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KR-theory
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involution. A real vector bundle over a real space X is defined to be a complex vector bundle E over X that is also a real space, such that the natural maps fromList of things named after Charles Hermite (405 words) [view diff] exact match in snippet view article find links to article
smoothly varying positive-definite Hermitian form on each fiber of a complex vector bundle Hermitian matrix, a square matrix with complex entries that is equalPontryagin class (1,906 words) [view diff] exact match in snippet view article find links to article
homotopy type and Pontryagin classes. The Pontryagin classes of a complex vector bundle π : E → X {\displaystyle \pi :E\to X} is completely determined byHiggs bundle (548 words) [view diff] exact match in snippet view article find links to article
{\displaystyle \operatorname {deg} } being the usual notion of degree for a complex vector bundle over a Riemann surface. Hitchin system Hitchin, Nigel (1987). "TheQuillen metric (2,063 words) [view diff] exact match in snippet view article find links to article
operators parametrised by the space of unitary connections on a complex vector bundle over a compact Riemann surface. In this section the constructionTodd class (1,168 words) [view diff] exact match in snippet view article find links to article
{\displaystyle \operatorname {td} (E)} where E {\displaystyle E} is a complex vector bundle on a topological space X {\displaystyle X} , it is usually possibleProjective bundle (1,373 words) [view diff] exact match in snippet view article find links to article
). Let X be a complex smooth projective variety and E a complex vector bundle of rank r on it. Let p: P(E) → X be the projective bundle of E.Euler class (2,003 words) [view diff] exact match in snippet view article find links to article
Stiefel-Whitney class as "the Euler class, ignoring orientation". Any complex vector bundle E of complex rank d can be regarded as an oriented, real vectorG-structure on a manifold (2,576 words) [view diff] exact match in snippet view article find links to article
and only if it is isomorphic to the underlying real bundle of a complex vector bundle. This is then a reduction along the inclusion GL(n,C) → GL(2n,R)Quantum cohomology (1,800 words) [view diff] exact match in snippet view article find links to article
is the first Chern class of the tangent bundle TX, regarded as a complex vector bundle by choosing any almost complex structure compatible with ω. ThusConnection (vector bundle) (8,674 words) [view diff] exact match in snippet view article
\nabla _{v}(fs)=df(v)s+f\nabla _{v}s.} Remark. In the case of a complex vector bundle, the above definition is still meaningful, but is usually takenHitchin's equations (1,350 words) [view diff] exact match in snippet view article find links to article
{\displaystyle P} , and let Φ {\displaystyle \Phi } be a section of the complex vector bundle ad P C ⊗ T 1 , 0 ∗ Σ {\displaystyle {\text{ad}}P^{\mathbb {C} }\otimesComplex geometry (3,677 words) [view diff] exact match in snippet view article find links to article
bundle in terms of characteristic classes of the underlying smooth complex vector bundle. One major theme in complex geometry is classification. Due to theGauge theory (mathematics) (11,468 words) [view diff] exact match in snippet view article
In this case the moduli space of Yang–Mills connections over a complex vector bundle E {\displaystyle E} admits various rich interpretations, and theKobayashi–Hitchin correspondence (4,434 words) [view diff] exact match in snippet view article find links to article
biholomorphism, and the moduli space of Hermite–Einstein metrics on the complex vector bundle E {\displaystyle E} up to gauge transformation. One direction ofSpinor (9,924 words) [view diff] exact match in snippet view article find links to article
every symplectic manifold) has a Spinc structure. Likewise, every complex vector bundle on a manifold carries a Spinc structure. A number of Clebsch–GordanNonabelian Hodge correspondence (5,131 words) [view diff] exact match in snippet view article find links to article
isomorphism between these objects. In the following, fix a smooth complex vector bundle E {\displaystyle E} . Every Higgs bundle will be considered to have