Then, we have Z U log p q dVωU ≤ 1 e +CUVol(X, ω)(logCU + logC)

A MEMORANDUM ON THE INVARIANCE OF PLURIGENERA

(PRIVATE NOTE)

OSAMU FUJINO

Proposition 1. LetX be a complex manifold. Letp, qbe smooth non- negative functions on X such that q 6= 0 almost everywhere. Assume that

Vol(X, ω) = Z

X

1dVω <∞

and

Z

X

p

qdVω ≤C

holds for some constant C ≥1, wheredV_ω is a volume form on X. Let U ⊂X be any open set of X and dVωU a volume form on U such that dV_ωU ≤C_UdV_ω for some constant C_U ≥1. Then, we have

Z

U

log p

q

dVωU ≤ 1

e +CUVol(X, ω)(logCU + logC).

Moreover, letai be a smooth nonnegative function onX for1≤i≤k such that ai 6= 0 almost everywhere for 1≤ i≤ k−1. We put a0 ≡1 and a=ak. Assume that

Z

X

a_i+1

ai

dV_ω ≤C

holds for 0≤i≤k−1. Then 1

k Z

U

loga dVωU ≤ 1

e +CUVol(X, ω)(logCU+ logC).

Date: 2006/3/28, Version 1.3.

The arguments in this note are indispensable when we obtain local uniform supre- mum norm estimates from the globalL²-estimates. I like intrinsic formulations.

1

2 OSAMU FUJINO

Proof. By Jensen’s inequality, Z

U

log p

q

dVωU

Vol(U, ωU) ≤log Z

U

p q

dVωU

Vol(U, ωU)

= log

1 Vol(U, ωU)

+ log

Z

U

p qdVωU

≤log

1 Vol(U, ωU)

+ log

Z

X

p

qCUdVω

≤log

1 Vol(U, ωU)

+ logCU + logC, where Vol(U, ωU) =R

U1dVωU <∞. Therefore, we have Z

U

log p

q

dVωU ≤Vol(U, ωU) log

1 Vol(U, ωU)

+ Vol(U, ωU)(logCU + logC).

Lemma 2. We consider g(x) = xlog ¹_x

for x > 0. Then g(x) ≤ g(¹_e) = ¹_e for any x > 0. Moreover, g(x) >0 for x < 1, g(1) = 0, and g(x)<0 for x >1. We note that g(x)→0 as x→0.

Proof. We have g(x) = −xlogx and g⁰(x) = −logx− 1. Thus, we

obtain the desired properties.

Therefore, we obtain Z

U

log p

q

dVωU ≤ 1

e +CUVol(X, ω)(logCU + logC).

The latter statement is obvious.

Theorem 3. InProposition 1, we further assume thatlogais a quasi- psh function. Let Y be a relatively compact open set of X. Then there exists a positive constant C⁰ such that

sup

x∈Y

1

kloga(x)≤C⁰ <∞.

Proof. Let Wα b Uα b X be relatively compact open sets of X for 1 ≤ α ≤ N. Assume that Y ⊂ SN

α=1Wα and Uα can be seen as a domain in Cⁿ for any α, where n = dimX, and L can be trivialized on each Uα. Let dVω^α be the Euclidean volume form on Uα ⊂Cⁿ. We note that we can assume that dV_ωα ≤C_αdV_ω onU_α for some constant Cα ≥ 1 (if we need, we can shrink Uα). Take a point x ∈ Y. Then there is an α such that x ∈ Wα. On each Uα, we can further assume that loga = uα +vα, where uα is a psh function and vα is a smooth

A MEMO ON THE INVARIANCE OF PLURIGENERA 3

function. For every x ∈ W_α, there exists an open polydisk U_x whose center is x such thatUx bUα and

Vol(Ux, ω_α) = Z

U^x

1dVω^α =C0.

Note that the positive constant C0 is independent of x ∈Wα. By the sub-mean-value property of uα, we have

uα(x)≤ 1 Vol(Ux, ωα)

Z

Ux

uαdVωα = 1 C0

Z

Ux

uαdVωα. Note that

1

kvα(x)≤C1

for any x∈Wα and

1 kC0

Z

Ux

|vα|dVωα ≤C2

for any Ux, where C1 and C2 are positive constants independent of x and Ux. Thus, we obtain

1

k loga(x) = 1

kuα(x) + 1 kvα(x)

≤ 1 kC0

Z

U^x

uαdVωα +C1

≤ 1 kC0

Z

U^x

loga dVωα +C1+C2

≤ 1 C0

1

e +CαVol(X, ω)(logCα+ logC)

+C1+C2

=: C^α. Thus we have

sup

x∈Wα

1

kloga(x)≤C^α.

Then we obtain the upper bound C⁰ = maxαC^α <∞.

Remark 4. LetL be a holomorphic line bundle onX andha smooth hermitian metric on L. Let sj be a holomorphic section of L onX for 1 ≤ j ≤ l. Then logPl

j=1|sj|²_h is a quasi-psh function on X, where

|sj|h is the pointwise norm of s_j with respect to h. For geometric applications, we use Theorem 3 for a=Pl

j=1|sj|²_h.

Acknowledgments. I was partially supported by The Sumitomo Foun- dation and by the Grant-in-Aid for Young Scientists (A) ]17684001 from JSPS.

4 OSAMU FUJINO

Graduate School of Mathematics, Nagoya University, Chikusa-ku Nagoya 464-8602 Japan

E-mail address: [email protected]