A product formula for volumes of varieties

Yujiro Kawamata

arXiv:0704.1014·math.AG·May 23, 2007·3 cites

A product formula for volumes of varieties

Yujiro Kawamata

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TL;DR

This paper presents a new product formula for calculating the volumes of algebraic varieties, utilizing semipositivity properties to simplify the process.

Contribution

It introduces a novel product formula for volumes of varieties based on semipositivity, advancing the understanding of their geometric properties.

Findings

01

Derived a new volume formula for algebraic varieties

02

Simplified volume calculations using semipositivity

03

Enhanced theoretical framework for algebraic geometry

Abstract

A simple application of the semipositivity.

Equations10

v (X) = lim sup \frac{dim H ^{0} ( X , m K _{X} )}{m ^{d} / d !}

v (X) = lim sup \frac{dim H ^{0} ( X , m K _{X} )}{m ^{d} / d !}

\frac{v ( X )}{d _{X} !} \geq \frac{v ( Y )}{d _{Y} !} \frac{v ( F )}{d _{F} !}

\frac{v ( X )}{d _{X} !} \geq \frac{v ( Y )}{d _{Y} !} \frac{v ( F )}{d _{F} !}

rank Im (S^{m} S^{k} (f_{*} O_{X} (m_{1} K_{X / Y})) \to f_{*} O_{X} (k m_{1} m K_{X / Y})) \geq dim H^{0} (F, k m L) \geq (v (F) - 2 ϵ) \frac{( k m _{1} m ) ^{d_{F}}}{d _{F} !}

rank Im (S^{m} S^{k} (f_{*} O_{X} (m_{1} K_{X / Y})) \to f_{*} O_{X} (k m_{1} m K_{X / Y})) \geq dim H^{0} (F, k m L) \geq (v (F) - 2 ϵ) \frac{( k m _{1} m ) ^{d_{F}}}{d _{F} !}

dim H^{0} (X, k m_{1} m K_{X}) \geq dim H^{0} (Y, k (m_{1} - m_{0}) m K_{Y}) \times (v (F) - 2 ϵ) \frac{( k m _{1} m ) ^{d_{F}}}{d _{F} !} \geq (v (Y) - ϵ) \frac{( k ( m _{1} - m _{0} ) m ) ^{d_{Y}}}{d _{Y} !} (v (F) - 2 ϵ) \frac{( k m _{1} m ) ^{d_{F}}}{d _{F} !} \geq (v (Y) - 2 ϵ) (v (F) - 2 ϵ) \frac{( k m _{1} m ) ^{d_{X}}}{d _{Y} ! d _{F} !}

dim H^{0} (X, k m_{1} m K_{X}) \geq dim H^{0} (Y, k (m_{1} - m_{0}) m K_{Y}) \times (v (F) - 2 ϵ) \frac{( k m _{1} m ) ^{d_{F}}}{d _{F} !} \geq (v (Y) - ϵ) \frac{( k ( m _{1} - m _{0} ) m ) ^{d_{Y}}}{d _{Y} !} (v (F) - 2 ϵ) \frac{( k m _{1} m ) ^{d_{F}}}{d _{F} !} \geq (v (Y) - 2 ϵ) (v (F) - 2 ϵ) \frac{( k m _{1} m ) ^{d_{X}}}{d _{Y} ! d _{F} !}

\frac{( v ( Y ) - ϵ )}{( v ( Y ) - 2 ϵ )} \geq (\frac{m _{1}}{m _{1} - m _{0}})^{d_{Y}} .

\frac{( v ( Y ) - ϵ )}{( v ( Y ) - 2 ϵ )} \geq (\frac{m _{1}}{m _{1} - m _{0}})^{d_{Y}} .

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Taxonomy

TopicsPolynomial and algebraic computation · Advanced Differential Equations and Dynamical Systems · Algebraic Geometry and Number Theory

Full text

A product formula for volumes of varieties

Yujiro Kawamata

The volume $v(X)$ of a smooth projective variety $X$ is defined by

[TABLE]

where $d=\dim X$ . This is a birational invariant.

Theorem 0.1.

Let $f:X\to Y$ be a surjective morphism of smooth projective varieties with connected fibers. Assume that both $Y$ and the general fiber $F$ of $f$ are varieties of general type. Then

[TABLE]

where $d_{X}=\dim X$ , $d_{Y}=\dim Y$ and $d_{F}=\dim F$ .

Proof.

Let $H$ be an ample divisor on $Y$ . There exists a positive integer $m_{0}$ such that $m_{0}K_{Y}-H$ is effective.

Let $\epsilon$ be a positive integer. By Fujita’s approximation theorem ([1]), after replacing a birational model of $X$ , there exists a positive integer $m_{1}$ and ample divisors $L$ on $F$ such that $m_{1}K_{F}-L$ is effective and $v(\frac{1}{m_{1}}L)>v(F)-\epsilon$ .

By Viehweg’s weak positivity theorem ([2]), there exists a positive integer $k$ such that $S^{k}(f_{*}\mathcal{O}_{X}(m_{1}K_{X/Y})\otimes\mathcal{O}_{Y}(H))$ is generically generated by global sections for a positive integer $k$ . $k$ is a function on $H$ and $m_{1}$ .

We have

[TABLE]

for sufficiently large $m$ .

Then

[TABLE]

if we take $m_{1}$ large compared with $m_{0}$ such that

[TABLE]

∎

Remark 0.2.

If $X=Y\times F$ , then we have an equality in the formula. We expect that the equality implies the isotriviality of the family.

Bibliography2

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] Fujita, Takao. Approximating Zariski decomposition of big line bundles . Kodai Math. J. 17 (1994), no. 1, 1–3.
2[2] Viehweg, Eckart. Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces . Algebraic varieties and analytic varieties (Tokyo, 1981), 329–353, Adv. Stud. Pure Math., 1, North-Holland, Amsterdam, 1983.

TL;DR

Contribution

Findings

Abstract

Peer Reviews

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Taxonomy

A product formula for volumes of varieties

Theorem 0.1**.**

Proof.

Remark 0.2**.**

Theorem 0.1.

Remark 0.2.