A numerical criterion for very ample line bundles

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Niveau: Supérieur, Licence, Bac+2
A numerical criterion for very ample line bundles Jean-Pierre Demailly Universite de Grenoble I, Institut Fourier, BP 74, Laboratoire associe au C.N.R.S. n˚ 188, F-38402 Saint-Martin d'Heres Abstract. — Let X be a projective algebraic manifold of dimension n and let L be an ample line bundle over X . We give a numerical criterion ensuring that the adjoint bundle KX + L is very ample. The sufficient conditions are expressed in terms of lower bounds for the intersection numbers Lp ·Y over subvarieties Y of X . In the case of surfaces, our criterion gives universal bounds and is only slightly weaker than I. Reider's criterion. When dimX ≥ 3 and codimY ≥ 2, the lower bounds for Lp · Y involve a numerical constant which depends on the geometry of X . By means of an iteration process, it is finally shown that 2KX +mL is very ample form ≥ 12nn. Our approach is mostly analytic and based on a combination of Hormander's L2 estimates for the operator ∂, Lelong number theory and the Aubin-Calabi-Yau theorem. Table of contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  • over subvarieties

  • ample property

  • l2 ·

  • closed positive

  • aubin-calabi-yau theorem

  • positive currents

  • bundle over

  • nef line

  • any vector bundle

  • very ample


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A numerical criterion for very ample line bundles

Jean-Pierre Demailly
Universit´ de Grenoble I,
Institut Fourier, BP 74,
Laboratoire associ´ au C.N.R.S. n˚188,
F-38402 Saint-Martin d’H`res

Abstract. —LetXbe a projective algebraic manifold of dimensionnand letL
be an ample line bundle overX. Wegive a numerical criterion ensuring that the
adjoint bundleKX+LThe sufficient conditions are expressed inis very ample.
p
terms of lower bounds for the intersection numbersL∙Yover subvarietiesYofX.
In the case of surfaces, our criterion gives universal bounds and is only slightly
weaker than I. Reider’s criterion.WhendimX≥3andcodimY≥2, the lower
p
bounds forL∙Yinvolve a numerical constant which depends on the geometry
ofXmeans of an iteration process, it is finally shown that. By2KX+mLis very
n
ample form≥12n. Ourapproach is mostly analytic and based on a combination
2
of H¨rmander’sLestimates for the operator∂, Lelong number theory and the
Aubin-Calabi-Yau theorem.

Table of contents

1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .p. 2
2. Singular hermitianmetrics on holomorphic line bundles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .p. 5
3. Basic results on Lelong numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .p. 6
2
4.Lestimates and existence of holomorphic sections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .p. 8
5. Aubin-Calabi-Yautheorem and convexity inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .p. 12
6. Mass concentration in the Monge-Amp`re equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .p. 14
7. Choice of the logarithmic singularities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .p. 17
8. Upper bound for the 1-codimensional polar components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .p. 19
9. Approximation of closed positive (1,1)-currents by divisors. . . . . . . . . . . . . . . . . . . . . . . . . . . .p. 21
10. Self-intersection inequality for closed positive currents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .p. 25
11. Proof of the criterion inarbitrary dimension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .p. 33
12. Universal bounds for very ample linebundles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .p. 38
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .p. 43

Key words:Very ample line bundle, numerical criterion, plurisubharmonic
function, closed positive current, Lelong number, Monge-Amp`re equation,
AubinCalabi-Yau theorem, intersection theory.

A.M.S. Classification:32J25, 32L10, 32C30

1

1. Introduction

LetLbe a holomorphic line bundle over a projective algebraic manifoldXof
dimensionndenote the canonical line bundle of. WeXbyKXand use an additive
1⋆
notation for the group Pic(X) =H(X,O). Theoriginal motivation of this work
was to study the following tantalizing conjecture of Fujita [Fu 88]:ifL∈Pic(X)
is ample, thenKX+ (n+ 2)Lis very ample; the constantnwould then be+ 2
n
optimal sinceKX+ (n+ 1)L=OXis not very ample whenXand= IPL=O(1).
Although such a sharp result seems at present out of reach, a consequence of our
results will be that 2KX+mLis always very ample forLample andmlarger
than some universal constant depending only onn.

Questions of this sort play a very important role in the classification theory
of projective varieties.In his pioneering work [Bo 73], Bombieri proved the
existence of pluricanonical embeddings of low degree for surfaces of general type.
More recently, for an ample line bundleLover an algebraic surfaceS, I. Reider
[Rei 88] obtained a sharp numerical criterion ensuring that the adjoint line bundle
KS+Lis spanned or very ample; in particular,KS+ 3Lis always spanned
andKS+ 4Lvery ample.Reider’s method was further developed by Catanese
[Ca 88], Sakai [Sa 88] and Beltrametti-Francia-Sommese [BFS 89], who studied
the existence of higher order embeddings viasapproach is based on-jets. Reider’s
the construction of rank two vector bundles associated to some 0-cycles in special
position with respect to the linear system|KS+L|and a use of Bogomolov’s
inequality for stable vector bundles.Unfortunately, these methods do not apply
in dimension≥3 and no similar general result was available.In a somewhat
different context, Fujita [Fu 87] proved thatKX+ (n+ 2)LThisis always ample.
result is obtained via Mori’s theory of extremal rays [Mo 82] and the cone theorem
of Kawamata (cf.[Ka 84], [KMM 87]), but the arguments are purely numerical
and give apparently no insight on the very ample property.

Our purpose here is to explain a completely different analytic approach
which is applicable in arbitrary dimension.Let us first recall a few usual notations
that will be used constantly in the sequel:
Z
(1.1)L1∙ ∙ ∙Lp∙Y=c1(L1)∧. . .∧c1(Lp)
Y

denotes the intersection product ofpline bundlesL1, . . . , Lpover ap-dimensional
p
subvarietyY⊂X. IncaseL1=. . .=Lpwe write insteadL∙Yand in case
Y=Xwe omitYSimilar notations will be used for divisors.in the notation.
Recall that a line bundle (or a IR-divisor)LoverXis said to be numerically
effective,neffor short, ifL∙C≥0 for every curveC⊂X; in this caseLis said
n
to bebigifL >0. Moregenerally, a vector bundleEis said to be nef if the

associated line bundleOE(1) is nef overP(E) = projective space of hyperplanes
m
inE; any vector bundleEsuch that some symmetric powerS Eis spanned by
its global sections is nef.In this context, we shall prove:

2

Main Theorem. —LetXbe a projectiven-fold and letLbe a big nef
line bundle overX. Supposethat there is a numbera≥0such thatT X⊗ O(aL)
is nef.ThenKX+Lis spanned at each point of a given subsetΞofX(resp.
separates all points inΞ, resp.generatess-jets at any point ofΞ )provided that
n nn n
L >σ0withσ0=n(resp.σ0= 2n, resp.σ0= (n+s) ), and that there exists
a sequence0 =β1< .. .< βn≤1such that any subvarietyY⊂Xof codimension
p= 1,2, . . . , n−1intersectingΞsatisfies
X
n−p−1−1p j
L∙Y >(β−β). . .(β−β)β)
p+1 1p+1pSj(a σp−j
0≤j≤p−1
p p
withS(β)
0= 1,Sj(β) =elementary symmetric function of degreejinβ1, . . . , βp
and
p/n
σ0
n
σp= 1−1−L ,σp∈]σ0p/n, σ0[.
n
L

The expression “separation of points” used here includes infinitesimal
n
separation, that is, generation of 1-jets at each point (the constantσ0= (n+ 1)
n
corresponding tos= 1 can therefore be replaced by the smaller value 2n). Infact,
our proof also gives sufficient conditions for the generation of jets corresponding
to arbitrary 0-dimensional subschemes (Ξ,OΞ) ofX, simply by changing the value
ofσ0; for example, if (Ξ,OΞ) is a local complete intersection, the constantσ0can
n0
be taken equal ton h(Ξ,OΞ) ;unfortunately, this value is in general far from
being optimal.Notice that the numberainvolved in the hypothesis onT Xneed
not be an integer nor even a rational number:the hypothesis then simply means
⋆ ⋆
that any real divisor associated toOT X(1) +La πis nef overP(T X).
As the notation is rather complicated, it is certainly worth examining the
particular case of surfaces and 3-folds.IfXis a surface, we haveσ0= 4 (resp.
2
σ0= 8, resp.σ0= (2+s) ), and we takeβ1= 0,β2= 1.This gives only two
conditions, namely
2
(1.2)σL >0, L∙C > σ1
for every curveCIn that case, the proof shows that the assumptionintersecting Ξ.
on the existence ofaThese bounds are not very far from thoseis unnecessary.
obtained with Reider’s method, although they are not exactly as sharp.IfXis
3
a 3-fold, we haveσ0= 27 (resp.σ0= 54, resp.σ0= (3+s) ), and we take
β1= 0< β2=β < β3= 1.Therefore our condition is that there existsβ∈]0,1[
such that
3 2−1−1
(1.3)L >σ0, L∙S > βσ1, L∙C >(1−β) (σ2+β aσ1)
for every curveCor surfaceSintersecting Ξ.
In general, we measure the “amount of ampleness” of a nef line bundleL
on a subset Ξ⊂Xby the number
p1/p
(1.4)µΞ(L) =min min(L∙Y),
1≤p≤ndimY=p, Y∩Ξ6=∅

3

whereYruns over allp-dimensional subvarieties ofXThe Nakai-intersecting Ξ.
Moishezon criterion tells us thatLis ample if and only ifµX(L)>0. Aneffective
version of this criterion can be easily deduced from the Main Theorem:in fact, a
suitable choice of the constantsβpin terms ofa,σ0andµΞ(L) yields:

Corollary 1.—LetLbe a big nef line bundle overXsuch that
T X⊗ O(aL)is nef for somea≥0, and letΞbe an arbitrary subset ofX. Then
the line bundleKX+mLspans(generatesresp. separatespoints, resp.s-jets)on
Ξas soon as

n1k1 11o
1
+ +. . .+
n−k n−k n−1n−2k+1
m >maxBnσ0,(Bnσ0) (aµΞ(L))
µΞ(L)1≤k≤n−2
n nn
withσ0=n(resp.σ0= 2n, resp.σ0= (n+s) )and with a constantBn<2.005
depending only onn(table(11.11)contains the first values ofBn).

WhenLis ample, the numberaalways exists and we haveµΞ(L)≥1 for
any choice of Ξ.We thus get an explicit lower boundm0depending only onn, a
such thatKX+mLis spanned or very ample form≥m0. Unfortunately,these
lower bounds are rather far from Fujita’s expected conditionsm≥nand+ 1
n
m≥nhowever that the lower bound for+ 2respectively. ObserveLin the Main
n
Theorem is optimal:ifXand= IPL=O(1), thenKX=O(−n−1) soKX+nL
n n
is not spanned, although (nL) =n=σ0. SimilarlyKX+ (n+s)Ldoes not

n
n n+1
generates-jets, although(n+s)L= (n+s) =σ0. WhenX⊂IP isthe
n-dimensional quadric andL=OX(1), thenKX+nL=OXis not very ample,
n n
although (nL2) =n=σ0.
Another unsatisfactory feature is that our bounds depend on the geometry
ofXthrough the numbera, while the case of curves or surfaces suggests that they
should not.In fact, our proof uses a rather delicate self-intersection inequality for
closed positive currents, and this inequality (which is essentially optimal) depends
in a crucial way on a bound for the “negative part” ofT Xfollows that new. It
ideas of a different nature are certainly necessary to get universal bounds for
the very ampleness ofKX+Lan elementary argument shows that. However,
T X⊗ O(KX+nF) is always nef whenFis very ample (see lemma 12.1).This
observation combined with an iteration of the Main Theorem finally leads to a
universal result.Corollary 2 below extends in particular Bombieri’s result on
pluricanonical embeddings of surfaces of general type to arbitrary dimensions (at
least whenKXis supposed to be ample, see 12.10 and 12.11), and can be seen as
an effective version of Matsusaka’s theorem ([Ma 72], [KoM 83]):

Corollary 2. —IfLis an ample line bundle overX, then2KX+mL
is very ample, resp.generatess-jets, when(m−1)µX(L) +s >2Cnσ0with a
constantCn<3depending only onn(see table(12.3))particular,. In2KX+mL
n
is very ample form >4Cnnand generates highers-jets form >2Cnσ0.

Our approach is based on three rather powerful analytic tools.First, we
2
use H¨rmander’sLestimates for the operator∂with singular plurisubharmonic

4