# 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 ) to the projection space decided by L. This ) of L

${\displaystyle R(X,L)=\bigoplus _{d=0}^{\infty }H^{0}(X,L^{\otimes d})}$

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,${\displaystyle -\infty}$ 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).

## 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 ) 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).

${\displaystyle N(M)= m\geq 1\mid P_{m}(M)\geq 1}$

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,${\displaystyle m\in N(M)}$  It is and I multiplex m-and represent it ${\displaystyle \Phi _{mK}}$  I am defined as the representation of the は next.

{\displaystyle {\begin{aligned}\Phi _{mK}:&M\longrightarrow \ \ \ \ \ \ \mathbb {P} ^{N}\\&z\ \ \ \mapsto \ \ (\varphi _{0}(z):\varphi _{1}(z):\cdots :\varphi _{N}(z))\end{aligned}}}

Here,${\displaystyle \varphi _{i}}$  Oh, it is a base of H0(M, Km). Then,${\displaystyle \Phi _{mK}}$  の image ${\displaystyle \Phi _{mK}(M)}$  Oh,${\displaystyle \mathbb {P} ^{N}}$  I am defined as a の part manifold.

For a certain m,${\displaystyle \Phi _{mk}\colon M\rightarrow W=\Phi _{mK}(M)\subset \mathbb {P} ^{N}}$  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).

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,${\displaystyle \Phi _{m_{1}K}\colon M\longrightarrow W_{m_{1}}(M)}$ ${\displaystyle \Phi _{m_{2}K}:M\longrightarrow W_{m_{2}}(M)}$  But, I showed that it became 双有理同値. This is 双有理写像 ${\displaystyle \varphi \colon W_{m_{1}}\longrightarrow W_{m_{2}}(M)}$  But, I mean that I exist.

Furthermore,${\displaystyle M}$  It is に 双有理同値 ${\displaystyle M^{*}}$  と,${\displaystyle W_{m_{1}}}$ ${\displaystyle W_{m_{1}}}$  It is 双有理同値 for both の ${\displaystyle W^{*}}$  Choose をうまく,

${\displaystyle \Phi :M^{*}\longrightarrow W^{*}}$

But, it is 双有理写像,${\displaystyle \Phi}$  の fiber is a single connection ${\displaystyle \Phi}$  の public fiber

${\displaystyle M_{w}^{*}\Phi ^{-1}(w),\ \ w\in W*}$

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 ― ${\displaystyle f\colon V\rightarrow W}$  Assume it fiber space from を m dimension manifold V to n dimension manifold W; each fiber ${\displaystyle V_{w}=f^{-1}(w)}$  It is said that it is は connection. Then

${\displaystyle \kappa (V)\geq \kappa (V_{w})+\kappa (W).}$

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.

## 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