Differential Graded Cohomology and Lie Algebras of Holomorphic Vector Fields

The continuous cohomology of Lie algebras of C-vector fields has proven to be a subject of great geometrical interest: One of its most famous applications is the construction of the Virasoro algebra as the universal central extension of the Lie algebra of vector fields on the circle. So there is the natural problem of calculating the continuous cohomology of the Lie algebra of holomorphic vector fields on a complex manifold. We solve the problem completely for arbitrary complex manifolds (up to (singular) cohomology calculations of some mapping spaces) combining a method of [Kawazumi] with hypercohomology techniques. Our main interest is in compact complex manifolds: Here, the Lie algebra of holomorphic vector fields seems to be too small to be interesting for compact Riemann surfaces of genus g it is of dimension 3 for g = 0, 1 for g = 1 and 0 for g ≥ 2. However, treating the holomorphic vector fields as a sheaf rather than taking brutally global sections proves to reveal a richer cohomology theory, as first remarked by [Feigin].