# Geometricity of the Hodge filtration on the $\infty$-stack of perfect complexes over $X_{DR}$

Abstract : We construct a locally geometric $\infty$-stack $M_{Hod}(X,Perf)$ of perfect complexes with $\lambda$-connection structure on a smooth projective variety $X$. This maps to $A ^1 / G_m$, so it can be considered as the Hodge filtration of its fiber over 1 which is $M_{DR}(X,Perf)$, parametrizing complexes of $D_X$-modules which are $O_X$-perfect. We apply the result of Toen-Vaquie that $Perf(X)$ is locally geometric. The proof of geometricity of the map $M_{Hod}(X,Perf) \rightarrow Perf(X)$ uses a Hochschild-like notion of weak complexes of modules over a sheaf of rings of differential operators. We prove a strictification result for these weak complexes, and also a strictification result for complexes of sheaves of $O$-modules over the big crystalline site.
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Preprints, Working Papers, ...

Cited literature [132 references]

https://hal.archives-ouvertes.fr/hal-00011204
Contributor : Carlos Simpson <>
Submitted on : Tuesday, April 29, 2008 - 9:30:00 AM
Last modification on : Friday, January 12, 2018 - 1:51:33 AM
Long-term archiving on: Tuesday, September 21, 2010 - 4:53:03 PM

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### Citation

Carlos Simpson. Geometricity of the Hodge filtration on the $\infty$-stack of perfect complexes over $X_{DR}$. 2008. ⟨hal-00011204v2⟩

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