Dolbeault cohomology
In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology (named after Pierre Dolbeault) is an analog of de Rham cohomology for complex manifolds. Let M be a complex manifold. Then the Dolbeault cohomology groups H^{p,q}(M,C) depend on a pair of integers p and q and are realized as a subquotient of the space of complex differential forms of degree (p,q).
Contents
Construction of the cohomology groups
Let Ω^{p,q} be the vector bundle of complex differential forms of degree (p,q). In the article on complex forms, the Dolbeault operator is defined as a differential operator on smooth sections
Since
this operator has some associated cohomology. Specifically, define the cohomology to be the quotient space
Dolbeault cohomology of vector bundles
If E is a holomorphic vector bundle on a complex manifold X, then one can define likewise a fine resolution of the sheaf of holomorphic sections of E. This is therefore a recollection of the sheaf cohomology of .
Dolbeault's theorem
Dolbeault's theorem is a complex analog^{[1]} of de Rham's theorem. It asserts that the Dolbeault cohomology is isomorphic to the sheaf cohomology of the sheaf of holomorphic differential forms. Specifically,
where Ω^{p} is the sheaf of holomorphic p forms on M.
A version for logarithmic forms has also been established.^{[2]}
Proof
Let be the fine sheaf of forms of type . Then the -Poincaré lemma says that the sequence
is exact. Like any long exact sequence, this sequence breaks up into short exact sequences. The long exact sequences of cohomology corresponding to these give the result, once one uses that the higher cohomologies of a fine sheaf vanish.
Footnotes
- ^ In contrast to de Rham cohomology, Dolbeault cohomology is no longer a topological invariant because it depends closely on complex structure.
- ^ Navarro Aznar, V. (1987), "Sur la théorie de Hodge-Deligne", Inventiones Mathematicae, 90 (1): 11–76, doi:10.1007/bf01389031, Section 8
References
- Dolbeault, P. (1953). "Sur la cohomologie des variétés analytiques complexes". C. R. Acad. Sci. Paris. 236: 175–277.
- Wells, R.O. (1980). Differential Analysis on Complex Manifolds. Springer-Verlag. ISBN 0-387-90419-0.