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Holomorphic vector bundle

A holomorphic vector bundle is a complex vector bundle E \to X over a complex manifold X, equipped with a holomorphic structure such that the total space E is a complex manifold and the projection \pi: E \to X is a holomorphic map.^[1] Equivalently, it consists of an open cover \{U_i\} of X with holomorphic trivializations \phi_i: \pi^{-1}(U_i) \to U_i \times \mathbb{C}^r (for rank r), where the transition functions g_{ij}: U_i \cap U_j \to \mathrm{GL}(r, \mathbb{C}) are holomorphic.^[2] This structure ensures that the fibers E_x \cong \mathbb{C}^r vary holomorphically with the base point x \in X. Holomorphic vector bundles form a cornerstone of complex geometry, providing the framework for analyzing families of complex vector spaces parametrized by complex manifolds.^[3] They generalize tangent bundles and line bundles, enabling the study of geometric invariants like Chern classes, which measure topological and analytic properties, and support the definition of holomorphic sections—maps s: X \to E that are holomorphic when composed with local trivializations.^[4] In algebraic geometry, they underpin the theory of coherent sheaves and projective embeddings via ample bundles, as in Kodaira's theorem.^[5] Key developments include Grothendieck's classification of holomorphic bundles over the Riemann sphere as direct sums of line bundles of specific degrees,^[6] and the notion of stability, which characterizes bundles admitting Hermitian-Einstein metrics that minimize the Yang-Mills functional and is vital for moduli problems.^[7] These objects also admit compatible connections, such as the Chern connection, facilitating the integration of differential geometry with complex analysis.^[8]

Definition and Structure

Local trivializations

A holomorphic vector bundle over a complex manifold X is a complex vector bundle equipped with a holomorphic structure, meaning that it admits an atlas of local trivializations related by holomorphic transition maps.^[9] More precisely, let E \to X be a complex vector bundle of rank r, with total space E a complex manifold and projection \pi: E \to X a holomorphic submersion such that each fiber E_x = \pi^{-1}(x) is a complex vector space isomorphic to \mathbb{C}^r.^[9] The holomorphic structure is induced by requiring that E is locally holomorphically trivial.^[10] To elaborate, consider an open cover \{U_i\}_{i \in I} of X. For each i, the restriction of the bundle over U_i, denoted E|_{U_i}, is holomorphically trivial, meaning there exists a biholomorphic map \phi_i: \pi^{-1}(U_i) \to U_i \times \mathbb{C}^r such that the following diagram commutes:

\begin{CD} \pi^{-1}(U_i) @>{\phi_i}>> U_i \times \mathbb{C}^r \\ @V{\pi}VV @VV{\mathrm{pr}_1}V \\ U_i @= U_i, \end{CD}

where \mathrm{pr}_1 is the projection onto the first factor.^[9] The collection \{(U_i, \phi_i)\}_{i \in I} forms a holomorphic atlas of trivializations, provided the transition maps are holomorphic.^[10] On overlaps U_i \cap U_j \neq \emptyset, the transition maps g_{ij}: U_i \cap U_j \to \mathrm{GL}(r, \mathbb{C}) are defined by \phi_j \circ \phi_i^{-1}(u, v) = (u, g_{ij}(u) \cdot v) for u \in U_i \cap U_j and v \in \mathbb{C}^r.^[9] These maps must be holomorphic functions, satisfying the cocycle condition g_{ij}(u) \cdot g_{jk}(u) = g_{ik}(u) on triple overlaps U_i \cap U_j \cap U_k, and g_{ji} = g_{ij}^{-1}.^[9] This ensures the bundle is consistently glued across the cover while preserving the holomorphic structure.^[10] The holomorphic atlas implicitly defines the sheaf of holomorphic sections of the bundle, denoted \mathcal{O}_E, where over each U_i, sections correspond to holomorphic maps s_i: U_i \to \mathbb{C}^r via \phi_i, and compatibility on overlaps requires s_j(u) = g_{ij}(u) \cdot s_i(u).^[9] This sheaf structure captures the global holomorphic sections as those that glue holomorphically from the local frames.^[10]

Transition functions and holomorphic structure

A holomorphic vector bundle over a complex manifold M is constructed by endowing a smooth complex vector bundle with a holomorphic structure. This is achieved by selecting an open cover \{U_i\}_{i \in I} of M and associating to each pair i, j a holomorphic transition function g_{ij}: U_i \cap U_j \to \mathrm{GL}(r, \mathbb{C}), where r is the rank of the bundle. These functions must satisfy the cocycle condition g_{ik} = g_{ij} g_{jk} on triple overlaps U_i \cap U_j \cap U_k, ensuring consistent gluing of the local trivializations \pi^{-1}(U_i) \cong U_i \times \mathbb{C}^r. The total space E then inherits a complex manifold structure from these biholomorphic trivializations, with the projection \pi: E \to M being holomorphic.^[8]^[4] The transition functions uniquely determine the holomorphic structure up to isomorphism. Specifically, two collections of transition functions \{g_{ij}\} and \{g'_{ij}\} define isomorphic bundles if there exist holomorphic maps h_i: U_i \to \mathrm{GL}(r, \mathbb{C}) such that g'_{ij} = h_i g_{ij} h_j^{-1} on U_i \cap U_j, meaning the collections differ by a holomorphic coboundary. This equivalence relation classifies holomorphic vector bundles in terms of non-abelian Čech cohomology, where the isomorphism classes correspond to elements in H^1(M, \mathrm{GL}(r, \mathcal{O}_M)), with \mathcal{O}_M the sheaf of holomorphic functions on M; though the explicit cocycle description provides the concrete gluing data.^[11]^[8] With this structure in place, sections of the bundle acquire a notion of holomorphy. A section s: U \to E over an open subset U \subset M is holomorphic if it is a holomorphic map satisfying \pi \circ s = \mathrm{id}_U, and locally, in a trivialization over U_i, it takes the form

s(z) = \sum_{k=1}^r f_k(z) \, e_k,

where \{e_k\}_{k=1}^r is a local holomorphic frame and each f_k: U_i \to \mathbb{C} is a holomorphic function. This local expression ensures compatibility across overlaps via the transition functions, as s on U_j transforms accordingly to maintain holomorphy.^[4]^[11] In contrast to algebraic vector bundles, which are defined on algebraic varieties using regular (e.g., polynomial) transition functions within the algebraic geometry framework, holomorphic vector bundles reside in the complex analytic category. Here, the transition functions are merely holomorphic, enabling constructions on general complex manifolds that may not admit algebraic structures.^[12]

Sections and Sheaves

Sheaf of holomorphic sections

Given a holomorphic vector bundle E \to X over a complex manifold X, the sheaf \mathcal{E} of holomorphic sections is constructed by assigning to each open subset U \subset X the \mathbb{C}-vector space \Gamma(U, \mathcal{E}) consisting of all holomorphic maps s: U \to E|_U such that the projection \pi \circ s = \mathrm{id}_U and, locally over trivializing open sets U_\alpha \subset U, s corresponds to a holomorphic function with values in \mathbb{C}^r via the trivialization \psi_\alpha: \pi^{-1}(U_\alpha) \to U_\alpha \times \mathbb{C}^r.^[13]^[14] The sheaf \mathcal{E} satisfies the standard sheaf axioms: for any open cover \{U_i\}_{i \in I} of an open set U \subset X, the restriction maps \rho_{U, U_i}: \Gamma(U, \mathcal{E}) \to \Gamma(U_i, \mathcal{E}), defined by projecting sections along the bundle fibers, are holomorphic linear maps satisfying \rho_{U, U_i} \circ \rho_{V, U} = \rho_{V, U_i} for U_i \subset U \subset V; moreover, sections over the U_i glue uniquely to a section over U if they agree on pairwise intersections U_i \cap U_j, where agreement is determined by the holomorphic transition functions g_{\alpha\beta}: U_\alpha \cap U_\beta \to \mathrm{GL}(r, \mathbb{C}) of the bundle such that the local representations match via s_\alpha = g_{\alpha\beta} s_\beta.^[13]^[15] The sheaf \mathcal{E} is naturally a sheaf of \mathcal{O}_X-modules, where \mathcal{O}_X is the structure sheaf of holomorphic functions on X, with the module structure given by pointwise multiplication: for f \in \Gamma(U, \mathcal{O}_X) and s \in \Gamma(U, \mathcal{E}), (f \cdot s)(x) = f(x) \cdot s(x) in the fiber E_x. Since E is locally trivial of finite rank r, \mathcal{E} is locally free of rank r over \mathcal{O}_X and hence a coherent \mathcal{O}_X-module sheaf.^[10]^[14] For a holomorphic morphism f: Y \to X of complex manifolds, the direct image sheaf f_* \mathcal{E} (or pushforward sheaf) is defined by (f_* \mathcal{E})(U) = \Gamma(f^{-1}(U), \mathcal{E}) for open U \subset X, equipped with the natural \mathcal{O}_X-module structure induced by composition with f, providing an algebraic framework to study sections of \mathcal{E} pulled back to Y. This construction preserves coherence and is functorial in the category of holomorphic vector bundles and morphisms.^[13]^[10]

Global holomorphic sections

The space of global holomorphic sections of a holomorphic vector bundle E \to X over a complex manifold X, denoted H^0(X, E) or \Gamma(X, E), consists of all holomorphic maps s: X \to E satisfying \pi \circ s = \mathrm{id}_X, where \pi: E \to X is the bundle projection.^[8] Locally, over a trivializing open cover \{U_\alpha\} with transition functions g_{\alpha\beta}: U_\alpha \cap U_\beta \to \mathrm{GL}_r(\mathbb{C}), a section s is represented by holomorphic functions s_\alpha: U_\alpha \to \mathbb{C}^r such that s_\alpha = g_{\alpha\beta} s_\beta on overlaps, and holomorphicity requires that each s_\alpha lies in the kernel of the \bar{\partial}-operator, defined componentwise as \bar{\partial} s_\alpha = (\bar{\partial} s_{\alpha,1}, \dots, \bar{\partial} s_{\alpha,r}).^[8] On a compact complex manifold X, the space \Gamma(X, \mathcal{E}) of global sections of the sheaf \mathcal{E} of holomorphic sections of E is finite-dimensional as a complex vector space, since \mathcal{E} is coherent and the cohomology groups of coherent sheaves on compact manifolds are finite-dimensional by the Cartan-Serre theorem.^[16] This finiteness follows from the coherence of the structure sheaf \mathcal{O}_X, established by Oka's theorem, which ensures that \mathcal{E} admits finite locally free resolutions, implying bounded dimensionality of global sections.^[17] For the trivial bundle E = X \times \mathbb{C}^r, the global holomorphic sections are precisely the r-tuples of global holomorphic functions on X.^[8] In contrast, for an ample line bundle L on a projective manifold X of dimension n, the space of sections \Gamma(X, L^k) grows richly with k, satisfying \dim \Gamma(X, L^k) \sim C k^n for some constant C > 0 independent of k, by the asymptotic Hirzebruch-Riemann-Roch theorem, reflecting the positivity that allows embedding X via high powers.^[18] Geometrically, for a line bundle L, the zero locus of a non-zero global holomorphic section s \in \Gamma(X, L) defines a divisor on X, whose associated line bundle is isomorphic to L, providing a correspondence between effective divisors and sections up to scaling.^[19]

Examples

Trivial and direct sum bundles

A trivial holomorphic vector bundle of rank r over a complex manifold X is the product bundle E = X \times \mathbb{C}^r equipped with the projection map \mathrm{pr}_1: (x, v) \mapsto x.^[20] The transition functions with respect to any trivializing cover are the identity matrices in \mathrm{GL}(r, \mathbb{C}), ensuring the holomorphic structure is induced directly from that of X.^[20] The holomorphic sections of such a bundle are precisely the holomorphic maps s: X \to \mathbb{C}^r, which assign to each point an element of the fiber in a holomorphic manner.^[20] The direct sum of two holomorphic vector bundles E and F over the same base X, denoted E \oplus F, has total space consisting of pairs (e, f) with e \in E and f \in F such that \pi_E(e) = \pi_F(f), and projection onto X given by this common base point.^[20] If E and F have ranks r and s, then E \oplus F has rank r + s, with fibers E_x \oplus F_x at each x \in X.^[20] The holomorphic structure on E \oplus F is defined using transition functions that are block-diagonal matrices, combining the transition functions of E and F along the diagonal.^[20] Holomorphic sections of E \oplus F are pairs (s_E, s_F) where s_E and s_F are holomorphic sections of E and F, respectively.^[20] The Hom bundle \mathrm{Hom}(E, F) between holomorphic vector bundles E and F over X has fibers consisting of linear maps between E_x and F_x, and in particular the endomorphism bundle \mathrm{End}(E) = \mathrm{Hom}(E, E) can be identified with the tensor product E^* \otimes E, where E^* is the dual bundle.^[20] The holomorphic structure on E^* uses transition functions that are the adjoints (inverse transposes) of those for E, ensuring the tensor product inherits a natural holomorphic structure via the product of transition functions.^[20] A holomorphic vector bundle is trivial if and only if it admits a global holomorphic frame, that is, r global holomorphic sections that are linearly independent at every point and span the fiber.^[20] Necessary conditions for triviality include the base manifold being parallelizable (for the tangent bundle case) or all Chern classes of the bundle vanishing.^[20]

Tangent and cotangent bundles

The holomorphic tangent bundle T^{1,0}X of a complex manifold X of dimension n provides a fundamental example of a holomorphic vector bundle of rank n. Given a holomorphic atlas \{ (U_\alpha, z^\alpha) \} for X, where z^\alpha = (z^1_\alpha, \dots, z^n_\alpha) are local holomorphic coordinates on U_\alpha, the bundle admits a local trivialization \phi_\alpha: \pi^{-1}(U_\alpha) \to U_\alpha \times \mathbb{C}^n over each chart, with the fiber at p \in U_\alpha spanned by the basis \left\{ \frac{\partial}{\partial z^i_\alpha} \big|_p \right\}_{i=1}^n.^[21] On overlapping charts U_\alpha \cap U_\beta, with coordinate change z^j_\alpha = h^j_{\alpha\beta}(z^\beta) given by a biholomorphism h_{\alpha\beta}: z^\beta(U_\alpha \cap U_\beta) \to z^\alpha(U_\alpha \cap U_\beta), the transition functions for T^{1,0}X are the entries of the Jacobian matrix g_{\alpha\beta}(z^\beta) = \left( \frac{\partial z^j_\alpha}{\partial z^k_\beta} \right)_{j,k=1}^n, valued in \mathrm{GL}_n(\mathbb{C}). These functions are holomorphic because h_{\alpha\beta} is biholomorphic, so each partial derivative \frac{\partial z^j_\alpha}{\partial z^k_\beta} is a holomorphic function of z^\beta, as ensured by the chain rule applied to the composition of holomorphic maps.^[21] The holomorphic cotangent bundle \Omega^1_X, also of rank n, is the dual vector bundle (T^{1,0}X)^*. Its local trivializations are induced by the dual basis \{ dz^i_\alpha \} over U_\alpha, and the transition functions are the inverse transpose matrices g_{\alpha\beta}^{*}(z^\beta) = \left( g_{\beta\alpha}(z^\alpha) \right)^{-T}, which remain holomorphic since the original Jacobians are invertible holomorphic matrices.^[22] Global holomorphic sections of T^{1,0}X are precisely the holomorphic vector fields on X, which in local coordinates take the form \sum_{i=1}^n f_i(z) \frac{\partial}{\partial z^i} with each f_i holomorphic on the chart. Similarly, holomorphic sections of \Omega^1_X are the holomorphic 1-forms, locally expressed as \sum_{j=1}^n g_j(z) \, dz^j with g_j holomorphic.^[21]^[22] The higher exterior powers \Omega^p_X = \bigwedge^p \Omega^1_X for $1 \leq p \leq n form holomorphic vector bundles of rank \binom{n}{p}, whose fibers consist of decomposable p-vectors in the alternating algebra on the cotangent fibers. The transition functions for \Omega^p_X are induced by the action of the cotangent transition matrices on the exterior power, yielding expressions involving pluri-differentials such as dz^{j_1} \wedge \cdots \wedge dz^{j_p} = \det\left( \frac{\partial z^{j_k}_\alpha}{\partial z^{\ell_m}_\beta} \right)_{k,m} \, dw^{\ell_1} \wedge \cdots \wedge dw^{\ell_p} over suitable minors, with holomorphicity preserved by the determinant of holomorphic matrices. In particular, for p = n, \Omega^n_X is the canonical line bundle, with transition functions given by the determinant \det(g_{\alpha\beta}^{*}).^[22]^[11] On \mathbb{C}^n, both T^{1,0}\mathbb{C}^n and \Omega^1_{\mathbb{C}^n} are holomorphically trivial bundles, isomorphic to \mathbb{C}^n \times \mathbb{C}^n, with global sections comprising entire vector fields \sum f_i(z) \frac{\partial}{\partial z^i} and entire 1-forms \sum g_j(z) \, dz^j where the f_i, g_j are entire functions. On a Riemann surface (complex dimension 1), T^{1,0}X reduces to the holomorphic tangent line bundle, while \Omega^1_X is the canonical line bundle whose global sections are the holomorphic differentials, forming a vector space of dimension equal to the genus of X.^[11]^[23]

Dolbeault Apparatus

Dolbeault operators

In the context of a holomorphic vector bundle E \to X over a complex manifold X, the Dolbeault operator \bar{\partial}_E is defined as the (0,1)-part of the exterior derivative d acting on the space A^{0,q}(X, E) of smooth sections of \Lambda^{0,q} T^*X \otimes E. This operator is a \mathbb{C}-linear differential \bar{\partial}_E: A^{0,q}(X, E) \to A^{0,q+1}(X, E) of bidegree (0,1) that satisfies the nilpotency condition \bar{\partial}_E^2 = 0 and the graded Leibniz rule \bar{\partial}_E(f \sigma) = \bar{\partial} f \wedge \sigma + (-1)^q f \bar{\partial}_E \sigma for a smooth function f on X and a section \sigma \in A^{0,q}(X, E). When f is holomorphic, \bar{\partial} f = 0, so the rule simplifies to \bar{\partial}_E(f \sigma) = f \bar{\partial}_E \sigma.^[10]^[24] Locally, over a trivializing open set U \subset X where E|_U \cong U \times \mathbb{C}^r, a section \sigma \in A^{0,q}(U, E) is represented by an \mathbb{C}^r-valued (0,q)-form s, and the action of \bar{\partial}_E takes the form

\bar{\partial}_E \sigma = \bar{\partial} s + \bar{\theta} \wedge s,

where \bar{\theta} \in A^{0,1}(U, \mathfrak{gl}(r, \mathbb{C})) is the (0,1)-connection form determined by the holomorphic transition functions of E. On overlaps U \cap V, the forms \bar{\theta}_U and \bar{\theta}_V are related by the transformation law \bar{\theta}_U = g_{UV} \bar{\theta}_V g_{UV}^{-1} + \bar{\partial} g_{UV} \cdot g_{UV}^{-1}, ensuring the operator is globally well-defined without invoking a full connection on E.^[10]^[24] The space of holomorphic sections of E, denoted \mathcal{O}(X, E), is precisely the kernel of \bar{\partial}_E acting on A^{0,0}(X, E), i.e., \mathcal{O}(X, E) = \ker(\bar{\partial}_E: A^{0,0}(X, E) \to A^{0,1}(X, E)). This identifies the sheaf of holomorphic sections with the zeroth cohomology of the Dolbeault complex (A^{0,\bullet}(X, E), \bar{\partial}_E), which resolves the structure sheaf \mathcal{O}_E.^[10] The holomorphic structure on E is integrable, meaning \bar{\partial}_E^2 = 0 holds globally, if and only if the local (0,1)-connection forms satisfy the Maurer-Cartan equation

\bar{\partial} \bar{\theta} + \bar{\theta} \wedge \bar{\theta} = 0

on each trivializing set; this ensures consistency under the Newlander-Nirenberg theorem that the distribution defined by the kernel of \bar{\theta} is integrable.^[10]^[24]

Cohomology of holomorphic vector bundles

The Dolbeault cohomology groups of a holomorphic vector bundle E over a complex manifold X are defined as H^{p,q}(X, E) = \ker(\bar{\partial}_E : A^{p,q}(X, E) \to A^{p,q+1}(X, E)) / \operatorname{im}(\bar{\partial}_E : A^{p,q-1}(X, E) \to A^{p,q}(X, E)), where A^{p,q}(X, E) denotes the space of smooth global sections of the bundle \Lambda^{p,0}T^*X \otimes \overline{\Lambda^{0,q}T^*X} \otimes E and \bar{\partial}_E is the extension of the Dolbeault operator to E-valued forms.^[25] These groups can also be computed as the sheaf cohomology \check{H}^q(X, \mathcal{A}^{p,E}), where \mathcal{A}^{p,E} is the sheaf of smooth E-valued (p,q)-forms on X.^[25] By Dolbeault's theorem, for a complex manifold X, there is an isomorphism of vector spaces H^{p,q}(X, E) \cong H^q(X, \Omega^p \otimes E), where \Omega^p is the sheaf of holomorphic p-forms on X.^[25]^[26] Additionally, Serre duality establishes a natural pairing between H^q(X, E) and H^{n-q}(X, E^* \otimes K_X), where n = \dim X, E^* is the dual bundle, and K_X is the canonical bundle, yielding \dim H^q(X, E) = \dim H^{n-q}(X, E^* \otimes K_X).^[27] For line bundles L on a compact Riemann surface X of genus g, the Riemann-Roch theorem provides a dimension formula \dim H^0(X, L) - \dim H^1(X, L) = \deg L + 1 - g, which computes the space of global holomorphic sections when higher cohomology vanishes, such as for sufficiently positive degree.^[28]

Forms and Cohomology

Sheaves of bundle-valued forms

In the context of a holomorphic vector bundle E over a complex manifold X, the sheaf \mathcal{A}^{p,q}_E is defined as the sheaf of smooth sections of the vector bundle \Lambda^p [T^*X](/page/T-X) \otimes \Lambda^q \overline{T}^*X \otimes E.^[15] This sheaf generalizes the structure sheaf of smooth functions on X by incorporating the bundle E and the decomposition of the cotangent bundle into holomorphic and anti-holomorphic parts according to the complex structure. If E is holomorphic, there exists a subsheaf consisting of holomorphic E-valued (p,q)-forms, which are the sections annihilated by the \overline{\partial} operator in the appropriate sense.^[15] Locally, on a coordinate chart U \subset X where E is trivialized as U \times \mathbb{C}^r, the sections of \mathcal{A}^{p,q}_E over U take the form \sum_{I,J} dz^I \wedge d\overline{z}^J \otimes \sigma_{I,J}, where I and J are multi-indices of lengths p and q respectively, and each \sigma_{I,J} is a smooth local section of the trivial bundle over U.^[15] Under transition functions between overlapping trivializations, the representation transforms via the holomorphic transition maps acting on the E-valued coefficients \sigma_{I,J}, combined with the change of coordinates affecting the differentials dz^I and d\overline{z}^J. This ensures that \mathcal{A}^{p,q}_E is well-defined globally as a sheaf on X. The sheaves \mathcal{A}^{0,q}_E for q \geq 0 form part of the Dolbeault complex $0 \to \mathcal{O}_X \otimes E \to \mathcal{A}^{0,1}_E \to \mathcal{A}^{0,2}_E \to \cdots, which provides a fine resolution of the sheaf of holomorphic sections \mathcal{O}_X \otimes E (also denoted \mathcal{O}(E)).^[29] This resolution is exact on the level of sheaves, allowing the computation of sheaf cohomology via the hypercohomology of the complex.^[15] Note that the degree (0,0) case corresponds to the sheaf of smooth E-valued functions, which contains the sheaf of holomorphic sections as a subsheaf. At the level of germs and stalks, for a point x \in X, the stalk (\mathcal{A}^{p,q}_E)_x is isomorphic to the tensor product of the stalk of smooth (p,q)-forms at x with the fiber E_x.^[15] This reflects the local triviality of both the form bundles and E, where sections near x behave like matrix-valued forms tensored with the vector space E_x.^[29]

de Rham and Dolbeault cohomology comparison

The de Rham cohomology of a holomorphic vector bundle E over a complex manifold X is defined using the complex of smooth E-valued differential forms \Omega^\bullet(X, E), equipped with the total exterior derivative d = \partial + \bar{\partial}. The resulting cohomology groups H^k_{\mathrm{dR}}(X, E) are isomorphic to H^k(X, \mathbb{C}) \otimes_{\mathbb{C}} \mathbb{C}^r, where r = \mathrm{rank}(E), capturing topological invariants of the base X scaled by the rank of E.^[20]^[30] In contrast, the Dolbeault cohomology of E arises from the \bar{\partial}-complex of smooth E-valued (p,q)-forms, yielding groups H^{p,q}_{\bar{\partial}}(X, E) that are isomorphic to the sheaf cohomology H^q(X, \Omega^p \otimes E), reflecting the holomorphic structure of E.^[20]^[30] On a compact Kähler manifold, the \partial \bar{\partial}-lemma provides a precise link between these theories: for any d-closed E-valued form of bidegree (p,q), exactness with respect to d, \partial, or \bar{\partial} is equivalent to exactness with respect to \partial \bar{\partial}. This implies the Hodge decomposition H^k_{\mathrm{dR}}(X, E) \cong \bigoplus_{p+q=k} H^{p,q}_{\bar{\partial}}(X, E), decomposing the topological cohomology into purely holomorphic components.^[20] Bott-Chern cohomology offers an intermediate refinement, defined via the double complex (\Omega^{p,q}(X, E), \partial, \bar{\partial}) as H^{p,q}_{\mathrm{BC}}(X, E) = \ker(d \colon \Omega^{p,q}(X, E) \to \Omega^{p+q+1}(X, E)) / \operatorname{im}(\partial \bar{\partial} \colon \Omega^{p-1,q-1}(X, E) \to \Omega^{p,q}(X, E)). Unlike de Rham or Dolbeault cohomology, Bott-Chern groups need not vanish even when both H^\bullet_{\mathrm{dR}}(X, E) and H^{\bullet,\bullet}_{\bar{\partial}}(X, E) do, highlighting obstructions that blend analytic and topological features of the holomorphic structure. On compact Kähler manifolds, however, Bott-Chern cohomology is isomorphic to both de Rham and Dolbeault cohomologies.^[20] On compact complex manifolds without a Kähler structure, de Rham cohomology primarily detects the topology of the base manifold X (such as its Betti numbers), scaled by the rank of E, while Dolbeault cohomology probes the existence of holomorphic sections or extensions, often vanishing in higher degrees by Kodaira vanishing theorems under suitable metric conditions. For instance, if E is ample on a projective manifold, H^q(X, \Omega^p \otimes E) = 0 for p+q > \dim X and suitable positivity, whereas de Rham groups remain non-trivial if the topology demands it.^[20]^[30]

Line Bundles and Classification

The Picard group

A holomorphic line bundle over a complex manifold X is a rank-one holomorphic vector bundle, locally trivialized by holomorphic sections with transition functions g_{ij}: U_i \cap U_j \to \mathbb{C}^* satisfying the Čech cocycle condition g_{ij} g_{jk} = g_{ik} on triple overlaps U_i \cap U_j \cap U_k, where \{U_i\} is an open cover of X.^[31] These transition functions ensure the bundle's sections glue holomorphically across the cover.^[31] Two holomorphic line bundles are isomorphic if their transition functions differ by a coboundary, meaning there exist holomorphic functions h_i: U_i \to \mathbb{C}^* such that the new transition functions satisfy g'_{ij} = h_i g_{ij} h_j^{-1}, or equivalently, g'_{ij} g_{ij}^{-1} = h_i h_j^{-1}.^[31] The set of isomorphism classes of holomorphic line bundles on X, equipped with the tensor product operation, forms an abelian group known as the Picard group \operatorname{Pic}(X).^[32] This group is isomorphic to the first Čech cohomology group H^1(X, \mathcal{O}_X^*), where \mathcal{O}_X^* is the sheaf of nowhere-vanishing holomorphic functions on X.^[32] The cohomology captures the obstruction to trivializing the bundle globally, with cocycles corresponding to transition functions modulo coboundaries.^[33] For example, on the projective line \mathbb{CP}^1, the Picard group is \operatorname{Pic}(\mathbb{CP}^1) \cong \mathbb{Z}, generated by the isomorphism class of the tautological line bundle \mathcal{O}(1), whose powers \mathcal{O}(d) correspond to line bundles of degree d.^[33] More generally, \operatorname{Pic}(X) is isomorphic to the quotient of the group of divisors \operatorname{Div}(X) by the subgroup of principal divisors \operatorname{Prin}(X), linking line bundles to meromorphic divisors on X.^[33]

Classification via cohomology

The isomorphism classes of rank-r holomorphic vector bundles over a complex manifold X form a pointed set in bijection with the non-abelian sheaf cohomology group H^1(X, \mathrm{GL}(r, \mathcal{O}_X)), where \mathcal{O}_X denotes the structure sheaf of holomorphic functions and the base point corresponds to the trivial bundle. This classification stems from describing such bundles via open covers with holomorphic transition functions valued in \mathrm{GL}(r, \mathbb{C}), where two cocycles define isomorphic bundles if they differ by a coboundary (holomorphic gauge transformations). For rank 1, this specializes to the Picard group \mathrm{Pic}(X) \cong H^1(X, \mathcal{O}_X^\times), parametrizing line bundles up to isomorphism.^[34] Obstructions to the triviality of a holomorphic vector bundle E lie in H^1(X, \mathrm{GL}(r, \mathcal{O}_X)); specifically, E is holomorphically trivial if and only if its classifying cocycle is cohomologous to the constant identity function, allowing a global frame of r nowhere-vanishing holomorphic sections. More generally, extensions of holomorphic vector bundles, captured in short exact sequences $0 \to F \to E \to Q \to 0, are classified by elements of \mathrm{Ext}^1(Q, F) \cong H^1(X, \mathrm{Hom}(Q, F)), measuring how E fails to split holomorphically. These cohomology groups provide analytic invariants distinguishing non-isomorphic bundles, with vanishing higher cohomology H^i(X, \mathrm{GL}(r, \mathcal{O}_X)) = 0 for i \geq 2 on Stein manifolds ensuring no further obstructions.^[35] On projective varieties, where the holomorphic and algebraic categories coincide, stable isomorphism classes of holomorphic vector bundles are parametrized by moduli spaces, which are quasi-projective algebraic varieties encoding bundles up to S-equivalence (semistable limits). For the projective line \mathbb{P}^1, the Birkhoff–Grothendieck theorem asserts that every holomorphic vector bundle decomposes uniquely (up to ordering) as a direct sum E \cong \bigoplus_{i=1}^r \mathcal{O}_{\mathbb{P}^1}(d_i) of line bundles, with integers d_i determined by the degrees; this provides a complete algebraic classification extensible to the analytic setting. In cases of fixed topological type (e.g., trivial underlying smooth bundle), the space of holomorphic structures is an open subset of a Grassmannian parametrizing compatible complex structures, though global classification relies on these cohomology obstructions rather than finite-dimensional Grassmannians alone.^[6] For compact Riemann surfaces, a key analytic relation links holomorphic vector bundles to unitary representations: by the Narasimhan–Seshadri theorem, stable holomorphic bundles of degree zero correspond precisely to irreducible projective unitary representations of the fundamental group \pi_1(X, x_0) up to conjugacy, realized via flat unitary connections on the underlying smooth bundle. This correspondence highlights how cohomology classifies the holomorphic structures, while stability ensures projectivity of the moduli space, bridging algebraic and representation-theoretic viewpoints without relying on metrics.^[36]

Metrics and Connections

Hermitian metrics

A Hermitian metric on a holomorphic vector bundle E \to X, where X is a complex manifold, is a smooth assignment to each point x \in X of a positive definite Hermitian inner product h_x: E_x \times E_x \to \mathbb{C} on the fiber E_x \cong \mathbb{C}^r, such that the assignment x \mapsto h_x varies smoothly with x.^[37]^[8] This metric provides a way to measure lengths and angles within each fiber, respecting the complex structure in the sense that it is defined on the underlying smooth complex vector bundle. The inner product h_x is sesquilinear, meaning it is \mathbb{C}-linear in the first argument and conjugate-linear in the second, with h_x(\sigma, \tau) = \overline{h_x(\tau, \sigma)} for sections \sigma, \tau, and positive definite, satisfying h_x(\sigma, \sigma) > 0 for all nonzero \sigma \in E_x.^[37] In a local trivialization of E over an open set U \subset X, with holomorphic frame \{e_k\}_{k=1}^r, any sections \sigma = \sum f_k e_k and \tau = \sum g_l e_l satisfy h(\sigma, \tau) = \sum_{k,l} \overline{f_k} \, h_{kl} \, g_l, where (h_{kl}) is a smooth positive definite Hermitian matrix on U with h_{kl} = \overline{h_{lk}}.^[8] Such a metric induces a smooth unitary frame locally, orthonormal with respect to h, though this frame need not align with the holomorphic structure beyond the smooth variation. On a paracompact complex manifold X, every holomorphic vector bundle admits a Hermitian metric, constructed via a partition of unity subordinate to a locally finite open cover by trivializing sets.^[37] Specifically, local Hermitian metrics on each trivialization are glued globally by weighting with the partition functions \{\lambda_i\}, yielding h = \sum_i \lambda_i h_i, which is smooth and positive definite everywhere.^[38] This existence relies on the paracompactness of X, ensuring the availability of such partitions of unity.

Chern connections and curvature

Given a holomorphic vector bundle E \to X over a complex manifold X, equipped with a Hermitian metric h, there exists a unique connection \nabla, called the Chern connection, that is compatible with both the holomorphic structure and the metric. This means \nabla^{0,1} = \bar{\partial}_E and \nabla h = 0, ensuring that the connection preserves the metric in the sense that h(\nabla s, t) + h(s, \nabla t) = 0 for sections s, t \in \Gamma(E).^[39] Locally, in a holomorphic frame where the metric is represented by the matrix h = (h_{\alpha \bar{\beta}}), the (1,0)-part of the connection form is \theta = h^{-1} \partial h, so \nabla = d + \theta with the (0,1)-part being \bar{\partial}.^[40] The curvature form \Omega of the Chern connection is defined by \Omega = \nabla^2, which acts on sections as \Omega(s) = \nabla^2 s. In terms of the connection forms, it computes locally as \Omega = \bar{\partial} \theta + \theta \wedge \theta, and due to the holomorphic compatibility, \Omega is a (1,1)-form taking values in \operatorname{End}(E).^[39] More explicitly, in a holomorphic frame, \Omega = \bar{\partial} (h^{-1} \partial h), reflecting the metric's influence on the bundle's geometry.^[40] This curvature satisfies the Bianchi identity d_\nabla \Omega = 0, ensuring its closedness in de Rham cohomology.^[39] The Chern classes of E, which are topological invariants in H^{2k}(X, \mathbb{Z}), arise from the curvature via Chern-Weil theory. Specifically, the k-th Chern class is represented by the cohomology class c_k(E) = \left[ \left( \frac{i}{2\pi} \right)^k \operatorname{Tr}(\wedge^k \Omega) \right], where \wedge^k \Omega denotes the k-th exterior power in the Chern character or via the total class \det\left( \operatorname{Id} + \frac{i}{2\pi} \Omega \right) = \sum c_k(E).^[39] These classes are independent of the choice of Hermitian metric and connection, capturing essential topological features of the bundle.^[39] A special class of Hermitian metrics on E are the Hermitian-Einstein metrics, for which the curvature satisfies the condition \operatorname{Ric}(\Omega) = \lambda \operatorname{Id} for some constant \lambda, where \operatorname{Ric}(\Omega) is the trace of \Omega contracted with a Kähler form on X. Such metrics are linked to the stability of the holomorphic vector bundle, as established in foundational results on the existence for stable bundles over compact Kähler manifolds.

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