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On the universal $L_\infty$-algebroid of linear foliations
Differential Geometry · Submitted 7 Jul 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim This $L_\infty$-algebroid provides invariants of the singular foliations, and also provides a constant-rank replacement of the singular foliation.
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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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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 $\mathfrak {gl}(V)$. | contribution 0.89 | 0.96 | 0.38 | asserted 1.00 |
| s1 | This $L_\infty$-algebroid provides invariants of the singular foliations, and also provides a constant-rank replacement of the singular foliation. | interpretation 0.49 | 0.88 | 0.49 | asserted 1.00 |
| s2 | 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. | method 0.99 | 0.96 | 0.09 | asserted 1.00 |
| s3 | We then generalize these constructions to a vector bundle $E$, where the role of the origin is now taken by the zero section $L$. | contribution 0.71 | 0.94 | 0.06 | asserted 0.99 |
| s4 | 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. | result 0.70 | 0.95 | 0.23 | asserted 1.00 |
| s5 | 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. | contribution 0.80 | 0.96 | 0.11 | asserted 0.98 |
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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.
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For the definition and important properties, we refer to [ref] .
The modular class of a singular foliation · Introduction · 0.99
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