  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
$M_{Hod}(X,Perf) \to Perf(X)$ 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.
