Iitaka dimension
In algebra geometry, Iitaka dimension (Iitaka dimension) of straight line bunch L on algebra manifold X is the dimension of the image of the Yuri representation () to the projection space decided by L. This section ring () of L
1 is smaller than の dimension.
Iitaka dimensions of L are always lower than a level of X. If L is not effective, the Iitaka dimension of L is common, That I am defined, or と is merely minus number (I might define it as -1 by the early documents). The Iitaka dimension of L may be called an L-dimension, and, on the other hand, the dimension of factor D is called a D-dimension. The Iitaka dimension was introduced by Shigeru Iitaka (1970, 1971).
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Big straight line bunch
If a straight line bunch is big, I say that Iitaka dimension is biggest. In other words, I say that Iitaka dimension is equal to the dimension of the basic manifold. The property to be big is 双有理不変量. f: If Y → X is 双有理写像 of the manifold, and L is a big straight line bunch on X, f*L is a big straight line bunch on Y.
All abundant straight line bunches are big straight line bunches.
The big straight line bunch may not decide 双有理同型射 of X and the image. For example, the standard bunch is big when I assume C a super oval curve (e.g., a curve of number of the kinds 2), but the Yuri representation that it decides is not 双有理同型. On the other hand, it is coating of 2:1 of the standard curve (as for this Yuri probability curve ) of C.
Kodaira dimension
The Iitaka dimension of the standard bunch of the smooth manifold is called Kodaira dimension.
Iitaka expected
I think in a double bare algebra manifold as follows.
I assume K a standard bunch on M. I express a dimension of Masanori cutting H0(M, Km) of Km in Pm(M) and call it number of the m-classes (m-genus).
N(M) becomes the set of the all positive integer when the number of the m-classes is not zero distantly. When N(M) is not an empty set, It is and I multiplex m-and represent it I am defined as the representation of the は next.
Here, Oh, it is a base of H0(M, Km). Then, の image Oh, I am defined as a の part manifold.
For a certain m, I assume it を m- multiplex representation. W is double bare manifolds buried in projection space PN here.
In the case of the curved surface that is Kodaira dimension κ(M) = 1, W mentioned above becomes curve C (κ(C) = 0) which is an oval curve. I expand this fact in a general dimension and want to get analytic fiber structure showing in the figure of the top right corner.
双有理写像 But, the number of m-multiplex kinds representation brings a commutation diagram described in the left figure when given. This, I mean であることを that is the number of m-multiplex kinds representation is 双有理不変.
When Kodaira dimension κ(M) satisfies 1≤κ(M)≤n-1 in n dimension compact double bare manifolds M, as for Iitaka, enough big m1 and m2 exist and, と But, I showed that it became 双有理同値. This is 双有理写像 But, I mean that I exist.
Furthermore, It is に 双有理同値 と, と It is 双有理同値 for both の Choose をうまく,
But, it is 双有理写像, の fiber is a single connection の public fiber
I can do it so that の Kodaira dimension is 0.
I call the fiber structure mentioned above Iitaka fiber space (Iitaka fiber space). Curved surface S (n = 2 = dim(S)) の case, W* become the algebra curve, and the fiber structure is dimension 1, and the Kodaira dimension of the general fiber is 0 that is an oval curve. Therefore, S is an oval curved surface. These facts are extendable to general dimension n. Therefore, as for the study of high-dimensional 双有理幾何学, fiber is dismantled with the study of the part of κ = - ∞, 0, n by a study of the fiber space of κ = 0.
The next formula (called Iitaka expectation (Iitaka conjecture)) by Iitaka is important in the classification of an algebra manifold or compact double bare manifolds.
Iitaka expectation ― Assume it fiber space from を m dimension manifold V to n dimension manifold W; each fiber It is said that it is は connection. Then
This expectation is deciphered only partially. As a deciphered example, I may be the モアシェゾン manifold. The classification theory removes Iitaka expectation, and it may be said that it is the effort that it is going to lead a theorem to be the equivalent or the generalization to to be three-dimensional manifold V being Abel manifold and κ(V) = 0 and q(V) = 3. The very small model program may be derived from this expectation, too.
Allied item
References
- Iitaka, Shigeru (1970), "On D-dimensions of algebraic varieties," it is Proc. Japan Acad. 46: 487–489, doi: 10.3792/pja/1195520260, MR 0285532
- Iitaka, Shigeru (1971), "On D-dimensions of algebraic varieties.," it is J. Math. Soc. It is 356–373, doi: Japan 23 10.2969/jmsj/02320356, MR 0285531
- Ueno, Kenji (1975), Classification theory of algebraic varieties and compact complex spaces, Lecture Notes in Mathematics, 439, Springer-Verlag, MR 0506253
- Iitaka, 茂 (1972), "number of the kinds and classification I of the algebra manifold," it is mathematics (Mathematical Society of Japan) 24 (1): 14-27
- Iitaka, 茂 (1977), "number of the kinds and classification II of the algebra manifold," it is mathematics (Mathematical Society of Japan) 29 (4): 334-349
- Iitaka, 茂 (1982), "various kinds of 双有理幾何 and Kodaira dimensions," it is mathematics (Mathematical Society of Japan) 34 (4): 289-300
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