On the universal $L_\infty$-algebroid of linear foliations

We compute an $L_\infty$-algebroid structure on a projective resolution of some classes of singular foliations on a vector space $V$ induced by the linear action of some Lie subalgebra of $\mathfrak {gl}(V)$. This $L_\infty$-algebroid provides invariants of the singular foliations, and also provides a constant-rank replacement of the singular foliation. We do this by first explicitly constructing projective resolutions of the singular foliations induced by the natural linear actions of endomorphisms of $V$ preserving a subspace $W\subset V$, the Lie algebra of traceless endomorphisms, and the symplectic Lie algebra of endomorphisms of $V$ preserving a non-degenerate skew-symmetric bilinear form $\omega$, and then computing the $L_\infty$-algebroid structure. We then generalize these constructions to a vector bundle $E$, where the role of the origin is now taken by the zero section $L$. We then show that the fibers over a singular point of a projective resolution of any singular foliation can be computed directly from the foliation, without needing the projective resolution. For linear foliations, we also provide a way to compute the action of the isotropy Lie algebra in the origin on these fibers, without needing the projective resolution.