(15.28) Corollary.Let X be a projective manifold and La line bundle onX.
(i) L is pseudo-effective if and only ifL·C>0 for all curvesC with nef normal sheaf NC. (ii) If Lis big, then L·C >0for all curves C with nef normal sheaf NC.
Corollary 15.28 (i) strenghtens results from [PSS99]. It is however not yet clear whetherMNSis equal to the closed cone of curves with ample normal bundle (although we certainly expect this to be true). The most important special case of Theorem 15.23 is
(15.29) Theorem.If X is a projective manifold, thenKX is pseudo-effective(i.e. KX ∈ ENS), if and only ifX is not uniruled (i.e. not covered by rational curves).
Proof. IfX is covered by rational curvesCt, then it is well-known that the normal bundleNCt is nef for a general member Ct, thus
KX·Ct=KCt·Ct−NCt·Ct6−2,
andKX cannot be pseudo-effective. Conversely, ifKX∈ E/ NS, Theorem 15.23 shows that there is a mobile curve Ctsuch thatKX·Ct<0. The standard “bend-and-break” lemma of Mori theory then produces a covering family
Γt of rational curves withKX·Γt<0, soX is uniruled.
Notice that the generalized abundance conjecture 15.16 would then yield the stronger result :
(15.30) Conjecture. Let X be a projective manifold. If X is not uniruled, then KX is a Q-effective divisor and κ(X) = num(KX)>0.
16. Super-canonical metrics and abundance
16.A. Construction of super-canonical metrics
LetX be a compact complex manifold and (L, hL,γ) a holomorphic line bundle overX equipped with a singular hermitian metric hL,γ =e−γhL with satisfiesR
e−γ <+∞ locally onX, wherehL is a smooth metric onL. In fact, we can more generally consider the case where (L, hL,γ) is a “hermitian R-line bundle”; by this we mean that we have chosen a smooth reald-closed (1,1) formαL onX (whoseddc cohomology class is equal toc1(L)), and a specific currentTL,γ representing it, namelyTL,γ =αL+ddcγ, such thatγ is a locally integrable function satisfying R
e−γ < +∞. An important special case is obtained by considering a klt (Kawamata log terminal) effective divisor ∆. In this situation∆ =P
cj∆j with cj ∈ R, and if gj is a local generator of the ideal sheaf O(−∆j) identifying it to the trivial invertible sheafgjO, we take γ=P
cjlog|gj|2,TL,γ=P
cj[∆j] (current of integration on∆) andαLgiven by any smooth representative of the sameddc-cohomology class; the klt condition precisely means that
(16.1)
Z
V
e−γ = Z
V
Y|gj|−2cj <+∞
on a small neighborhoodV of any point in the support |∆| =S
∆j (condition (16.1) implies cj <1 for every j, and this in turn is sufficient to imply ∆ klt if ∆ is a normal crossing divisor; the line bundle L is then the real line bundleO(∆), which makes sens as a genuine line bundle only ifcj ∈Z). For each klt pair (X, ∆) such that KX +∆ is pseudo-effective, H. Tsuji [Tsu07a, Tsu07b] has introduced a “super-canonical metric” which generalizes the metric introduced by Narasimhan and Simha [NS68] for projective algebraic varieties with ample canonical divisor. We take the opportunity to present here a simpler, more direct and more general approach.
We assume from now on thatKX+Lispseudo-effective, i.e. that the classc1(KX) +{αL}is pseudo-effective, and under this condition, we are going to define a “super-canonical metric” onKX+L. Select an arbitrary smooth hermitian metricωonX. We then find induced hermitian metricshKX onKXandhKX+L=hKXhLonKX+L, whose curvature is the smooth real (1,1)-form
α=ΘKX+L,hKX+L=ΘKX,ω+αL.
A singular hermitian metric onKX+Lis a metric of the formhKX+L,ϕ=e−ϕhKX+Lwhereϕis locally integrable, and by the pseudo-effectivity assumption, we can find quasi-psh functionsϕsuch thatα+ddcϕ>0. The metrics onLandKX+Lcan now be “subtracted” to give rise to a metric
in view of the fact that ϕ is locally bounded from above and of the assumption R
e−γ < +∞. Observe that condition (16.2) can always be achieved by subtracting a constant to ϕ. Now, we can generalize Tsuji’s super-canonical metrics on klt pairs (cf. [Tsu07b]) as follows.
(16.3) Definition. Let X be a compact complex manifold and let (L, hL) be a hermitian R-line bundle on X associated with a smooth real closed (1,1) form αL. Assume that KX +L is pseudo-effective and that L is equipped with a singular hermitian metric hL,γ =e−γhL such that R
In particular, this gives a definition of the super-canonical metric on KX +∆ for every klt pair (X, ∆) such thatKX+∆is pseudo-effective, and as an even more special case, a super-canonical metric onKX whenKX is pseudo-effective.
In the sequel, we assume thatγhas analytic singularities, otherwise not much can be said. The mean value in-equality then immediately shows that the quasi-psh functionsϕinvolved in definition (16.3) are globally uniformly bounded outside of the poles ofγ, and therefore everywhere onX, hence the envelopesϕcan= supϕϕare indeed well defined and bounded above. As a consequence, we get a “super-canonical” currentTcan=α+ddcϕcan>0 andhKX+L,cansatisfies
It is easy to see that in Definition (16.3) the supremum is a maximum and that ϕcan = (ϕcan)∗ everywhere, so that taking the upper semicontinuous regularization is not needed. In fact ifx0∈X is given and we write
(ϕcan)∗(x0) = lim sup properties of quasi-psh functions in L1 topology imply the existence of a subsequence of (ϕν) converging in L1 and almost everywhere to a quasi-psh limit ϕ. Since R
Xeϕν−γdVω 6 1 holds true for every ν, Fatou’s lemma ϕcan=ϕ∗can. By elaborating on this argument, one can infer certain regularity properties of the envelope.
(16.5) Theorem ([BmD09]). Let X be a compact complex manifold and (L, hL) a holomorphic R-line bundle such that KX +L is big. Assume that L is equipped with a singular hermitian metric hL,γ = e−γhL with
16. Super-canonical metrics and abundance 103
analytic singularities such that R
e−γ <+∞ (klt condition). Denote byZ0 the set of poles of a singular metric h0=e−ψ0hKX+L with analytic singularities on KX+L and by Zγ the poles of γ (assumed analytic). Then the associated super-canonical metric hcan is continuous on Xr(Z0∪Zγ).
In fact, using the regularization techniques of [Dem94a], it is shown in [BmD09] thathcan possesses some com-putable logarithmic modulus of continuity. In order to shorten the exposition, we will only give a proof of the continuity in the algebraic case, using approximation by pluri-canonical sections.
(16.6) Algebraic version of the super-canonical metric. Since the klt condition is open andKX+Lis assumed to be big, we can always perturb L a little bit, and after blowing-up X, assume that X is projective and that (L, hL,γ) is obtained as a sum ofQ-divisors
L=G+∆
where∆ is klt andGis equipped with a smooth metrichG (from whichhL,γ is inferred, with∆ as its poles, so that ΘL,hL,γ =ΘG,LG+ [∆]). Clearly this situation is “dense” in what we have been considering before, just as Qis dense inR. In this case, it is possible to give a more algebraic definition of the super-canonical metricϕcan, following the original idea of Narasimhan-Simha [NS68] (see also H. Tsuji [Tsu07a]) – the case considered by these authors is the special situation where G= 0,hG= 1 (and moreover∆= 0 andKX ample, for [NS68]). In fact, ifmis a large integer which is a multiple of the denominators involved inGand∆, we can consider sections
σ∈H0(X, m(KX+G+∆)).
We view them rather as sections ofm(KX+G) with poles along the support|∆|of our divisor. Then (σ∧σ)1/mhG
is a volume form with integrable poles along|∆|(this is the klt condition for∆). Therefore one can normalizeσ
by requiring that Z
X
(σ∧σ)1/mhG= 1.
Each of these sections defines a singular hermitian metric on KX+L = KX+G+∆, and we can take the regularized upper envelope
(16.7) ϕalgcan=
sup
m,σ
1
mlog|σ|2hm
KX+L
∗
of the weights associated with a smooth metrichKX+L. It is clear thatϕalgcan6ϕcan since the supremum is taken on the smaller set of weightsϕ=m1log|σ|2hm
KX+L, and the equalities eϕ−γdVω=|σ|2/mhm
KX+Le−γdVω= (σ∧σ)1/me−γhL= (σ∧σ)1/mhL,γ = (σ∧σ)1/mhG
imply R
Xeϕ−γdVω 6 1. We claim that the inequality ϕalgcan 6 ϕcan is an equality. The proof is an immediate consequence of the following statement based in turn on the Ohsawa-Takegoshi theorem and the approximation technique of [Dem92].
(16.8) Proposition. With L =G+∆, ω,α=ΘKX+L,hKX+L,γ as above and KX +L assumed to be big, fix a singular hermitian metrice−ϕhKX+L of curvatureα+ddcϕ>0, such thatR
Xeϕ−γdVω61. Then ϕis equal to a regularized limit
ϕ=
lim sup
m→+∞
1
mlog|σm|2hm
KX+L
∗
for a suitable sequenceσm∈H0(X, m(KX+G+∆))with R
X(σm∧σm)1/mhG61.
Proof. By our assumption, there exists a quasi-psh functionψ0 with analytic singularity setZ0 such that α+ddcψ0>ε0ω >0
and we can assumeR
Ceψ0−γdVω < 1 (the strict inequality will be useful later). Form >p>1, this defines a singular metric exp(−(m−p)ϕ−pψ0)hmKX+L onm(KX+L) with curvature>pε0ω, and therefore a singular metric
hL′ = exp(−(m−p)ϕ−pψ0)hmKX+Lh−K1X
onL′ = (m−1)KX+mL, whose curvatureΘL′,hL′ >(pε0−C0)ωis arbitrary large ifpis large enough. Let us fix a finite covering ofX by coordinate balls. Pick a pointx0and one of the coordinate ballsBcontainingx0. By the Ohsawa-Takegoshi extension theorem applied on the ballB, we can find a sectionσB ofKX+L′=m(KX+L) which has norm 1 at x0 with respect to the metric hKX+L′ and R
B|σB|2h
KX+L′dVω 6 C1 for some uniform constant C1 depending on the finite covering, but independent of m, p, x0 . Now, we use a cut-off function θ(x) with θ(x) = 1 near x0 to truncate σB and solve a ∂-equation for (n,1)-forms with values in L to get a global section σ on X with |σ(x0)|hKX+L′ = 1. For this we need to multiply our metric by a truncated factor exp(−2nθ(x) log|x−x0|) so as to get solutions of ∂ vanishing at x0. However, this perturbs the curvature by bounded terms and we can absorb them again by takingplarger. In this way we obtain
(16.9) Takingp >1, the H¨older inequality for congugate exponentsm, mm−1 implies
Z
Xeϕ−γdVω61 and another application of H¨older’s inequality. Since klt is an open condition and limp→+∞R forplarge enough. On the other hand
|σ(x0)|2h
16. Super-canonical metrics and abundance 105
(the latter equality occurring if{xq}is suitably chosen with respect to ϕ). In the other direction, (16.9) implies a mean value estimate
KX+L is plurisubharmonic after we correct the non necessarily positively curved smooth metrichKX+L by a factor of the form exp(C6|z−x|2), hence the mean value inequality shows that
(16.12) Remark.We can rephrase our results in slightly different terms. In fact, let us put ϕalgm = sup
σ
1
mlog|σ|2hm
KX+L, σ∈H0(X, m(KX+G+∆)), with normalized sectionsσsuch that R
X(σ∧σ)1/mhG= 1. Thenϕalgm is quasi-psh (the supremum is taken over a compact set in a finite dimensional vector space) and by passing to the regularized supremum over all σand allϕin (16.10) we get
Asϕcan is bounded from above, we find in particular 06ϕcan−ϕalgm 6 C
m(|ψ0(x)|+ 1).
This implies that (ϕalgm) converges uniformly toϕcanon every compact subset ofX ⊂Z0, and in this way we infer again (in a purely qualitative manner) thatϕcan is continuous onXrZ0. Moreover, we also see that in (16.7) the upper semicontinuous regularization is not needed on X rZ0; in caseKX+L is ample, it is not needed at all and we have uniform convergence of (ϕalgm) towards ϕcan on the whole ofX. Obtaining such a uniform convergence when KX+Lis just big looks like a more delicate question, related e.g. to abundance ofKX+L on those subvarietiesY where the restriction (KX+L)|Y would be e.g. nef but not big.
(16.13) Generalization. In the general case whereLis aR-line bundle andKX+Lis merely pseudo-effective, a similar algebraic approximation can be obtained. We take instead sections
σ∈H0(X, mKX+⌊mG⌋+⌊m∆⌋+pmA)
We then find again ϕcan = ϕalgcan, with an almost identical proof – though we no longer have a sup in the envelope, but just a lim sup. The analogue of Proposition (16.8) also holds true in this context, with an appropriate sequence of sectionsσm∈H0(X, mKX+⌊mG⌋+⌊m∆⌋+pmA).
(16.16) Remark. It would be nice to have a better understanding of the super-canonical metrics. In caseX is a curve, this should be easier. In factX then has a hermitian metricωwith constant curvature, which we normalize by requiring thatR
Xω = 1, and we can also supposeR
Xe−γω = 1. The classλ=c1(KX+L)>0 is a number
and we takeα=λω. Our envelope isϕcan= supϕwhereλω+ddcϕ>0 andR
Xeϕ−γω61. Ifλ= 0 thenϕmust be constant and clearlyϕcan= 0. Otherwise, ifG(z, a) denotes the Green function such thatR
XG(z, a)ω(z) = 0 andddcG(z, a) =δa−ω(z), we find
ϕcan(z)>sup
a∈X
λG(z, a)−log Z
z∈X
eλG(z,a)−γ(z)ω(z)
by taking already the envelope overϕ(z) =λG(z, a)−Const. It is natural to ask whether this is always an equality, i.e. whether the extremal functions are always given by one of the Green functions, especially whenγ= 0.
16.B. Invariance of plurigenera and positivity of curvature of super-canonical metrics
The concept of super-canonical metric can be used to give a very interesting result on the positivity of relative pluricanonical divisors, which itself can be seen to imply the invariance of plurigenera. The main idea is due to H. Tsuji [Tsu07a], and some important details were fixed by Berndtsson and P˘aun [BnP09], using techniques inspired from their results on positivity of direct images [Bnd06], [BnP08].
(16.17) Theorem.Letπ:X →S be a deformation of projective algebraic manifolds over some irreducible complex space S (π being assumed locally projective over S). Let L → X be a holomorphic line bundle equipped with a hermitian metric hL,γ of weight γ such that iΘL,hL,γ >0 (i.e. γ is plurisubharmonic), and R
Xte−γ <+∞, i.e.
we assume the metric to be klt over all fibers Xt=π−1(t). Then the metric defined on KX+L as the fiberwise super-canonical metric has semi-positive curvature overX. In particular,t7→h0(Xt, m(KXt+L↾Xt))is constant for all m >0.
Once the metric is known to have a plurisuharmonic weight on the total space of X, the Ohsawa-Takegoshi theorem can be used exactly as at the end of the proof of lemma (12.3). Therefore the final statement is just an easy consequence. The cases when L= OX is trivial or when L↾Xt =O(∆t) for a family of klt Q-divisors are especially interesting.
Proof. (Sketch) By our assumptions, there exists (at least locally over S) a relatively ample line bundle A overX. We have to show that the weight of the global super-canonical metric is plurisubharmonic, and for this, it is enough to look at analytic disks∆→S. We may thus as well assume thatS=∆is the unit disk. Consider the super-canonical metric hcan,0 over the fiber X0. The approximation argument seen above (see (16.9) and remark (16.13)) show thathcan,0 has a weightϕcan,0which is a regularized upper limit
ϕalgcan,0=
lim sup
m→+∞
1
mlog|σm|2 ∗
defined by sectionsσm∈H0(X0, m(KX0+L↾X0) +pmA↾X0) such that Z
X0
|σ|2e−(m−pm)ϕcan,0−pmψ0dVω6C2.
with the suitable weights. Now, by section 12, these sections extend to sectionseσmdefined on the whole familyX, satisfying a similarL2estimate (possibly with a slightly larger constantC2′ under control). If we set
Φ=
lim sup
m→+∞
1
mlog|eσm|2 ∗
,
thenΦis plurisubharmonic by construction, andϕcan>Φby the defining property of the super-canonical metric.
Finally, we also have ϕcan,0 =Φ↾X0 from the approximation technique. It follows easily that ϕcan satisfies the mean value inequality with respect to any disk centered on the central fiberX0. Since we can consider arbitrary
analytic disks∆→S, the plurisubharmonicity ofϕcanfollows.