In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold X such that the total space E is a complex manifold and...

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space...

vector bundle is a (holomorphic or algebraic) vector bundle that is stable in the sense of geometric invariant theory. Any holomorphic vector bundle may...

complex vector bundle is canonically oriented; in particular, one can take its Euler class. A complex vector bundle is a holomorphic vector bundle if X is a...

complex geometry, the holomorphic tangent bundle of a complex manifold M {\displaystyle M} is the holomorphic analogue of the tangent bundle of a smooth manifold...

In mathematics, a Higgs bundle is a pair ( E , φ ) {\displaystyle (E,\varphi )} consisting of a holomorphic vector bundle E and a Higgs field φ {\displaystyle...

tangent bundle is a way of organising these. More formally, in algebraic topology and differential topology, a line bundle is defined as a vector bundle of...

functions of several complex variables, and holomorphic constructions such as holomorphic vector bundles and coherent sheaves. Application of transcendental...

Birkhoff–Grothendieck theorem classifies holomorphic vector bundles over the complex projective line. In particular every holomorphic vector bundle over C P 1 {\displaystyle...

Chern–Weil theory that computes topological invariants of vector bundles and principal bundles on a smooth manifold M in terms of connections and curvature...

connection) is a Chern connection associated to an inner product on a holomorphic vector bundle over a Kähler manifold that satisfies an analogue of Einstein's...

applied this new theory vector bundles to develop a notion of slope stability. Define the degree of a holomorphic vector bundle E → ( X , ω ) {\displaystyle...

gauge theory is the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should not be confused...

Hirzebruch–Riemann–Roch theorem applies to any holomorphic vector bundle E on a compact complex manifold X, to calculate the holomorphic Euler characteristic of E in sheaf...

same duality statement for X a compact complex manifold and E a holomorphic vector bundle. Here, the Serre duality theorem is a consequence of Hodge theory...

In mathematics, vector bundles on algebraic curves may be studied as holomorphic vector bundles on compact Riemann surfaces, which is the classical approach...

{\displaystyle (E,\Phi )} where E → X {\displaystyle E\to X} is a holomorphic vector bundle and Φ : E → E ⊗ Ω 1 {\displaystyle \Phi :E\to E\otimes {\boldsymbol...

any noncritical value of a holomorphic map. Smooth complex algebraic varieties are complex manifolds, including: Complex vector spaces. Complex projective...

Hermitian metrics on a holomorphic vector bundle. In particular, if the base manifold is Kähler and the vector bundle is its tangent bundle, then the Chern connection...

canonical bundle is anti-ample Matsusaka's big theorem Divisorial scheme: a scheme admitting an ample family of line bundles Holomorphic vector bundle Kodaira...

Röhrl (1956), states moreover that every holomorphic vector bundle on X is trivial. In particular, every line bundle is trivial, so H 1 ( X , O X ∗ ) = 0...

theorem, proved by Narasimhan and Seshadri (1965), says that a holomorphic vector bundle over a Riemann surface is stable if and only if it comes from...

associated bundle E = P × GL ⁡ ( n , C ) C n {\displaystyle E=P\times _{\operatorname {GL} (n,\mathbb {C} )}\mathbb {C} ^{n}} . This is a holomorphic vector bundle...

information. Coherent sheaves can be seen as a generalization of vector bundles. Unlike vector bundles, they form an abelian category, and so they are closed under...

{\displaystyle \Sigma } . A pair consisting of a holomorphic vector bundle E {\displaystyle E} with a holomorphic endomorphism-valued ( 1 , 0 ) {\displaystyle...

bundle Ω {\displaystyle \Omega } on V {\displaystyle V} . Over the complex numbers, it is the determinant bundle of the holomorphic cotangent bundle T...

consider D, the sheaf of differential operators.) fractional ideal holomorphic vector bundle generic freeness Vakil, Math 216: Foundations of algebraic geometry...

the theory of holomorphic vector bundles (more generally coherent analytic sheaves) on X coincide with that of algebraic vector bundles. Chow's theorem...

{\partial }}:\Omega ^{p,q-1}\to \Omega ^{p,q})}}.} If E is a holomorphic vector bundle on a complex manifold X, then one can define likewise a fine resolution...

d{\bar {z}}^{n}.} One can also consider a hermitian metric on a holomorphic vector bundle. The most important class of Hermitian manifolds are Kähler manifolds...