mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 solution set, set of solutions of a system of polynomial equations over the real number, ...

''V'' (which is nonsingular), or of a

complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...

, is the graded ring :

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

of sections of powers of the canonical bundle ''K''. Its ''n''th graded component (for

n\geq 0

) is: :

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

that is, the space of sections of the ''n''-th

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

''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 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 organis ...

''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. 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. 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, 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