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Differential Graded Cohomology and Lie algebras of Holomorphic Vector Fields
Mathematical Physics · Submitted 20 Jun 1998
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Feigin - an isomorphism between the differential graded cohomology of the space of global sections of $g$ and the hypercohomology of the sheaf of continuous cochain complexes of $Lie$.
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The abstract, sentence by sentence
The Dolbeault resolution of the sheaf of holomorphic vector fields $Lie$ on a complex manifold $M$ relates $Lie$ to a sheaf of differential graded Lie algebras, known as the Fr\"olicher-Nijenhuis algebra $g$. We establish - following B. L. Feigin - an isomorphism between the differential graded cohomology of the space of global sections of $g$ and the hypercohomology of the sheaf of continuous cochain complexes of $Lie$. We calculate this cohomology up to the singular cohomology of some mapping space. We use and generalize results of N. Kawazumi on complex Gelfand-Fuks cohomology. Applications are - again following B. L. Feigin - in conformal field theory, and in the theory of deformations of complex structures. In an erratum to this paper, we admit that the sheaf of continuous cochains of a sheaf of vector fields with values in the ground fields does not make much sense. The most important cochains (like evaluations in a point or integrations over the manifold) do not come from sheaf homomorphisms. The main result of the above article (theorem 7) remains true.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The Dolbeault resolution of the sheaf of holomorphic vector fields $Lie$ on a complex manifold $M$ relates $Lie$ to a sheaf of differential graded Lie algebras, known as the Fr\"olicher-Nijenhuis algebra $g$. | background 1.00 | 0.09 | 0.06 | asserted 1.00 |
| s1 | We establish - following B. L. | contribution 0.83 | 0.77 | 0.04 | unclear 0.62 |
| s2 | Feigin - an isomorphism between the differential graded cohomology of the space of global sections of $g$ and the hypercohomology of the sheaf of continuous cochain complexes of $Lie$. | result 0.85 | 0.88 | 0.57 | asserted 1.00 |
| s3 | We calculate this cohomology up to the singular cohomology of some mapping space. | result 0.78 | 0.92 | 0.09 | asserted 1.00 |
| s4 | We use and generalize results of N. | method 0.82 | 0.64 | 0.03 | asserted 0.88 |
| s5 | Kawazumi on complex Gelfand-Fuks cohomology. | method 0.52 | 0.14 | 0.03 | unclear 0.59 |
| s6 | Applications are - again following B. L. | contribution 0.53 | 0.50 | 0.04 | unclear 0.51 |
| s7 | Feigin - in conformal field theory, and in the theory of deformations of complex structures. | interpretation 0.54 | 0.36 | 0.05 | asserted 0.82 |
| s8 | In an erratum to this paper, we admit that the sheaf of continuous cochains of a sheaf of vector fields with values in the ground fields does not make much sense. | limitation 1.00 | 0.78 | 0.05 | asserted 0.88 |
| s9 | The most important cochains (like evaluations in a point or integrations over the manifold) do not come from sheaf homomorphisms. | limitation 0.98 | 0.71 | 0.06 | asserted 1.00 |
| s10 | The main result of the above article (theorem 7) remains true. | result 0.61 | 0.55 | 0.35 | asserted 1.00 |
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Original abstract
The Dolbeault resolution of the sheaf of holomorphic vector fields $Lie$ on a complex manifold $M$ relates $Lie$ to a sheaf of differential graded Lie algebras, known as the Fr\"olicher-Nijenhuis algebra $g$. We establish - following B. L. Feigin - an isomorphism between the differential graded cohomology of the space of global sections of $g$ and the hypercohomology of the sheaf of continuous cochain complexes of $Lie$. We calculate this cohomology up to the singular cohomology of some mapping space. We use and generalize results of N. Kawazumi on complex Gelfand-Fuks cohomology. Applications are - again following B. L. Feigin - in conformal field theory, and in the theory of deformations of complex structures. In an erratum to this paper, we admit that the sheaf of continuous cochains of a sheaf of vector fields with values in the ground fields does not make much sense. The most important cochains (like evaluations in a point or integrations over the manifold) do not come from sheaf homomorphisms. The main result of the above article (theorem 7) remains true.
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Feigin back in the 1980's but there has been no proof even in the case of curves, despite some work for holomorphic vector fields and continuous cohomology [ref] [ref] , [ref] .
Gelfand-Fuchs cohomology in algebraic geometry and factorization algebras · Introduction · 0.95
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