In mathematics, the pluricanonical ring of an

algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers ...

''V'' (which is

non-singular In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In ...

), or of a

complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a ...

, is the

graded ring In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the s ...

R(V,K)=R(V,K_V) \,

of sections of powers of the

canonical bundle In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''. Over the complex numbers, ...

''K''. Its ''n''th graded component (for

n\geq 0

) is: :

R_n := H^0(V, K^n),\

that is, the space of

sections Section, Sectioning or Sectioned may refer to: Arts, entertainment and media * Section (music), a complete, but not independent, musical idea * Section (typography), a subdivision, especially of a chapter, in books and documents ** Section sig ...

of the ''n''-th

tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an e ...

''K''^''n'' of the canonical bundle ''K''. The 0th graded component

R_0

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 In algebraic geometry, the Kodaira dimension ''κ''(''X'') measures the size of the canonical model of a projective variety ''X''. Igor Shafarevich, in a seminar introduced an important numerical invariant of surfaces with the notation ''� ...

of ''V''. One can define an analogous ring for any

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 organisin ...

''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.

Properties

Birational invariance

The canonical ring and therefore likewise the Kodaira dimension is a birational invariant: 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 In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety ''V'' has a resolution, a non-singular variety ''W'' with a proper birational map ''W''→''V''. For varieties over fields of characteris ...

. 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 generated. This is considered a major step in the

Mori program In algebraic geometry, the minimal model program is part of the birational classification of algebraic varieties. Its goal is to construct a birational model of any complex projective variety which is as simple as possible. The subject has its ori ...

. proved this conjecture.

The plurigenera

The dimension :

P_n = h^0(V, K^n) = \operatorname\ H^0(V, K^n)

is the classically defined ''n''-th ''plurigenus'' of ''V''. The pluricanonical divisor

K^n

, via the corresponding

linear 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. These arose first in the fo ...

, gives a map to projective space

\mathbf(H^0(V, K^n)) = \mathbf^

, called the ''n''-canonical map. The size of ''R'' is a basic invariant of ''V'', and is called the Kodaira dimension.

Notes

References

* * {{Citation , first1=Phillip , last1=Griffiths , authorlink=Phillip Griffiths , first2=Joe , last2=Harris , author-link2=Joe Harris (mathematician) , title=Principles of Algebraic Geometry , series=Wiley Classics Library , publisher=Wiley Interscience , year=1994 , isbn=0-471-05059-8 , page=573 Algebraic geometry Birational geometry Structures on manifolds