Cauchy–Riemann operator of the base manifold. This operator is well-defined on all of because on an overlap of two trivialisations with holomorphic transition function , if where is a local frame for on , then , and so because the transition functions are holomorphic. This leads to the following definition: A Dolbeault operator on a smooth complex vector bundle is an -linear operator such that (Cauchy–Riemann condition) , (Leibniz rule) For any section and function on , one has . By an application of the Newlander–Nirenberg theorem, one obtains a converse to the construction of the Dolbeault operator of a holomorphic bundle: Theorem: Given a Dolbeault operator on a smooth complex vector bundle , there is a unique holomorphic structure on such that is the associated Dolbeault operator as constructed above. With respect to the holomorphic structure induced by a Dolbeault operator , a smooth section is holomorphic if and only if . This is similar morally to the definition of a smooth or complex manifold as a ringed space. Namely, it is enough to specify which functions on a topological manifold are smooth or complex, in order to imbue it with a smooth or complex structure. Dolbeault operator has local inverse in terms of homotopy operator. The sheaves of forms with values in a holomorphic vector bundle If denotes the sheaf of differential forms of type , then the sheaf of type forms with values in can be defined as the tensor product These sheaves are fine, meaning that they admit partitions of unity. A fundamental distinction between smooth and holomorphic vector bundles is that in the latter, there is a canonical differential operator, given by the Dolbeault operator defined above: Cohomology of holomorphic vector bundles If is a holomorphic vector bundle, the cohomology of is defined to be the sheaf cohomology of . In particular, we have the space of global holomorphic sections of . We also have that parametrizes the group of extensions of the trivial line bundle of by , that is, exact sequences of holomorphic vector bundles . For the group structure, see also Baer sum as well as sheaf extension. By Dolbeault's theorem, this sheaf cohomology can alternatively be described as the cohomology of the chain complex defined by the sheaves of forms with values in the holomorphic bundle . Namely we have The Picard group In the context of complex differential geometry, the