2017년 6월 11일 일요일

Iitaka dimension

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 (English version) to the projection space decided by L. This section ring (English version) 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 YX 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 (English version)) of C.

Kodaira dimension

The Iitaka dimension of the standard bunch of the smooth manifold is called Kodaira dimension.

Iitaka expected

 
The number of m-multiplex kinds representation from double bare manifolds M to W causes fiber structure.

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.

 
The m-multiplex representation is 双有理不変量. Pm(M) = Pm(W).

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.

 
双有理写像 ψ in the projection space: Existence of Wm1Wm2

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,   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,   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 

This article is taken from the Japanese Wikipedia Iitaka dimension

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