Bergman kernels and local holomorphic Morse inequalities. (English) Zbl 1066.32002

Berman, Robert

Math. Z. 248, No. 2, 325-344 (2004).

J.-P. Demailly’s holomorphic Morse inequalities [Ann. Inst. Fourier 35, No. 4, 189–229 (1985; Zbl 0565.58017)] asymptotically estimate the dimension of the Dolbeault cohomology group associated to a high tensor power of a Hermitian holomorphic line bundle over a complex manifold. The author gives a relatively elementary proof of these inequalities. The new idea is to localize and to estimate the Bergman kernel function asymptotically by a model kernel in \(\mathbb{C}^n\).

Reviewer: Harold P. Boas (College Station)

MSC:

32A25	Integral representations; canonical kernels (Szegő, Bergman, etc.)
32L10	Sheaves and cohomology of sections of holomorphic vector bundles, general results
32L20	Vanishing theorems

Keywords:

Bergman kernel function; Demailly’s inequalities; model kernel

Citations:

Zbl 0565.58017