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An existence theorem for tempered solutions of $$\mathcal {D}$$-modules on complex curves. (English) Zbl 1155.32018
The problem of existence of solutions for ordinary linear differential equations is classical and studied by many. The author proves an existence theorem for tempered holomorphic solutions in the subanalytic category. One consequence is the following. For a holonomic $${\mathcal D}$$-module $${\mathcal M}$$ on a complex curve $$X$$, the (tempered) solutions give a complex ($$\text{Sol}^t{{\mathcal M}}$$) $$\text{Sol\,} {\mathcal M}$$ on the subanalytic site $$X_{\text{sa}}$$. Then, it is proved that there is a natural isomorphism between $$H^1(\text{Sol} ^t{{\mathcal M}})$$ and $$H^1(\text{Sol\,} {\mathcal M})$$. Another consequence is the $$\mathbb R$$-constructibility of $$\text{Sol} ^t({\mathcal M})$$ in the sense of sheaves on $$X_{\text{sa}}$$. This is conjectured to be true in any dimension by M. Kashiwara and P. Schapira [Astérisque No. 284, 143–164 (2003; Zbl 1053.35009)].

##### MSC:
 32S40 Monodromy; relations with differential equations and $$D$$-modules (complex-analytic aspects) 34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms 32B20 Semi-analytic sets, subanalytic sets, and generalizations
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