# 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).

## Table of contents

## 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 H^{0}(M, K^{m}) of K^{m}* in

*P*(

_{m}*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 *H*^{0}(M, K^{m}). 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 **P**^{N} 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 *m*_{1} and *m*_{2} 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: Japa*n**23**10.2969/jmsj/02320356, MR 0285531 - Ueno, Kenji (1975),
*Classification theory of algebraic varieties and compact complex space*s, 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)**2**4 (1): 14-27 - Iitaka, 茂 (1977), "number of the kinds and classification II of the algebra manifold," it is
*mathematics*(Mathematical Society of Japan)**2**9 (4): 334-349 - Iitaka, 茂 (1982), "various kinds of 双有理幾何 and Kodaira dimensions," it is
*mathematics*(Mathematical Society of Japan)**3**4 (4): 289-300

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