the canonical bundle of a non-singular algebraic variety V {\displaystyle V} of dimension n {\displaystyle n} over a field is the line bundle Ω n =...

of canonical coordinates in classical mechanics may be generalized to a more abstract 20th century definition of coordinates on the cotangent bundle of...

also tautological bundles on a projective bundle of a vector bundle as well as a Grassmann bundle. The older term canonical bundle has dropped out of...

compute the canonical bundle of a minimal elliptic surface f: X → S. Over the complex numbers, Kodaira proved the following canonical bundle formula: K...

the canonical bundle in the ordinary case. It is sometimes also called the pure spinor bundle, as its sections are pure spinors. The canonical bundle is...

:TM\rightarrow M} is the canonical projection. Pushforward (differential) Unit tangent bundle Cotangent bundle Frame bundle Musical isomorphism The disjoint...

denoted h 0 ( X , L ) {\displaystyle h^{0}(X,L)} . Let K denote the canonical bundle on X. Then, the Riemann–Roch theorem states that h 0 ( X , L ) − h...

power of the canonical bundle of M {\displaystyle M} is trivial. M {\displaystyle M} has a finite cover that has trivial canonical bundle. M {\displaystyle...

) {\displaystyle R(V,K)=R(V,K_{V})\,} of sections of powers of the canonical bundle K. Its nth graded component (for n ≥ 0 {\displaystyle n\geq 0} ) is:...

element of a set partition Canonical one-form, a special 1-form defined on the cotangent bundle T*M of a manifold M Canonical symplectic form, the exterior...

vector bundle E* is the Hom bundle Hom(E, R × X) of bundle homomorphisms of E and the trivial bundle R × X. There is a canonical vector bundle isomorphism...

cotangent bundle. That bundle can always be endowed with a certain differential form, called the canonical one-form. This form gives the cotangent bundle the...

, the canonical bundle K X {\displaystyle K_{X}} means the line bundle Ω n {\displaystyle \Omega ^{n}} . Thus sections of the canonical bundle are algebro-geometric...

in H 0 ( X , L ) {\displaystyle H^{0}(X,L)} , canonically associated to the basepoint-free line bundle L. This morphism has the property that L is the...

the theory of complex manifolds, the adjunction formula relates the canonical bundle of a variety and a hypersurface inside that variety. It is often used...

dimensional varieties (under the name of canonical dimension), and later named it after Kunihiko Kodaira. The canonical bundle of a smooth algebraic variety X...

tangent bundle TX is an orientable vector bundle). A special set of coordinates can be defined on the cotangent bundle; these are called the canonical coordinates...

a compact connected complex manifold of dimension 2 with а trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means...

Poincaré one-form, the canonical one-form, or the symplectic potential. A similar object is the canonical vector field on the tangent bundle. To define the tautological...

in terms of the canonical bundle of X {\displaystyle X} : c 1 ( X ) < 0 {\displaystyle c_{1}(X)<0} if and only if the canonical bundle K X {\displaystyle...

Chern class of the tangent bundle) in H2(X, R). It follows that a compact Kähler–Einstein manifold X must have canonical bundle KX either anti-ample, homologically...

locally Noetherian scheme whose local rings are all Gorenstein. The canonical line bundle is defined for any Gorenstein scheme over a field, and its properties...

(L)\subset L\otimes K} with K {\displaystyle K} the canonical bundle over the Riemann surface M. Then a Higgs bundle ( E , φ ) {\displaystyle (E,\varphi )} is stable...

projective spaces are Fano varieties, because the canonical bundle is anti-ample and this line bundle has no non-zero global sections, so the geometric...

its canonical bundle is big, but the rational map it determines is not a birational isomorphism. Instead, it is a two-to-one cover of the canonical curve...

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

curvature form of the canonical line bundle (Moroianu 2007, Chapter 12). The canonical line bundle is the top exterior power of the bundle of holomorphic Kähler...

modern definition is that a projective variety X is minimal if the canonical line bundle KX has nonnegative degree on every curve in X; in other words, KX...

If E is a vector bundle over a topological space X, then the projection map from E to X is the structure map. In topology, a canonical map is a function...

simply a line bundle (viewed as a complex in degree −dim(X)); this line bundle is called the canonical bundle of X. Using the canonical bundle, Serre duality...