Multiplier ideal sheaves and analytic methods in algebraic geometry

Jean-Pierre Demailly

Description

Multiplier ideal sheaves and analytic methods in algebraic geometry Jean-Pierre Demailly Universite de Grenoble I, Institut Fourier Lectures given at the ICTP School held in Trieste, Italy, April 24 – May 12, 2000 Vanishing theorems and effective results in Algebraic Geometry Contents 0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. Preliminary Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2. Lelong Numbers and Intersection Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3. Holomorphic Vector Bundles, Connections and Curvature . . . . . . . . . . . . . . . . . . . 21 4. Bochner Technique and Vanishing Theorems .

• dzi ?


• been brought


• vanishing theorems


• skoda's estimates


• has been known


• effort has


• condi- tion can


• l2 section


• local holomorphic

Sujets

• Trieste
• Université Grenoble-I
• Demailly
• Bundle
• Schneider

Informations

• Publié par: pefav
• Nombre de lectures: 75
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Multiplier ideal sheaves and analytic methods in
algebraic geometry
Jean-Pierre Demailly
Universit´e de Grenoble I, Institut Fourier
Lectures given at the ICTP School held in Trieste, Italy, April 24 – May 12, 2000
Vanishing theorems and effective results in Algebraic Geometry
Contents
0. Introduction ................................................................ 1
1. Preliminary Material ........................................................4
2. Lelong Numbers and Intersection Theory ...................................12
3. Holomorphic Vector Bundles, Connections and Curvature ...................21
4. Bochner Technique and Vanishing Theorems ................................26
25. L Estimates and Existence Theorems ......................................31
6. Numerically Effective and Pseudoeffective Line Bundles .....................40
7. A Simple Algebraic Approach to Fujita’s Conjecture ........................47
8. Holomorphic Morse Inequalities ............................................56
9. Effective Version of Matsusaka’s Big Theorem ..............................58
10. Positivity Concepts for Vector Bundles ....................................63
211. Skoda’s L Estimates for Surjective Bundle Morphisms ....................71
212. The Ohsawa-Takegoshi L Extension Theorem .............................80
13. Invariance of Plurigenera of Varieties of General Type .....................96
14. Subadditivity of Multiplier Ideal Sheaves and Zariski Decomposition ......104
15. Hard Lefschetz Theorem with Multiplier Ideal Sheaves ....................109
References ...................................................................122
0.Introduction
Transcendental methodsofalgebraicgeometryhavebeenextensivelystudied sincea
verylongtime,startingwiththeworkofAbel,JacobiandRiemanninthenineteenth
century. More recently, in the period 1940-1970, the work of Hodge, Hirzebruch,
Kodaira, Atiyah revealed still deeper relations between complex analysis, topology,
PDE theory and algebraic geometry. In the last ten years, gauge theory has proved
to be a very efficient toolfor the study of many important questions: moduli spaces,
stable sheaves, non abelian Hodge theory, low dimensional topology...
Our main purpose here is to describe a few analytic tools which are useful to
study questions such as linear series and vanishing theorems for algebraic vector
bundles. One of the early successes of analytic methods in this context is Kodaira’s2 J.-P. Demailly, Multiplier ideal sheaves and analytic methods
use of the Bochner technique in relation with the theory of harmonic forms, during
the decade 1950-60. The idea is to represent cohomology classes by harmonic forms
and to prove vanishing theorems by means of suitable a priori curvature estimates.
TheprototypeofsuchresultsistheAkizuki-Kodaira-Nakanotheorem(1954):ifX is
a nonsingular projective algebraic variety and L is a holomorphic line bundle on X
pqwithpositivecurvature, thenH (X,Ω ⊗L)= 0 forp+q> dimX (throughout theX
p p ⋆ n ⋆paperwesetΩ =Λ T andK =Λ T ,n= dimX,viewingtheseobjectseitherXX X X
asholomorphicbundles oraslocallyfreeO -modules).Itisonlymuch laterthatanX
algebraic proof of this result has been proposed by Deligne-Illusie, via characteristic
p methods, in 1986.
A refinement of the Bochner technique used by Kodaira led, about ten years
2later, to fundamental L estimates due to H¨ormander [H¨or65], concerning solu-
tions of the Cauchy-Riemann operator. Not only vanishing theorems are proved,
but more precise information of a quantitative nature is obtained about solutions
2of∂-equations. The best way of expressing these L estimates is to use a geometric
settingfirst considered by Andreotti-Vesentini [AV65].More explicitly,suppose that
we have a holomorphic line bundle L equipped with a hermitian metric of weight
−2ϕe ,whereϕ isa (locallydefined) plurisubharmonic function; then explicitboundsR
2 2 −2ϕon the L norm |f| e of solutions is obtained. The result is still more usefulX
if the plurisubharmonic weight ϕ is allowed to have singularities. Following Nadel
[Nad89], we define the multiplier ideal sheaf I(ϕ) to be the sheaf of germs of holo-
2 −2ϕmorphicfunctionsf suchthat|f| e islocallysummable.ThenI(ϕ)isacoherent
qalgebraic sheaf overX andH (X,K ⊗L⊗I(ϕ)) = 0 for allq≥ 1 if the curvatureX
ofL is positive (as a current). This important result can be seen as a generalization
ofthe Kawamata-Viehweg vanishing theorem ([Kaw82],[Vie82]),which isone of the
cornerstones of higher dimensional algebraic geometry (especially of Mori’s minimal
model program).
In the dictionary between analytic geometry and algebraic geometry, the ideal
I(ϕ) plays a very important role, since it directly converts an analytic object into
an algebraic one, and, simultaneously, takes care of the singularities in a very ef-
ficient way. Another analytic tool used to deal with singularities is the theory of
positive currents introduced by Lelong [Lel57]. Currents can be seen as generaliza-
tions of algebraic cycles, and many classical results of intersection theory still apply
to currents. The concept of Lelong number of a current is the analytic analogue
of the concept of multiplicity of a germ of algebraic variety. Intersections of cycles
correspond to wedge products of currents (whenever these products are defined).
Besides the Kodaira-Nakano vanishing theorem, one of the most basic “effective
result” expected to hold in algebraic geometry is expressed in the following conjec-
ture of Fujita [Fuj87]: if L is an ample (i.e. positive) line bundle on a projective
n-dimensional algebraic varietyX, thenK +(n+1)L is generated by sections andX
K +(n+2)L is very ample. In the last decade, a lot of effort has been brought forX
the solution of this conjecture – and it seems indeed that a solution might finally
emerge in the first years or the third millenium – hopefully during this Summer
School! The first major results are the proof of the Fujita conjecture in the case of
surfacesbyReider[Rei88](thecaseofcurvesiseasyandhasbeenknownsinceavery
long time), and the numerical criterion for the very ampleness of 2K +L given inX
[Dem93b], obtained by means of analytic techniques and Monge-Amp`ere equations0. Introduction 3
with isolated singularities. Alternative algebraic techniques were developed slightly
later by Koll´ar [Kol92],Ein-Lazarsfeld [EL93],Fujita [Fuj93], Siu [Siu95, 96],Kawa-
mata [Kaw97] and Helmke [Hel97]. We will explain here Siu’s method because it
is technically the simplest method; one of the results obtained by this method is
3n+1the following effective result: 2K +mL is very ample for m ≥ 2 + . TheX n
basic idea is to apply the Kawamata-Viehweg vanishing theorem, and to combine
this with the Riemann-Roch formula in order to produce sections through a clever
inductionprocedure onthedimension ofthe baselociofthelinearsystemsinvolved.
Although Siu’s result is certainly not optimal, it is sufficient to obtain a nice
constructive proof of Matsusaka’s big theorem ([Siu93], [Dem96]). The result states
nthat there is an effective value m depending only on the intersection numbers L0
n−1andL ·K ,suchthatmLisveryampleform≥m .ThebasicideaistocombineX 0
results on thevery ampleness of 2K +mL togetherwiththe theoryof holomorphicX
Morseinequalities([Dem85b]).TheMorseinequalitiesareusedtoconstructsections
′ ′ofmL−K form large. Again this step can be made algebraic (following sugges-X
tions by F. Catanese and R. Lazarsfeld), but the analytic formulation apparently
has a wider range of applicability.
2In the next sections, we pursue the study of L estimates, in relation with the
Nullstellenstatzandwiththeextensionproblem.Skoda[Sko72b,Sko78]showedthatP
2thedivisionproblemf = g h canbesolvedholomorphicallywithverypreciseLj j
2 −pestimates, provided that the L norm of|f||g| is finite for some sufficiently large
exponent p (p > n = dimX is enough). Skoda’s estimates have a nice interpreta-
tion in terms of local algebra, and they lead to precise qualitative and quantitative
estimates in connection with the B´ezout problem. Another very important result
2is the L extension theorem by Ohsawa-Takegoshi [OT87, Ohs88], which has also
2been generalized later by Manivel [Man93]. The main statement is that every L
section f of a suitably positive line bundle defined on a subavariety Y ⊂ X can
2 ˜be extended to a L section f defined over the whole of X. The positivity condi-
tion can be understood in terms of the canonical sheaf and normal bundle to the
subvariety. The extension theorem turns outto have an incredible amount of impor-
tant consequences: among them, let us mention for instance Siu’s theorem [Siu74]
on the analyticity of Lelong numbers, the basic approximation theorem of closed
positive (1,1)-currents by divisors, the subadditivity propertyI(ϕ+ψ)⊂I(ϕ)I(ψ)
of multiplier ideals [DEL00], the restriction formula I(ϕ ) ⊂ I(ϕ) , .... A suit-|Y |Y
able combination o