2022/07/07 by Singh, Karandeep Jandu
#Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2207.03278
We compute an L_∞-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_∞-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⊂ V, the Lie algebra of traceless endomorphisms, and the symplectic Lie algebra of endomorphisms of V preserving a non-degenerate skew-symmetric bilinear form ω, and then computing the L_∞-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.