Pluricanonical ring
Encyclopedia
In mathematics
, the pluricanonical ring of an algebraic variety
V (which is non-singular), or of a complex manifold
, is the graded ring
of sections of powers of the canonical bundle
K. Its nth graded component (for
) is:
that is, the space of sections
of the n-th tensor product
Kn of the canonical bundle
K.
The 0th graded component
is sections of the trivial bundle, and is one dimensional as V is projective. The projective variety defined by this graded ring is called the canonical model of V, and the dimension of the canonical model, is called the Kodaira dimension
of V.
One can define an analogous ring for any line bundle
L over V; the analogous dimension is called the Iitaka dimension. A line bundle is called big if the Iitaka dimension equals the dimension of the variety.
: Any birational map between smooth compact complex manifolds induces an isomorphism between the respective canonical rings. As a consequence one can define the Kodaira dimension of a singular space as the Kodaira dimension of a desingularization. Due to the birational invariance this is well defined, i.e., independent of the choice of the desingularization.
. This is considered a major step in the Mori program.
and have announced proofs of this conjecture.
is the classically-defined n-th plurigenus of V. The pluricanonical divisor
, via the corresponding linear system of divisors
, gives a map to projective space
, called the n-canonical map.
The size of R is a basic invariant of V, and is called the Kodaira dimension
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the pluricanonical ring of an algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
V (which is non-singular), or of a complex manifold
Complex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
, is the graded ring
of sections of powers of the canonical bundle
Canonical bundle
In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n is the line bundle\,\!\Omega^n = \omegawhich is the nth exterior power of the cotangent bundle Ω on V. Over the complex numbers, it is the determinant bundle of holomorphic n-forms on V.This is the dualising...
K. Its nth graded component (for
) is:that is, the space of sections
Section (fiber bundle)
In the mathematical field of topology, a section of a fiber bundle π is a continuous right inverse of the function π...
of the n-th tensor product
Tensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
Kn of the canonical bundle
Canonical bundle
In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n is the line bundle\,\!\Omega^n = \omegawhich is the nth exterior power of the cotangent bundle Ω on V. Over the complex numbers, it is the determinant bundle of holomorphic n-forms on V.This is the dualising...
K.
The 0th graded component
is sections of the trivial bundle, and is one dimensional as V is projective. The projective variety defined by this graded ring is called the canonical model of V, and the dimension of the canonical model, is called the Kodaira dimensionKodaira dimension
In algebraic geometry, the Kodaira dimension κ measures the size of the canonical model of a projective variety V.The definition of Kodaira dimension, named for Kunihiko Kodaira, and the notation κ were introduced in the seminar.-The plurigenera:...
of V.
One can define an analogous ring for any line bundle
Line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example a curve in the plane having a tangent line at each point determines a varying line: the tangent bundle is a way of organising these...
L over V; the analogous dimension is called the Iitaka dimension. A line bundle is called big if the Iitaka dimension equals the dimension of the variety.
Birational invariance
The canonical ring and therefore likewise the Kodaira dimension is a birational invariantBirational invariant
In algebraic geometry, a birational invariant is a quantity or object that is well-defined on a birational equivalence class of algebraic varieties...
: Any birational map between smooth compact complex manifolds induces an isomorphism between the respective canonical rings. As a consequence one can define the Kodaira dimension of a singular space as the Kodaira dimension of a desingularization. Due to the birational invariance this is well defined, i.e., independent of the choice of the desingularization.
Fundamental conjecture of birational geometry
A basic conjecture is that the pluricanonical ring is finitely generatedFinitely generated
In mathematics, finitely generated may refer to:* Finitely generated group* Finitely generated monoid* Finitely generated abelian group* Finitely generated module* Finitely generated ideal* Finitely generated algebra* Finitely generated space...
. This is considered a major step in the Mori program.
and have announced proofs of this conjecture.
The plurigenera
The dimensionis the classically-defined n-th plurigenus of V. The pluricanonical divisor
, via the corresponding linear system of divisorsLinear system of divisors
In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family....
, gives a map to projective space
, called the n-canonical map.The size of R is a basic invariant of V, and is called the Kodaira dimension
Kodaira dimension
In algebraic geometry, the Kodaira dimension κ measures the size of the canonical model of a projective variety V.The definition of Kodaira dimension, named for Kunihiko Kodaira, and the notation κ were introduced in the seminar.-The plurigenera:...
.