Differential Graded Cohomology and Lie algebras of Holomorphic Vector Fields

Details Differential Graded Cohomology and Lie algebras of Holomorphic Vector Fields Differential Graded Cohomology and Lie algebras of Holomorphic Vector Fields

1998-06-20

arxiv.org/abs/math-ph/9806015v3

Comm. Math. Phys. 208 (1999), no. 2, 521-540

Physics

Mathematical Physics

19 pages, an erratum of 2 pages has been added

Scientific paper

10.1007/s002200050768

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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