Complex Analytic and Differential Geometry 2025, 15 - Holomorphic Vector Bundles

A vector bundle π: E → X is holomorphic for a complex manifold E, if the projection π is holomorphic and there is a covering (V_α) of X and a family of holomorphic trivializations θ_α: E_↑Vα → V_α × ℂ^r. H¹(X, O^☆) of isomorphic classes of holomorphic line bundles is the Picard group of X. Define the operator d'' as the canonical (0, 1)-connection of the holomorphic bundle E. The q-th cohomology group of the Dolbeault complex is denoted H^{p, q}(X, E) is the (p, q) Dolbeault cohomology group with values in E. H^{0, q}(X, E) ≡ H^q(X, O(E)), Ω^p_X is the vector bundle Λ^pT^*X or its sheaf of sections.

The unique hermitian connection D with D'' = d'' is the Chern connection of E. The curvature tensor of this connection is denoted by Θ(E), the Chern curvature of E. It is such that iΘ(E) ∈ C^∞_{1, 1}(X, Herm(E, E)), and if θ: E_↑Ω → Ω×ℂ^r is a holomorphic trivialization and H is the hermitian matrix representing the metric along the fibers of E, then iΘ(E) = id''(H'^-1 d' H') on Ω. For all points in X and all coordinate systems there is a holomorphic frame in its neighborhood with

A meromorphic section of a bundle E → X is a section s on an open dense subset of X, where all trivializations consists of meromorphic functions on V_α. For a line bundle E → X with meromorphic section s in E that doesn't vanish identically on any component of X, if Σm_jZ_j, then c₁(E)_ℝ = {Σ m_j[Z_j]} ∈ H²(X, ℝ). If X is any complex manifold, all hermitian line bundles E over M has a Chern curvature form i/2π Θ(E) is a closed real (1, 1)-form with De Rahm cohomology class is the image of an internal class. If ω is a C^∞ closed real (1, 1)-form so that the class {ω} &in H²_DR(X, ℝ) is the image of an integral class.

An exact sequence of holomorphic vector bundles 0 → S →^j E →^g Q → 0 has E as an extension of S by Q. It splits by some holomorphic h: Q → E which is a right inverse of the projection E → Q, with g ○ h = Id_Q. If C^∞ a hermitian metric on E is given and S, Q are endowed with the quotient metric. The adjoint homomorphisms j* and g* are C^∞ and can be described as orthogonal projections of E to S.

j*⊕g: E ≅ S ⊕ Q

Using it,

The correspondence {E} → {β*} induces a bijection from the set of isomorphism classes of extensions of S by Q onto H¹(X, Hom(Q, S)). {β*} vanishes iff the above sequence splits holomorphically.