7 A product formula for volumes of varieties
Yujiro Kawamata
October 28, 2018
The volume v(X) of a smooth projective variety X is defined by
v(X) = lim sup
dimH0(X,mKX)
md/d!
where d = dimX . This is a birational invariant.
Theorem 0.1. Let f : X → 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
v(Y )
v(F )
where dX = dimX, dY = dimY and dF = dimF .
Proof. Let H be an ample divisor on Y . There exists a positive integer m0
such that m0KY −H is effective.
Let ǫ be a positive integer. By Fujita’s approximation theorem ([1]),
after replacing a birational model of X , there exists a positive integer m1 and
ample divisors L on F such thatm1KF−L is effective and v(
L) > v(F )−ǫ.
By Viehweg’s weak positivity theorem ([2]), there exists a positive integer
k such that Sk(f
OX(m1KX/Y )⊗OY (H)) is generically generated by global
sections for a positive integer k. k is a function on H and m1.
We have
rank Im(SmSk(f
OX(m1KX/Y )) → f∗OX(km1mKX/Y ))
≥ dimH0(F, kmL)
≥ (v(F )− 2ǫ)
(km1m)
http://arxiv.org/abs/0704.1014v1
for sufficiently large m.
dimH0(X, km1mKX)
≥ dimH0(Y, k(m1 −m0)mKY )× (v(F )− 2ǫ)
(km1m)
≥ (v(Y )− ǫ)
(k(m1 −m0)m)
(v(F )− 2ǫ)
(km1m)
≥ (v(Y )− 2ǫ)(v(F )− 2ǫ)
(km1m)
dY !dF !
if we take m1 large compared with m0 such that
(v(Y )− ǫ)
(v(Y )− 2ǫ)
m1 −m0
)dY .
Remark 0.2. If X = Y × F , then we have an equality in the formula. We
expect that the equality implies the isotriviality of the family.
References
[1] Fujita, Takao. Approximating Zariski decomposition of big line bundles.
Kodai Math. J. 17 (1994), no. 1, 1–3.
[2] Viehweg, Eckart. Weak positivity and the additivity of the Kodaira di-
mension for certain fibre spaces. Algebraic varieties and analytic vari-
eties (Tokyo, 1981), 329–353, Adv. Stud. Pure Math., 1, North-Holland,
Amsterdam, 1983.
Department of Mathematical Sciences, University of Tokyo,
Komaba, Meguro, Tokyo, 153-8914, Japan
kawamata@ms.u-tokyo.ac.jp

  A simple application of the semipositivity.

7 A product formula for volumes of varieties
Yujiro Kawamata
October 28, 2018
The volume v(X) of a smooth projective variety X is defined by
v(X) = lim sup
dimH0(X,mKX)
md/d!
where d = dimX . This is a birational invariant.
Theorem 0.1. Let f : X → 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
v(Y )
v(F )
where dX = dimX, dY = dimY and dF = dimF .
Proof. Let H be an ample divisor on Y . There exists a positive integer m0
such that m0KY −H is effective.
Let ǫ be a positive integer. By Fujita’s approximation theorem ([1]),
after replacing a birational model of X , there exists a positive integer m1 and
ample divisors L on F such thatm1KF−L is effective and v(
L) > v(F )−ǫ.
By Viehweg’s weak positivity theorem ([2]), there exists a positive integer
k such that Sk(f
OX(m1KX/Y )⊗OY (H)) is generically generated by global
sections for a positive integer k. k is a function on H and m1.
We have
rank Im(SmSk(f
OX(m1KX/Y )) → f∗OX(km1mKX/Y ))
≥ dimH0(F, kmL)
≥ (v(F )− 2ǫ)
(km1m)
http://arxiv.org/abs/0704.1014v1
for sufficiently large m.
dimH0(X, km1mKX)
≥ dimH0(Y, k(m1 −m0)mKY )× (v(F )− 2ǫ)
(km1m)
≥ (v(Y )− ǫ)
(k(m1 −m0)m)
(v(F )− 2ǫ)
(km1m)
≥ (v(Y )− 2ǫ)(v(F )− 2ǫ)
(km1m)
dY !dF !
if we take m1 large compared with m0 such that
(v(Y )− ǫ)
(v(Y )− 2ǫ)
m1 −m0
)dY .
Remark 0.2. If X = Y × F , then we have an equality in the formula. We
expect that the equality implies the isotriviality of the family.
References
[1] Fujita, Takao. Approximating Zariski decomposition of big line bundles.
Kodai Math. J. 17 (1994), no. 1, 1–3.
[2] Viehweg, Eckart. Weak positivity and the additivity of the Kodaira di-
mension for certain fibre spaces. Algebraic varieties and analytic vari-
eties (Tokyo, 1981), 329–353, Adv. Stud. Pure Math., 1, North-Holland,
Amsterdam, 1983.
Department of Mathematical Sciences, University of Tokyo,
Komaba, Meguro, Tokyo, 153-8914, Japan
kawamata@ms.u-tokyo.ac.jp
