A Mapping Theorem for Derived Foliations
Victor Alfieri
TL;DR
This work develops a mapping theorem for derived foliations in characteristic zero, proving that a derived foliation on a base Y descends to a foliation on the derived mapping stack Map_S(X,Y) when X is proper, flat, and lci over S. The central technique builds a push-forward operation for foliations along proper lci morphisms by recasting foliations as equivariant perfect linear stacks over graded loop stacks and leveraging Beck-Chevalley formalism to preserve structure. A precise description of tangent data shows that leaves of the induced foliation correspond to families of maps with leaves aligned pointwise via the original foliation, enabling concrete applications to derived moduli spaces such as R M_{g,n}^{pre}(X) and R Hilb^{lci}(X). The results yield a systematic transfer of derived foliations to a broad class of derived moduli stacks and point toward future directions including shifted symplectic structures and Lagrangian foliations, with potential extensions to derived Artin settings and links to shifted Poisson theory.
Abstract
In this paper, we construct in characteristic zero a derived foliation on derived mapping stacks $\underline{\mathbf{Map}}_S(X,Y)$, for $S$ a base derived stack, $X$ a proper schematic, flat, and local complete intersection derived stack over $S$, and $Y$ a relative derived Deligne-Mumford stack over $S$, when $Y$ is equipped with a derived foliation relative to $S$. In the process, given a relative derived Deligne-Mumford stack $Z$ over a derived stack $X$, we will first show that the $\infty$-category of derived foliations over $Z$ relative to $X$ embeds as a full subcategory of derived stacks over $Z$ equipped with extra structure, and describe its essential image explicitly. We will then show that given a proper schematic, flat, and local complete intersection map of derived stacks $f : X \to Y$, the push-forward functor $f_*$ from derived stacks over $X$ to derived stacks over $Y$ preserves the preceding essential images, and thus defines a push-forward, from derived foliations over $Z$ relative to $X$, to derived foliations over $f_* Z$ relative to $Y$. The aforementioned result on derived mapping stacks is obtained as a special case of this statement. As example applications, given a smooth projective scheme $X$ equipped with a derived folation, we obtain derived folations on the derived moduli stacks $\mathbb R \overline{\mathbf M}_{g,n}(X)$ and $\mathbb R \mathbf{Hilb}^{lci}(X)$, which are respectively the derived enhancements of the moduli stack of families of stable curves of genus $g$ with $n$ marked points on $X$, and of the Hilbert scheme of closed subschemes of $X$.