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

Carlos Simpson (1)

1 JAD - Laboratoire Jean Alexandre Dieudonné

We construct a locally geometric $\infty$-stack $M_{Hod}(X,Perf)$ of perfect complexes on $X$ with $\lambda$-connection structure for 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 $\Dd_X$-modules which are $\Oo_X$-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.

Domaines: Géométrie algébrique [math.AG]

Carlos Simpson. Geometricity of the Hodge filtration on the $\infty$-stack of perfect complexes over $X_{DR}$. 2005. ⟨hal-00011204v1⟩