In the case of algebraic surfaces, the Minimal Model Program (MMP) was already initiated by Italian geometers at the turn of the XXth century, and was finally completed by Zariski and Kodaira for all complex surfaces. The case of higher
dimensions (starting with dimension 3) is a major endeavor of modern times, re-vitalized by Mori [41], Kawamata [29], [30], [31] and Shokurov [53], [54] among others (see also [33] for a good survey).
The basic question is to prove that every birational class of non uniruled al-gebraic varieties contains a “minimal” member X exhibiting mild singularities (“terminal singularities”), where “minimal” is taken in the sense of avoiding un-necessary blow-ups; minimality actually means thatKXis nef and not just pseudo-effective (pseudo-effectivity follows in general from Theorem 5.19). This requires performing certain birational transforms known as flips, and important questions are whether a) flips are indeed possible (“existence of flips”), b) the process termi-nates (“termination of flips”). Thanks to Kawamata [31] and Shokurov [53], [54], this has been proved in dimension 3 at the end of the 80’s. Very recently, C. Hacon and J. McKernan [23] announced that flips exist in dimension n, if one assumes that a slightly stronger version of MMP (involving log pairs with real divisors) holds true in dimension n−1. As a consequence, the existence of flips obtained by Shokurov [55] in 2003 would be achieved in dimension 4 via a more systematic method. Strongly related to these issues are the following fundamental questions
(i) Finiteness of the canonical ring: is thecanonical ring R=LH0(X, mKX) of a variety of general type always finitely generated ?
If true, Proj(R) of this graded ring R yields of course a “canonical model”
in the birational class ofX.
(ii) Boundedness of pluricanonical embeddings: is there a bound rn depending only on dimension dimX =n, such that the pluricanonical map ΦmKX of a variety of general type yields a birational embedding in projective space for m≥rn ?
(iii) Invariance of plurigenera: are plurigenerapm=h0(X, mKX) always invari-ant under deformation ?
These questions involve taking “limits” of divisors asm→+∞, and therefore transcendental methods are a strong contender in the arena. Question (ii) was indeed solved in the affirmative by H. Tsuji [64], [63] under the assumption that the MMP program is solved, and in general by S. Takayama [61], and Ch. Hacon-J. McKernan [22] by pursuing further Tsuji’s ideas. Question (iii) was completely settled by Y.T. Siu ([58] in the case of varieties of general type, and [59] for arbitrary varieties). Quite recently, M. Pˇaun gave a very elementary proof based merely on the Ohwawa-Takegoshi extension theorem, that we briefly sketch below.
Y.T. Siu’s work also gives strong support for the hope that (i) can be solved by a suitable combination of theL2existence theorems (Skoda’s division theorem being one of the main ingredients). The following is a very slight extension of results by M. Pˇaun [52] and B. Claudon [9], which are themselves based on the ideas of Y.T. Siu [59] and S. Takayama [62].
Theorem 6.1. Let π: X →∆ be a projective family over the unit disk, and let (Lj, hj)0≤j≤m−1 be (singular) hermitian line bundles with semipositive curvature currents iΘLj,hj ≥0 onX. Assume that
(i) the restriction of hj to the central fiber X0 is well defined (i.e. not identi-cally +∞).
(ii) additionally the multiplier ideal sheaf I(hj|X0)is trivial for1≤j≤m−1.
Then any section σ of O(mKX +P
Lj)|X0 ⊗ I(h0|X0) over the central fiber X0
extends toX.
We first state the technical version of the Ohsawa-TakegoshiL2extension theo-rem needed for the proof, which is a special case of Theotheo-rem 2.7 (see also Siu [59]).
Lemma 6.2. Let π:X →∆ be as before and let (L, h)be a (singular)hermitian line bundle with semipositive curvature currentiΘL,h≥0 onX. Letω be a global K¨ahler metric on X, and dVX, dVX0 the respective induced volume elements on X0 andX. Assume that hX0 is well defined. Then any holomorphic section uof O(KX+L)⊗ I(h|X0)extends into a sectioneuoverX satisfying an L2 estimate
Z
X
kuke 2ω⊗hdVX ≤C0
Z
X0
kuk2ω⊗hdVX0,
whereC0≥0is some universal constant (independent ofX,L,. . .).
Proof. We write hj = e−ϕj in terms of local plurisubharmonic weights. Fix an auxiliary line bundle A (which will later be taken to be sufficiently ample), and define inductively a sequence of line bundles Fp by puttingF0=Aand
Fp =Fp−1+KX+Lr ifp=mq+r, 0≤r≤m−1.
By construction we haveFp+m=Fp+mKX+P
jLj and
F0=A, F1=A+KX+L1, . . . , Fp=A+pKX+L1+. . .+Lp, 1≤p≤m−1.
The game is to construct inductively families of sections, say (eu(p)j )j=1...Np, ofFp
overX in such a way that
(a) forp= 0, . . . , m−1,Fp is generated by its sections (ue(p)j )j=1...Np;
(b) we have the m-periodicity relationsNp+m=Np andue(p)j is an extension of u(p)j :=σqu(r)j overX forp=mq+r, where u(r)j :=ue(r)j|X0, 0≤r≤m−1.
Property (a) can certainly be achieved by takingA ample enough so thatF0,. . ., Fm−1 are generated by their sections, and by choosing theue(p)j appropriately for p= 0, . . . , m−1. Now, by induction, we equipFp−1 with the tautological metric
|ξ|2/P|eu(p−1)j (x)|2, and Fp−KX = Fp−1+Lr with that metric multiplied by hr=e−ϕr ; it is clear that these metrics have semipositive curvature currents (the metric onFp itself if obtained by using a smooth K¨ahler metricω onX). In this setting, we apply the Ohsawa-Takegoshi theorem to the line bundleFp−1+Lrto
extendu(p)j into a sectionue(p)j overX. By construction the pointwise norm of that section in Fp|X0 in a local trivialization of the bundles involved is the ratio
|u(p)j |2 P
ℓ|u(p−1)ℓ |2e−ϕr,
up to some fixed smooth positive factor depending only on the metric induced by ω onKX. However, by the induction relations, we have
P
Since the sections (u(r)j ) generate their line bundle, the ratios involved are positive functions without zeroes and poles, hence smooth and bounded [possibly after shrinking the base disc ∆, as is permitted]. On the other hand, assumption (ii) and the fact that σ has coefficients in the multiplier ideal sheaf I(h0|X0) tell us that e−ϕr, 1≤r < m and |σ|2e−ϕ0 are locally integrable on X0. It follows that
for allp≥1 [of course, the integral certainly involves finitely many trivializations of the bundles involved, whereas the integrand expression is just local in each chart]. Inductively, the L2 extension theorem produces sectionsue(p)j ofFp overX such that
The next idea is to extract the limits ofp-th roots of these sections to get a singular hermitian metric onmKX+PLj. As the functionse−ϕr are locally bounded below (ϕr being psh), the H¨older inequality implies that
Z
Jensen’s inequality together with well known facts of potential theory now show that some subsequence of the sequence of plurisubharmonic functions
1 qlogP
j|eu(mq)j |2
which should be thought of as weights on the Q-line bundles
1
q(A+q(mKX+P Lj))
converges almost everywhere to the weightψof a singu-lar hermitian metricH with semi-positive curvature onmKX+P
Lj, in the form
OnX0 we have metricω⊗γ. Clearlyγ has a semipositive curvature current on X and in a local trivialization we have kσk2ω⊗γ is also locally integrable on X0 by the H¨older inequality. A new (and final) application of theL2 extension theorem to the hermitian line bundle (G, γ) implies thatσcan be extended to X. The theorem is proved.
The special case of the theorem obtained by taking all bundlesLj trivial tells us in particular that any pluricanonical sectionσofmKX overX0extends toX. By the upper semi-continuity oft7→h0(Xt, mKXt), this implies
Corollary 6.3. (Siu [59]) For any projective familyt7→Xtof algebraic varieties, the plurigenera pm(Xt) =h0(Xt, mKXt)do not depend on t.
At the moment, it should be observed that there are no purely algebraic proofs of the invariance of plurigenera, though Y. Kawamata [32] has given an algebraic proof in the case of varieties of general type.
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Universit´e de Grenoble I, Institut Fourier, 100 rue des Maths, BP 74, 38402 Saint-Martin d’H`eres
E-mail: demailly@fourier.ujf-grenoble.fr