Geometrie Analytique Analytic Geometry

Jean-Pierre Demailly

Description

Geometrie Analytique/Analytic Geometry Proprietes de semi-continuite de la cohomologie et de la dimension de Kodaira-Iitaka Jean-Pierre Demailly Resume – Soit X ? S un morphisme analytique propre et plat d'espaces complexes reduits, de fibres (Xt)t?S . Etant donne un faisceau E sur X de OX -modules localement libres, induisant sur les fibres une famille de faisceaux (Et ? Xt)t?S, nous montrons que les dimensions de cohomologie hq(t) = hq(Xt, Et) satisfont la propriete de semi-continuite suivante: pour tout q ≥ 0, la somme alternee hq(t)? hq?1(t) + . . . + (?1)qh0(t) est semi- continue superieurement pour la topologie de Zariski. En particulier, E etant suppose de rang 1, si le faisceau E0 au dessus d'une fibre X0, 0 ? S, a une dimension de Kodaira-Iitaka ?(E0) = d0 ≥ 0 et si h0(X0, E?m0 ) croıt plus vite que h1(X0, E ?m 0 ), alors ?(Et) ≥ d0 pour t voisin de 0. Semicontinuity properties of cohomology and of Kodaira-Iitaka dimension Abstra t – Let X ? S be a proper and flat morphism of complex spaces, and let (Xt)t?S be the fibres.

• semi-continuite de hq


• cohomologie


• fibre


• dimension de kodaira ?


• theorie du modele minimal


• topologie ordinaire


• famille de varietes complexes compactes

Sujets

• Demailly
• Schneider
• Homologie et cohomologie
• maths
• Mathématiques

Informations

• Publié par: pefav
• Nombre de lectures: 54
• Langue: Español

Extrait

G´eom´etrie Analytique/Analytic Geometry
Propri´et´es de semi-continuit´e de la cohomologie
et de la dimension de Kodaira-Iitaka
Jean-Pierre Demailly
– SoitX → S un morphisme analytique propre et plat d’espaces complexes
r´eduits, de fibres (X ) . Etant donn´e un faisceauE surX deO -modules localementt t∈S X
libres, induisant sur les fibres une famillede faisceaux (E → X ) , nous montrons quet t t∈S
q qlesdimensionsdecohomologieh (t)= h (X ,E )satisfontlapropri´et´edesemi-continuit´et t
q q−1 q 0suivante: pourtoutq≥ 0,lasommealtern´eeh (t)−h (t)+...+(−1) h (t)estsemi-
continue sup´erieurement pour la topologie de Zariski. En particulier, E ´etant suppos´e
de rang 1, si le faisceau E au dessus d’une fibre X , 0 ∈ S, a une dimension de0 0
⊗m ⊗m0 1Kodaira-Iitaka κ(E ) = d ≥ 0 et si h (X ,E ) croˆıt plus vite que h (X ,E ),0 0 0 00 0
alors κ(E )≥ d pour t voisin de 0.t 0
Semicontinuity properties of cohomology
and of Kodaira-Iitaka dimension
– Let X → S be a proper and flat morphism of complex spaces, and let
(X ) be the fibres. Given a sheaf E over X of locally free O -modules, inducingt t∈S X
on the fibres a family of sheaves (E → X ) , we show that the cohomology groupt t t∈S
q qdimensions h (t) = h (X ,E ) satisfy the following semicontinuity property: for everyt t
q q−1 q 0q≥0, the alternate sum h (t)−h (t)+...+(−1) h (t) is upper semicontinuous for
the Zariski topology. In particular, for E of rank 1, if the sheaf E over a fibre X ,0 0
⊗m00∈ S, has a Kodaira-Iitaka dimension κ(E )= d ≥ 0 and if h (X ,E ) grows faster0 0 0 0⊗m1than h (X ,E ), then κ(E )≥ d for t near 0.0 t 00
Abridged English version. – Let X → S be a proper and flat morphism of
reduced complex spaces, and let (X ) be the fibres. Given a sheafE overX oft t∈S
locally freeO -modules, inducing on the fibres a family of sheaves (E →X ) ,X t t t∈S
q qit is well-known that the cohomology group dimensions h (t) = h (X ,E ) aret t
uppersemicontinuous; thiswasfirstshownbyKodaira-Spencer[KoS57]inthecase
of a smooth deformation X → S. In fact, the following stronger semicontinuity
property (almost explicitly contained in H. Flenner’s results [Fle81a]) holds:
q q−1 q 0Theorem. – For everyq≥0, the alternate sumh (t)−h (t)+...+(−1) h (t)
is upper semicontinuous for the (analytic) Zariski topology.
In the above analytic situation, the proof is an easy consequence of the method
developed by Kiehl-Verdier [KiV71] to prove Grauert’s direct image theorem
[Gra60], modulo an elementary linear algebra argument. In fact, Kiehl-Verdier
qprovedthatthereexistslocallyonS acomplex(V ,d)oflocallyfreesheaveswhich
qadmits precisely the direct imagesR π E as itscohomology sheaves. Moreover, as⋆
qπ :X →S is flat, the fibre cohomology groups H (X ,E ) are obtained from thet t
q q qcomplexoffinitedimensionalvectorspaces(V ,d ),whereV =V ⊗ O /mt O S,t S,tt t S
q q q q+1
(see e.g. [DoV72], expos´e II-bis). IfZ denotes the kernel ofd :V →V , thent t t t
qqz (t):=dimZ isanupper semicontinuous functionfortheZariskitopology. Thist
is easily seen by looking at the rank of the minor determinants of the matrix of
q q q+1d :V →V . Since the truncated complex
q−1 q0 10→V →V →...→V →Z →0t t t t
1
bstr
e
R


esum
act
Ajadmits H (X ,E ), 0≤j≤q, as its sole cohomology groups, we infert t
q q−1 q 0 q q−1 q−2 q 0h (t)−h (t)+...+(−1) h (t) =z (t)−v +v +...+(−1) v ,
q qwhere v denotes the rank of V . The Theorem follows. A similar proof can be
obtained via harmonic forms and Hodge theory (but the semicontinuity is then
proved only for the “ordinary” topology on S). For rank 1 sheaves E , the uppert
1 0semicontinuity of h (t)−h (t) implies the following
Consequence. – If the sheaf E over a fibre X , 0 ∈ S, has a Kodaira-Iitaka0 0
dimension κ(E )=d ≥0 and if0 0
−d 0 ⊗m 1 ⊗m0limsupm h (X ,E )−h (X ,E ) > 0,0 00 0
m→+∞
then κ(E ) ≥ d for t near 0 in the Zariski topology. In particular, if X = Xt 0 0
is a nonsingular variety of Kodaira dimension κ(X) := κ(K ) = d such thatX 0
⊗m ⊗m0 1 d0h (X,K ) grows faster than h (X,K ) by a term of magnitude cm , thenX X
every small deformation X satisfies κ(X )≥d .t t 0
Easy examples show that κ(E ) is in general neither upper nor lower semicon-t
tinuous (evenwithrespect to theordinarytopology),withoutfurther assumptions
⊗m1on h (X ,E ). On the other hand, according to a more or less standard con-t t
jecture, it is expected that the Kodaira dimension κ(X ) should be constant fort
an arbitrary smooth deformation. I would like to thank S. Kosarew for point-
ing out such questions and for other stimulating discussions. In order to apply
the above results to deformations of varieties of general type, it would be inter-
esting to know whether every birational class of varieties of general type con-
tains a normal variety X with only Q-Gorenstein terminal singularities, such that
⊗m ⊗m0 1 dimXh (X,K )−h (X,K ) ≥ cm for m large. This is indeed the case if aX X
minimal modelX exists (i.e. ifK can be taken numerically effective). Moreover,X
in the above general setting, we conjecture that

−n q ⊗m q−1 ⊗m q 0 ⊗mlimsupm h (X ,E )−h (X ,E )+...+(−1) h (X ,E )t t tt t t
m→+∞
is an upper semicontinuous function of t (in the ordinary topology, say). We
expectthatthiswillfollowfromacarefulanalysisofthespectraltheoryofcomplex
Laplace-Beltrami operators, at least whenX →S is smooth.
1. Semi-continuit´e des dimensions de cohomologie. – Soient X et S des
espacesanalytiquescomplexesr´eduits, etsoitπ :X →S unmorphismeanalytique
propre et plat, de fibres (X ) . On suppose donn´e sur X un faisceau E det t∈S
O -modules localement libres, c’est-a`-dire une famille holomorphe (E →X )X t t t∈S
de faisceaux localement libres. On sait alors que les dimensions de cohomologie
q qh (t) = dimH (X ,E ) sont des fonctions semi-continues sup´erieurement de tt t
pour la topologie de Zariski (dans sa version analytique, c’est-`a-dire la topologie
dont les ouverts sont les compl´ementaires d’ensembles analytiques). Ce r´esultat
remonte aux travaux de Kodaira-Spencer [KoS57] dans le cas d’une d´eformation
2lisseX →S, la topologie consid´er´ee sur S ´etant la topologie ordinaire (de vari´et´e
diff´erentiable). Le cas g´en´eral peut se d´eduire de la d´emonstration du th´eor`eme
des images directes de Grauert [Gra60] (voir [KiV71], [FoK71], [DoV72], [Sch72]).
On a en fait le r´esultat de semi-continuit´e nettement plus fort suivant:
Th´eor`eme. – Pour tout q≥0, la somme altern´ee
q q−1 q 0h (t)−h (t)+...+(−1) h (t)
est une fonction semi-continue sup´erieurement de t dans la topologie de Zariski.
A notre grande surprise, ce r´esultat semble peu connu des sp´ecialistes (voir
toutefois [Pal76], pour un r´esultat de semi-continuit´e analogue de la “cohomologie
tangente”, et H. Flenner [Fle81a,b] pour des r´esultats plus forts concernant des
foncteurs cohomologiques g´en´eraux; voir aussi M. Schneider [Sch72]). Nous
redonnons ci-dessous une d´emonstration ind´ependante ´el´ementaire. Je tiens `a
remercier vivement S. Kosarew pour quelques r´ef´erences et conversations ayant
fortement contribu´e `a l’´elaboration des id´ees de cette note.
Preuve. La d´emonstration du th´eor`eme des images directes donn´ee dans [KiV71]
qfournit localement sur S un complexe (V ,d) de faisceaux localement libres dont
qla cohomologie calcule les images directes R π E. De plus, comme π :X →S est⋆
qplat,lacohomologiedesfibresH (X ,E )s’obtienta`partirducomplexed’espacest t
q q qvectorielsdedimensionfinie(V ,d ),ou`V =V ⊗ O /m (voirparexemplet O S,t S,tt t S
q q q q+1
[DoV72], expos´e II-bis). Si Z d´esigne le noyau du morphisme d : V → V ,t t t t
qqalors z (t) := dimZ est une fonction semi-continue sup´erieurement pour lat
topologie de Zariski, comme on le voit ais´ement en regardant le rang des mineurs
q q q+1de la matrice d´efinissant le morphisme d :V →V . Le complexe tronqu´e
q−1 q0 10→V →V →...→V →Z →0t t t t
jayant pour cohomologie les groupes H (X ,E ) d’indices 0≤j≤q, on obtientt t
q q−1 q 0 q q−1 q−2 q 0h (t)−h (t)+...+(−1) h (t) =z (t)−v +v +...+(−1) v ,
q qou` v d´esigne le rang deV . Le th