\(L^ 2\)-cohomology and index theorem for the Bergman metric. (English) Zbl 0532.58027

Donnelly, Harold; Fefferman, Charles

Ann. Math. (2) 118, 593-618 (1983).

Let \(\Omega\) be a strictly pseudoconvex domain in \({\mathbb{C}}^ n\) endowed with its Bergman metric. The \(L_ 2\) cohomology of \(\Omega\) is infinite dimensional in the middle degree and vanishes for all other degrees. Asymptotic expansions are given for the Schwartz kernels of the corresponding projections onto harmonic forms. This includes an index theorem for the Dolbeault complex of the Bergman metric.

MSC:

58J10	Differential complexes
32T99	Pseudoconvex domains
32A25	Integral representations; canonical kernels (Szegő, Bergman, etc.)
53C55	Global differential geometry of Hermitian and Kählerian manifolds
53C15	General geometric structures on manifolds (almost complex, almost product structures, etc.)
58J60	Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
58A14	Hodge theory in global analysis
35N15	\(\overline\partial\)-Neumann problems and formal complexes in context of PDEs
55N20	Generalized (extraordinary) homology and cohomology theories in algebraic topology
35C20	Asymptotic expansions of solutions to PDEs
58J20	Index theory and related fixed-point theorems on manifolds

Keywords:

Hermitian manifolds; constant holomorphic sectional curvature; Duhamel’s principle; contact transformations; Laplacian on forms; Schwartz kernels; harmonic forms; Dolbeault complex