page_title: Iitaka dimension
section_title: Iitaka conjecture
hierarchical_section_title: Iitaka dimension / Iitaka conjecture
caption_attribution_description: English: m-pluricanonical map is birational invariant
context_page_description: In algebraic geometry, the Iitaka dimension of a line bundle L on an algebraic variety X is the dimension of the image of the rational map to projective space determined by L. This is 1 less than the dimension of the section ring of L
  The Iitaka dimension of L is always less than or equal to the dimension of X. If L is not effective, then its Iitaka dimension is usually defined to be or simply said to be negative. The Iitaka dimension of L is sometimes called L-dimension, while the dimension of a divisor D is called D-dimension. The Iitaka dimension was introduced by Shigeru Iitaka.
context_section_description: Consider on complex algebraic varieties in the following.
  Let K be the canonical bundle on M. The dimension of H⁰(M,Kᵐ), holomorphic sections of Kᵐ, is denoted by Pₘ(M), called m-genus. Let
  then N(M) becomes to be all of the positive integer with non-zero m-genus. When N(M) is not empty, for  m-pluricanonical map  is defined as the map
  where  are the bases of H⁰(M,Kᵐ). Then the image of ,  is defined as the submanifold of .
  For certain  let  be the m-pluricanonical map where W is the complex manifold embedded into projective space Pᴺ.
  In the case of surfaces with κ(M)=1 the above W is replaced by a curve C, which is an elliptic curve (κ(C)=0). We want to extend this fact to the general dimension and obtain the analytic fiber structure depicted in the upper right figure.
  Given a birational map , m-pluricanonical map brings the commutative diagram depicted in the left figure, which means that , i.e. m-pluricanonical genus is birationally invariant.
  It is shown by Iitaka that given n-dimensional compact complex manifold M with its Kodaira dimension κ(M) satisfying 1 ≤ κ(M) ≤ n-1, there are enough large m₁,m₂ such that  and  are birationally equivalent, which means there are the birational map . Namely, the diagram depicted in the right figure is commutative.
  Furthermore, one can select  that is birational with  and  that is birational with both  and  such that
  is birational map, the fibers of  are simply connected and the general fibers of 
  have Kodaira dimension 0.
  The above fiber structure is called the Iitaka fiber space. In the case of the surface S (n = 2 = dim(S)), W* is the algebraic curve, the fiber structure is of dimension 1, and then the general fibers have the Kodaira dimension 0 i.e. elliptic curve. Therefore, S is the elliptic surface. These fact can be generalized to the general n. Therefore The study of the higher-dimensional birational geometry decompose to the part of κ=-∞,0,n and the fiber space whose fibers is of κ=0.
  The following additional formula by Iitaka, called Iitaka conjecture, is important for the classification of algebraic varieties or compact complex manifolds.
  Iitaka　Conjecture — Let  to be the fiber space from m-dimensional variety  to n-dimensional variety  and each fibers  connected. Then
  This conjecture has been only partly solved, for example in the case of 
  Moishezon manifolds. The classification theory might been said to be the effort to solve the Iitaka conjecture and lead another theorems that the three-dimensional variety V is abelian if and only if κ(V)=0 and q(V)=3 and its generalization so on. The minimal model program might be led from this conjecture.
caption: The m-pluricanonical map is birational invariant. Pm(M)=Pm(W)
exif: {}