Applications of the theory of L2 estimates and positive currents in algebraic geometry

Jean-Pierre Demailly

Description

Niveau: Supérieur, Doctorat, Bac+8
Applications of the theory of L2 estimates and positive currents in algebraic geometry Jean-Pierre Demailly Ecole d'ete de Mathematiques de Grenoble, June-July 2007 These notes are derived in part from the lectures “Multiplier ideal sheaves and analytic methods in algebraic geometry” given at the ICTP School held in Tri- este, Italy, April 24 – May 12, 2000 “Vanishing theorems and effective results in Algebraic Geometry”, as well as from a more detailed online book on Algebraic Geometry available at the author's home page Contents 0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. Preliminary Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2. Lelong Numbers and Intersection Theory . . . . . . . . . . . . . . . . . . . . . .

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• l2 estimates


• skoda's estimates


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• local holomorphic

Sujets

• Demailly
• Bundle
• Schneider

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 28
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

2Applications of the theory of L estimates
and positive currents in algebraic geometry
Jean-Pierre Demailly
´Ecole d’´et´e de Math´ematiques de Grenoble, June-July 2007
These notes are derived in part from the lectures “Multiplier ideal sheaves and
analytic methods in algebraic geometry” given at the ICTP School held in Tri-
este, Italy, April 24 – May 12, 2000 “Vanishing theorems and effective results in
Algebraic Geometry”, as well as from a more detailed online book on Algebraic
Geometry available at the author’s home page
http://www-fourier.ujf-grenoble.fr/˜demailly/
Contents
0. Introduction ................................................................ 1
1. Preliminary Material ........................................................5
2. Lelong Numbers and Intersection Theory ...................................13
3. Holomorphic Vector Bundles, Connections and Curvature ...................22
4. K¨ahler identities and Hodge Theory ........................................27
5. Bochner Technique and Vanishing Theorems ................................37
26. L Estimates and Existence Theorems ......................................39
7. Numerically Effective and Pseudoeffective Line Bundles .....................51
8. A Simple Algebraic Approach to Fujita’s Conjecture ........................58
9. Holomorphic Morse Inequalities ............................................67
10. Effective Version of Matsusaka’s Big Theorem .............................69
11. Positivity Concepts for Vector Bundles ....................................74
212. Skoda’s L Estimates for Surjective Bundle Morphisms ....................81
213. The Ohsawa-Takegoshi L Extension Theorem .............................90
14. Invariance of Plurigenera of Varieties of General Type ....................106
15. Subadditivity of Multiplier Ideal Sheaves and Zariski Decomposition ......113
16. Hard Lefschetz Theorem with Multiplier Ideal Sheaves ....................118
17. Nef and Pseudoeffective Cones in K¨ahler Geometry .......................130
18. Numerical Characterization of the K¨ahler Cone ...........................133
19. Cones of Curves ......................................................... 142
20. Duality Results ..........................................................144
21. Approximation of psh functions by logarithms of holomorphic functions ...146
22. Zariski Decomposition and Movable Intersections .........................150
23. The Orthogonality Estimate ............................................. 156
24. Proof of the Main Duality Theorem ......................................158
References ...................................................................1592 J.-P. Demailly, Complex analytic techniques in algebraic geometry
0.Introduction
Transcendental methods of algebraic geometry have been extensively studied since
a long time, starting with the work of Abel, Jacobi and Riemann in the nineteenth
century. More recently, in the period 1940-1970, the work of Hodge, Hirzebruch,
Kodaira,Atiyahrevealed deeper relationsbetween complexanalysis,topology,PDE
theory and algebraic geometry. In the last twenty years, gauge theory has proved to
be a very efficient tool for 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’s
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 geome-
try, the idealI(ϕ) plays a very important role, since it directly converts an analytic
object into an algebraic one, and, simultaneously, takes care of singularities in a
very efficient way.
Another analytic tool used to deal with singularities is the theory of positive
currents introduced by Lelong [Lel57]. Currents can be seen as generalizations of
algebraic cycles, and many classical results of intersection theory still apply to cur-
rents. 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 corre-
spond to wedge products of currents (whenever these products are defined).
Besides the Kodaira-Nakano vanishing theorem, one of the most basic “effective0. Introduction 3
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 equations
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 is3n+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]showedthat
P 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). Skod