To determine cohomology groups of holomorphic vector bundles
or more general coherent analytic sheaves on complex manifolds is very important
in several complex variables and complex geometry. For example, Cartan-Serre's theorem
B and Kodaira's vanishing theorem are fundamental respectively in the studies of
two classes of complex manifolds: Stein manifolds and projective algebraic manifolds.
Deformation and moduli space of complex structures are also closely related to the determination
of the cohomology groups of the bundles according to Kodaira-Spencer's
theory