Кодайра өлшемі: алгебралық геометриядағы маңызды инвариант

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 κ. Shigeru Iitaka extended it and defined the Kodaira dimension for higher dimensional varieties (under the name of canonical dimension), and later named it after Kunihiko Kodaira.

The following integers are equal if they are non negative. A good reference is , Theorem 2.1.33. The dimension of the Proj construction , a projective variety called the canonical model of X depending only on the birational equivalence class of X. (This is defined only if the canonical ring is finitely generated, which is true in characteristic zero and conjectured in general.) The dimension of the image of the d canonical mapping for all positive multiples d of some positive integer The transcendence degree of the fraction field of R, minus one; i. e. , where t is the number of algebraically independent generators one can find. The rate of growth of the plurigenera: that is, the smallest number κ such that is bounded. In Big O notation, it is the minimal κ such that
When one of these numbers is undefined or negative, then all of them are. In this case, the Kodaira dimension is said to be negative or to be Some historical references define it to be −1, but then the formula does not always hold, and the statement of the Iitaka conjecture becomes more complicated. For example, the Kodaira dimension of is for all varieties X.

The Kodaira dimension gives a useful rough division of all algebraic varieties into several classes. Varieties with low Kodaira dimension can be considered special, while varieties of maximal Kodaira dimension are said to be of general type. Geometrically, there is a very rough correspondence between Kodaira dimension and curvature: negative Kodaira dimension corresponds to positive curvature, zero Kodaira dimension corresponds to flatness, and maximum Kodaira dimension (general type) corresponds to negative curvature. The specialness of varieties of low Kodaira dimension is analogous to the specialness of Riemannian manifolds of positive curvature (and general type corresponds to the genericity of non positive curvature); see classical theorems, especially on Pinched sectional curvature and Positive curvature. These statements are made more precise below.

Let X be a variety of nonnegative Kodaira dimension over a field of characteristic zero, and let B be the canonical model of X, B = Proj R(X, KX); the dimension of B is equal to the Kodaira dimension of X. There is a natural rational map X – → B; any morphism obtained from it by blowing up X and B is called the Iitaka fibration. The minimal model and abundance conjectures would imply that the general fiber of the Iitaka fibration can be arranged to be a Calabi–Yau variety, which in particular has Kodaira dimension zero. Moreover, there is an effective Q divisor Δ on B (not unique) such that the pair (B, Δ) is klt, KB + Δ is ample, and the canonical ring of X is the same as the canonical ring of (B, Δ) in degrees a multiple of some d > 0. In this sense, X is decomposed into a family of varieties of Kodaira dimension zero over a base (B, Δ) of general type. (Note that the variety B by itself need not be of general type. For example, there are surfaces of Kodaira dimension 1 for which the Iitaka fibration is an elliptic fibration over P1.) Given the conjectures mentioned, the classification of algebraic varieties would largely reduce to the cases of Kodaira dimension , 0 and general type. For Kodaira dimension and 0, there are some approaches to classification. The minimal model and abundance conjectures would imply that every variety of Kodaira dimension is uniruled, and it is known that every uniruled variety in characteristic zero is birational to a Fano fiber space. The minimal model and abundance conjectures would imply that every variety of Kodaira dimension 0 is birational to a Calabi Yau variety with terminal singularities. The Iitaka conjecture states that the Kodaira dimension of a fibration is at least the sum of the Kodaira dimension of the base and the Kodaira dimension of a general fiber; see for a survey. The Iitaka conjecture helped to inspire the development of minimal model theory in the 1970s and 1980s. It is now known in many cases, and would follow in general from the minimal model and abundance conjectures.

Nakamura and Ueno proved the following additivity formula for complex manifolds Although the base space is not required to be algebraic, the assumption that all the fibers are isomorphic is very special. Even with this assumption, the formula can fail when the fiber is not Moishezon. Let π: V → W be an analytic fiber bundle of compact complex manifolds, meaning that π is locally a product (and so all fibers are isomorphic as complex manifolds). Suppose that the fiber F is a Moishezon manifold. Then