Analytic methods in algebraic geometry

Jean-Pierre Demailly

Description

Analytic methods in algebraic geometry Lectures by Jean-Pierre Demailly Universite de Grenoble I, Institut Fourier December 2009 A compilation of lectures given at various places (CIME 1994, Trieste 2000, Mahdia 2004, Grenoble 2007, Park City 2008, Beijing 2009 ...) Contents 0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. Preliminary material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 2. Lelong numbers and intersection theory . . . . . . . . . . . . . . . . . . . . . . . . . . .

• been brought


• given manifold


• step can


• skoda's estimates


• has been known


• l2 section


• problems via analytic methods


• local holomorphic

Sujets

• Université Grenoble-I
• Demailly

Informations

• Publié par: pefav
• Nombre de lectures: 9
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Analytic methods in algebraic geometry
Lectures by Jean-Pierre Demailly
Universit´e de Grenoble I, Institut Fourier
December 2009
Acompilationoflecturesgivenatvariousplaces(CIME1994,Trieste2000,Mahdia2004,
Grenoble 2007, Park City 2008, Beijing 2009 ...)
Contents
0. Introduction.....................................................................................1
1. Preliminary material.............................................................................4
2. Lelong numbers and intersection theory.........................................................12
3. Hermitian vector bundles, connections and curvature............................................21
4. Bochner technique and vanishing theorems......................................................26
25. L estimates and existence theorems............................................................31
6. Numerically effective and pseudo-effective line bundles...........................................39
7. A simple algebraic approach to Fujita’s conjecture...............................................50
8. Holomorphic Morse inequalities.................................................................59
9. Effective version of Matsusaka’s big theorem....................................................62
10. Positivity concepts for vector bundles..........................................................67
211. Skoda’s L estimates for surjective bundle morphisms..........................................74
212. The Ohsawa-Takegoshi L extension theorem..................................................84
13. Approximation of closed positive currents by analytic cycles....................................97
14. Subadditivity of multiplier ideals and Fujita’s approximate Zariski decomposition theorem.....117
15. Hard Lefschetz theorem with multiplier ideal sheaves..........................................122
16. Invariance of plurigenera of projective varieties................................................134
17. Numerical characterization of the Ka¨hler cone ................................................ 137
18. Structure of the pseudo-effective cone and mobile intersection theory..........................147
19. Super-canonical metrics and abundance.......................................................162
20. Siu’s analytic approach and P˘aun’s non vanishing theorem....................................169
References....................................................................................... 172
§0. Introduction
The main purpose of these notes is to describe analytic techniques which are use-
ful to study questions such as linear series, multiplier ideals and vanishing theorems for
algebraicvectorbundles. One century aftertheground-breaking workofRiemannonge-
ometric aspects of function theory, the general progress achieved in differential geometry
and global analysis on manifolds resulted into major advances in the theory of algebraic
and analytic varieties of arbitrary dimension. One central unifying concept is the con-
cept of positivity, which can ve viewed either in algebraic terms (positivity of divisors
and algebraic cycles), or in more analytic terms (plurisubharmonicity, hermitian connec-
tions withpositive curvature). In this direction, one of the most basic result is Kodaira’s2 Jean-Pierre Demailly, Analytic methods in algebraic geometry
vanishing theorem for positive vector bundles (1953-54), which is a deep consequence of
the Bochner technique and of the theory of harmonic forms initiated by W.V.D. Hodge
during the 1940’s. This method quickly led Kodaira to the well-known embedding the-
orem for projective varieties, a far reaching extension of Riemann’s characterization of
abelian varieties. Further refinements of the Bochner technique led ten years later to
2the theory of L estimates for the Cauchy-Riemann operator, (J.J. Kohn [Koh63, 64],
Andreotti-Vesentini [AV65], [Ho¨r65]). Not only vanishing theorems can be proved of re-
proved in that manner, but perhaps more importantly, extremely precise information of
a quantitative nature is obtained about solutions of ∂-equations, their zeroes, poles and
growth at infinity.
What makes the theory extremely flexible is the possibility to formulate existence
2theorems with a wide assortment of different L norms, namely norms of the formR
2 −2ϕ|f| e where ϕ is a plurisubharmonic or strictly plurisubharmonic function on theX
given manifold or variety X. Here, the weight ϕ need not be smooth, and it is on
the contrary extremely important to allow weights which have logarithmic poles of theP
2form ϕ(z) = clog |g | , where c > 0 and (g ) is a collection of holomorphic func-j j
tions possessing a common zero zet Z ⊂ X. Following Nadel [Nad89], one defines the
multiplier ideal sheaf (ϕ) to be the sheaf of germs of holomorphic functions f such
2 −2ϕthat |f| e is locally summable. Then (ϕ) is a coherent algebraic sheaf over X and
qH (X,K ⊗L⊗ (ϕ)) = 0 for all q > 1 if the curvature of L is positive as a current.X
This important result can be seen as a generalization of the Kawamata-Viehweg vanish-
ing theorem ([Kaw82], [Vie82]), which is one of the cornerstones of higher dimensional
algebraic geometry, especially in relation with Mori’s minimal model program.
In the dictionary between analytic geometry and algebraic geometry, the ideal (ϕ)
playsa very important role, since it directly converts ananalyticobject into analgebraic
one, and, simultaneously, takes care of the 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 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 re-
sult” expected to hold in algebraic geometry is expressed in the following conjecture of
Fujita [Fuj87]: if L is an ample (i.e. positive) line bundle on a projective n-dimensional
algebraic variety X, then K +(n+1)L is generated by sections and K +(n+2)L isX X
very ample. In the last decade, a lot of effort has been brought for 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 surfaces by Reider [Rei88] (the case of
curvesiseasyandhasbeenknownsinceaverylongtime),andthenumericalcriterionfor
the very ampleness of 2K +L given in [Dem93b], obtained by means of analytic tech-X
niques and Monge-Amp`ere equations with isolated singularities. Alternative algebraic
techniques were developed slightly later by Kolla´r [Kol92], Ein-Lazarsfeld [EL93], Fujita
[Fuj93], Siu [Siu95, 96], Kawamata [Kaw97] and Helmke [Hel97]. We will explain here
Siu’smethodbecauseitistechnicallythesimplestmethod; oneoftheresultsobtainedby3n+1
thismethodisthefollowingeffectiveresult: 2K +mL isveryampleform>2+ .X n
I
I
I
I§0. Introduction 3
ThebasicideaistoapplytheKawamata-Viehwegvanishingtheorem,andtocombinethis
with the Riemann-Roch formula in order to produce sections through a clever induction
procedure on the dimension of the base loci of the linear systems involved.
AlthoughSiu’sresultiscertainlynotoptimal,itissufficienttoobtainaniceconstruc-
tive proof of Matsusaka’s big theorem ([Siu93], [Dem96]). The result states that there
n n−1is an effective value m depending only on the intersection numbers L and L ·K ,0 X
such that mL is very ample for m > m . The basic idea is to combine results on the0
very ampleness of2K +mL together withthetheory of holomorphicMorse inequalitiesX
′ ′([Dem85b]). The Morse inequalities are used to construct sections of mL−K for mX
large. Again this step can be made algebraic (following suggestions by F. Catanese and
R.Lazarsfeld), buttheanalyticformulationapparentlyhasawiderrangeofapplicability.
2In the next sections, we pursue the study of L estimates, in relation with the Null-
stellenstatz and with the extension problem. Skoda [Sko72b, Sko78] showed that theP
2division problem f = g h can be solved holomorphically with very precise L esti-j j
2 −pmates, providedthattheL normof|f||g| isfiniteforsomesufficiently largeexponent
p (p > n = dimX is enough). Skoda’s estimates have a nice interpretation in terms of
local algebra, and they lead to precise qualitative and quantitative estimates in con-
2nection with the B´ezout problem. Another very important result is the L extension
theorem by Ohsawa-Takegoshi [OT87, Ohs88], which has also been generalized later by
2Manivel [Man93]. The main statement is that every L section f of a suitably posi-
2 ˜tive line bundle defined on a subavariety Y