aa r X i v : . [ m a t h . AG ] J u l Canonical volume forms on compactK¨ahler manifolds
Hajime TSUJIJune 29, 2007
Abstract
We construct a canonical singular hermitian metric with semipositive cur-vature current on the canonical line bundle of a compact K¨ahler manifoldwith pseudoeffective canonical bundle. The method of the construction isa modification of the one in [T].MSC: 14J15,14J40, 32J18
In [T], I have constructed the canonical singular hermitian metric on relativecanonical bundles of projective families whose general fiber has pseudoeffectivecanonical bundles.Although the construction in [T] works only for projective families, it is verylikely that the similar result holds also for K¨ahler families. The purpose ofthis short article is to show that this is the case up to certain extent, i.e., theconstruction of the metric also works for the K¨ahler case, but we need more workto show the semipositivity of the curvature of the singular hermitian metrics onthe families. The reason is that we cannot use the branched coverings in thecase of K¨ahler families. To overcome this difficulty, it seems to be necessaryto consider the variation of plurisubharmonic objects instead of holomorphicobjects. This “horizontal semipositivity” of the curvature is very important inmany applications such as the deformation invariance of plurigenera in the caseof K¨ahler families.Let X be a compact K¨ahler manifold of dimension n and let K X be thecanonical bundle of X . The purpose of this short note is to consruct a canonicalsingular hermitian metric on K X when K X is pseudoeffective. Theorem 1.1
Let X be a compact K¨ahler manifold with pseudoeffective K X .Then there exists a singular hermitian metric ˜ h can on K X such that1. ˜ h can is uniquely determined by X .2. ˜ h can is an AZD of K X , i.e., ˜ h can has the following properties :(a) The curvature current Θ ˜ h can of ˜ h can is semipositive in the sense ofcurrent on X . b) For every m ≧ , H ( X, O X ( mK X ) ⊗ I (˜ h mcan )) ≃ H ( X, O X ( mK X )) holds for every m ≧ . (cid:3) Remark 1.2 If X is projective, ˜ h can seems to be closely related to the super-canonical AZD ˆ h can in [T] (see Remark 2.4 below). (cid:3) We note that the bounded semipositive ( n, n ) form ˜ h − can is an invariantvolume form on X , i.e., the automorphism group of X preserves ˜ h − can . ˜ h can In this section, we shall prove Theorem 1.1.
The construction below is modeled after the one in [T].Let X be a compact K¨ahler manifold. Let ω be any K¨ahler form on X . Forevery ε ∈ (0 , h ε := inf { h | h is a singular hermitian metric on K X , Θ h + ε · ω ≧ , Z X h − = 1 } , where the infimum means the pointwise infimum. And set dV ε := h − ε And we set ˜ h can := the lower envelope of lim inf ε ↓ h ε and define d ˜ V can := ˜ h − can = the upper envelope of lim sup ε ↓ dV ε . Then since the upper limit of a sequence of plurisubharmonic function locallyuniformly bounded from above is again plurisubharmonic, if we take the up-per envelope of the limit (cf. ([L, p.26, Theorem 5]), we see that ˜ h can hassemipositive curvature in the sense of current. dV ε Let dµ be a C ∞ volume form on X . Lemma 2.1
There exists a positive constant C independent of ε ∈ (0 , suchthat dV ε ≦ C · dµ holds on X . (cid:3) roof of Lemma 2.1 . Let x be a point on X . Let ( U, z , · · · , z n ) be a localcoordinate neighbourhood such such that1. z ( x ) = · · · = z n ( x ) = 0.2. z , · · · , z n are holomorphic on a neighbourhood of the closure of U .3. U is biholomorphic to the unit open polydisk ∆ n in C n with center O viathe coordinate ( z , · · · , z n ).By the ∂ ¯ ∂ Poincar´e lemma, there exists a positive C ∞ function ϕ U such that ω | U = √− ∂ ¯ ∂ log ϕ U holds. By the construction, we may and do assume that ϕ U can be taken sothat ϕ is bounded on a neighbourhood of the closure of U .Let dV be a bounded uppersemicontinuous semipositive ( n, n ) form suchthat √− ∂ ¯ ∂ log dV + εω ≧ Z X dV = 1 (1)hold. We define the function a U by dV | U = a U · ( √− n n Y i =1 dz i ∧ d ¯ z i holds. Then since ε ∈ (0 , √− ∂ ¯ ∂ log( a U · e − ϕ U ) ≧ Z U e − ϕ U dV ≦ sup U e − ϕ U (3)holds. Hence combining (2) and (3), by the submeanvalue property of plurisub-harmonic functions, we see that dV ( x ) ≦ (sup U e − ϕ U ) · e ϕ U ( x ) ( √− π ) n · n Y i =1 dz i ∧ d ¯ z i holds. Hence since X is compact, moving x , we obtain Lemma 2.1. (cid:3) dV ε The lower estimate is also easy. Let h be any singular hermitian metric on K X with semipositive curvature current. We note that h − is a bounded semipositive( n, n ) from on X . Then by the definition of d ˜ V ε , we see that dV ε ≧ ( Z X h − ) − · h − holds. Hence letting ε tend to 0, we have the following lemma.3 emma 2.2 Let h be any singular hermitian metric on K X with semipositivecurvature current. Then we have that d ˜ V can ≧ ( Z X h − ) − · h − holds. (cid:3) Lemma 2.2 immediately implies the following lemma.
Lemma 2.3 ˜ h can is an AZD of K X , i.e.,1. Θ ˜ h can is semipositive in the sense of current.2. For every m ≧ , H ( X, O X ( mK X ) ⊗ I (˜ h mcan )) ≃ H ( X, O X ( mK X )) holds. (cid:3) Proof of Lemma 2.3 . Let σ be a nonzero element of H ( X, O X ( mK X )) forsome m ≧
0. Then we see that1 | σ | m := h ( h m ( σ, σ )) m is a singular hermitian metric on K X with semipositive curvature current, where h is any C ∞ hermitian metric on K X . Hence by Lemma 2.2, we see that˜ h can ≦ | Z X | σ | m | · | σ | m . This means that ˜ h mcan ( σ, σ ) ≦ | Z X | σ | m | m holds. Hence σ ∈ H ( X, O X ( mK X ) ⊗ I (˜ h mcan ))holds. Since σ is arbitrary, we complete the proof of Lemma 2.3. (cid:3) Remark 2.4