In mathematics, the canonical bundle of a non-singular algebraic variety of dimension over a field is the line bundle , which is the nth exterior power of the cotangent bundle on .
Over the complex numbers, it is the determinant bundle of the holomorphic cotangent bundle . Equivalently, it is the line bundle of holomorphic n-forms on .
This is the dualising object for Serre duality on . It may equally well be considered as an invertible sheaf.
The canonical class is the divisor class of a Cartier divisor on giving rise to the canonical bundle — it is an equivalence class for linear equivalence on , and any divisor in it may be called a canonical divisor. An anticanonical divisor is any divisor − with canonical.
The anticanonical bundle is the corresponding inverse bundle . When the anticanonical bundle of is ample, is called a Fano variety.
of canonical coordinates in classical mechanics may be generalized to a more abstract 20th century definition of coordinates on the cotangent bundle of...
the canonicalbundle in the ordinary case. It is sometimes also called the pure spinor bundle, as its sections are pure spinors. The canonicalbundle is...
also tautological bundles on a projective bundle of a vector bundle as well as a Grassmann bundle. The older term canonicalbundle has dropped out of...
:TM\rightarrow M} is the canonical projection. Pushforward (differential) Unit tangent bundle Cotangent bundle Frame bundle Musical isomorphism The disjoint...
compute the canonicalbundle of a minimal elliptic surface f: X → S. Over the complex numbers, Kodaira proved the following canonicalbundle formula: K...
) {\displaystyle R(V,K)=R(V,K_{V})\,} of sections of powers of the canonicalbundle K. Its nth graded component (for n ≥ 0 {\displaystyle n\geq 0} ) is:...
, the canonicalbundle K X {\displaystyle K_{X}} means the line bundle Ω n {\displaystyle \Omega ^{n}} . Thus sections of the canonicalbundle are algebro-geometric...
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...
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...
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...
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...
dimensional varieties (under the name of canonical dimension), and later named it after Kunihiko Kodaira. The canonicalbundle of a smooth algebraic variety X...
the theory of complex manifolds, the adjunction formula relates the canonicalbundle of a variety and a hypersurface inside that variety. It is often used...
a compact connected complex manifold of dimension 2 with а trivial canonicalbundle and irregularity zero. An (algebraic) K3 surface over any field means...
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 canonicalbundle over the Riemann surface M. Then a Higgs bundle ( E , φ ) {\displaystyle (E,\varphi )} is stable...
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...
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...
its canonicalbundle 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...
projective spaces are Fano varieties, because the canonicalbundle is anti-ample and this line bundle has no non-zero global sections, so the geometric...
Define the canonical line bundle K X {\displaystyle K_{X}} to be the bundle of n-forms on X, the top exterior power of the cotangent bundle: K X = Ω X...
log terminal singularities over a field k {\displaystyle k} if the canonicalbundle K X {\displaystyle K_{X}} is nef, then K X {\displaystyle K_{X}} is...
Global Information
