevski theorem for \({\mathcal E}_X\)-modules and for holomorphic functions in the category \(D^b(Y;p_Y).\) Namely, upon considering a complex analytic manifold \(X,\) a point \(p_X\in T^*X\) and a coherent left \({\mathcal E}_X\)-module \({\mathcal M},\) he proves that the object \(R{\mathcal H}om_{{\mathcal E}_X}({\mathcal M},{\mathcal O}_X)_{p_X}\) is well defined in \(D^b(X;p_X)\) (by proving, among other properties that, for instance, it depends only on \({\mathcal M}\) up to isomorphisms, and that it is functorial with respect to \({\mathcal M}).\) He next takes a morphism \(f\colon Y\rightarrow X\) of complex manifolds, a point \(p\in Y\times_XT^*X,\) considers \(p_X=f_\pi(p)\) and \(p_Y= ^t f'(p),\) and supposes that \({\mathcal M}\) is defined in a neighborhood of \(p_X\) and non-characteristic with respect to \(f.\) Upon denoting by \(\underline{f_p}^{-1}{\mathcal M}\) the inverse image of \({\mathcal M}\) by \(f,\) he constructs the natural morphism \(f_p^{-1}R{\mathcal H}om_{{\mathcal E}_X}({\mathcal M},{\mathcal O}_X)_{p_X}\rightarrow R{\mathcal H}om_{{\mathcal E}_Y}(\underline{f_p}^{-1}{\mathcal M},{\mathcal O}_Y)_{p_Y}\) in the category \(D^b(Y;p_Y)\) and proves that it is an isomorphism. In the final section, he gives an application.

MSC:
35A10 Cauchy-Kovalevskaya theorems
32C38 Sheaves of differential operators and their modules, \(D\)-modules
58J15 Relations of PDEs on manifolds with hyperfunctions
32A10 Holomorphic functions of several complex variables
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