A Mapping Theorem for Derived Foliations

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 $X$.