Picard group of the complex manifold is the group of isomorphism classes of holomorphic line bundles with group law given by tensor product and inversion given by dualization. It can be equivalently defined as the first cohomology group of the sheaf of non-vanishing holomorphic functions. Hermitian metrics on a holomorphic vector bundle Let E be a holomorphic vector bundle on a complex manifold M and suppose there is a hermitian metric on E; that is, fibers Ex are equipped with inner products <·,·> that vary smoothly. Then there exists a unique connection ∇ on E that is compatible with both complex structure and metric structure, called the Chern connection; that is, ∇ is a connection such that (1) For any smooth sections s of E, where π0,1 takes the (0, 1)-component of an E-valued 1-form. (2) For any smooth sections s, t of E and a vector field X on M, where we wrote for the contraction of by X. (This is equivalent to saying that the parallel transport by ∇ preserves the metric <·,·>.) Indeed, if u = (e1, …, en) is a holomorphic frame, then let and define ωu by the equation , which we write more simply as: If u' = ug is another frame with a holomorphic change of basis g, then and so ω is indeed a connection form, giving rise to ∇ by ∇s = ds + ω · s. Now, since , That is, ∇ is compatible with metric structure. Finally, since ω is a (1, 0)-form, the (0, 1)-component of is . Let be the curvature form of ∇. Since squares to zero by the definition of a Dolbeault operator, Ω has no (0, 2)-component and since Ω is easily shown to be skew-hermitian, it also has no (2, 0)-component. Consequently, Ω is a (1, 1)-form given by The curvature Ω appears prominently in the vanishing theorems for higher cohomology of holomorphic vector bundles; e.g., Kodaira's vanishing theorem and Nakano's vanishing theorem. Notes References See also Birkhoff–Grothendieck theorem Quillen metric Serre duality External links Splitting principle for holomorphic vector bundles Vector bundles Complex manifolds