Algebraic approximations of holomorphic maps from Stein domains to projective manifolds

Jean-Pierre Demailly

Description

Niveau: Supérieur, Licence, Bac+2
Algebraic approximations of holomorphic maps from Stein domains to projective manifolds Jean-Pierre Demailly 1 Laszlo Lempert 2 Universite de Grenoble I Purdue University Institut Fourier, BP 74 Department of Mathematics U.R.A. 188 du C.N.R.S. West Lafayette, IN 47907, U.S.A. 38402 Saint-Martin d'Heres, France Bernard Shiffman 3 Johns Hopkins University Department of Mathematics Baltimore, MD 21218, U.S.A. Key words: affine algebraic manifold, algebraic approximation, algebraic curve, complete pluripolar set, Eisenman metric, Hormander's L2-estimates for ∂, holo- morphic map, holomorphic retraction, holomorphic vector bundle, hyperbolic space, Kobayashi pseudodistance, Kobayashi-Royden pseudometric, Nash alge- braic map, Nash algebraic retraction, plurisubharmonic function, projective alge- braic manifold, quasi-projective variety, Runge domain, Stein manifold. A.M.S. Classification 1985: 32E30, 32H20, 14C30 Table of contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• line bundle


• projective algebraic


• compact subset


• nash algebraic


• kobayashi pseudodistance


• every relatively compact


• manifold


• ?0 ?


• subset ? ?

Sujets

• Johns Hopkins University
• Institut universitaire de France
• Demailly
• Line bundle
• Compact space

Informations

• Publié par: profil-vieg-2012
• Nombre de lectures: 25
• Langue: English

Extrait

Algebraic approximations of holomorphic maps from Stein domains to projective manifolds Jean-Pierre Demailly 1 zlasL´ermpLe´ot 2 Universite´deGrenobleIPurdueUniversity Institut Fourier, BP 74 Department of Mathematics U.R.A. 188 du C.N.R.S. West Lafayette, IN 47907, U.S.A. 38402Saint-Martind’H`eres,France Bernard Shiffman 3 Johns Hopkins University Department of Mathematics Baltimore, MD 21218, U.S.A.

Key words:affine algebraic manifold, algebraic approximation, algebraic curve, completepluripolarset,Eisenmanmetric,Ho¨rmander’sL 2 -estimates for∂, holo- morphic map, holomorphic retraction, holomorphic vector bundle, hyperbolic space, Kobayashi pseudodistance, Kobayashi-Royden pseudometric, Nash alge- braic map, Nash algebraic retraction, plurisubharmonic function, projective alge- braic manifold, quasi-projective variety, Runge domain, Stein manifold. A.M.S. Classification 1985:32E30, 32H20, 14C30

Table of contents 1. Introduction                                                                       p. 2 2. Holomorphic and Nash algebraic retractions                                           p. 8 3. Nash algebraic approximation on Runge domains in affine algebraic varieties          p. 12 4. Nash algebraic approximations omitting ample divisors                               p. 18 5. Exhaustion of Stein manifolds by Runge domains of affine algebraic manifolds        p. 26 References                                                                          p. 30

1 Research partially supported by Institut Universitaire de France 2 partially supported by National Science Foundation Grant No. DMS-9303479Research 3 Research partially supported by National Science Foundation Grant No. DMS-9204037 1

1. Introduction The present work, which was motivated by the study of the Kobayashi pseudodistance on algebraic manifolds, proceeds from the general philosophy that analytic objects can be approximated by algebraic objects under suitable restric- tions. Such questions have been extensively studied in the case of holomorphic functions of several complex variables and can be traced back to the Oka-Weil ap- proximation theorem (see [We35] and [Oka37]). The main approximation result of this work (Theorem 1.1) is used to show that both the Kobayashi pseudodistance and the Kobayashi-Royden infinitesimal metric on a quasi-projective algebraic manifoldZare computable solely in terms of the closed algebraic curves inZ (Corollaries 1.3 and 1.4). Our general goal is to show that algebraic approximation is always possible in the cases of holomorphic maps to quasi-projective manifolds (Theorems 1.1 and 4.1) and of locally free sheaves (Theorem 1.8 and Proposition 3.2). Since we deal with algebraic approximation, a central notion is that of Runge domain: By definition, an open set Ω in a Stein spaceYis said to be aRunge domainif Ω is Stein and if the restriction mapO(Y)→ O(Ω) has dense range. It is well known that Ω is a Runge domain inYif and only if the holomorphic hull with respect to O(Y) of any compact subsetK⊂Ω is contained in Ω. IfYis an affine algebraic variety, a Stein open set Ω⊂Yis Runge if and only if the polynomial functions onYare dense inO(Ω). Our first result given below concerns approximations of holomorphic maps by (complex) Nash algebraic maps. IfY,Zare quasi-projective (irreducible, reduced) algebraic varieties, a mapf: Ω→Zdefined on an open subset Ω⊂Yis said to beNash algebraiciffis holomorphic and the graph Γ f :=  (y, f(y))∈Ω×Z:y∈Ω  is contained in an algebraic subvarietyGofY×Zof dimension equal to dimY. Iffis Nash algebraic, then the imagefis contained in an algebraic subvariety(Ω) AofZwith dimA= dimf(Ω)≤dimY. (TakeA= pr 2 (G)⊂Z, after eliminating any unnecessary components ofG.) Theorem 1.1. —LetΩbe a Runge domain in an affine algebraic varietyS, and letf: Ω→Xbe a holomorphic map into a quasi-projective algebraic manifoldX. Then for every relatively compact domainΩ 0 ⊂⊂Ω, there is a sequence of Nash algebraic mapsf ν : Ω 0 →Xsuch thatf ν →funiformly onΩ 0 . Moreover, if there is an algebraic subvarietyA(not necessarily reduced)ofSand an algebraic morphismα:A→Xsuch thatf |A∩Ω =α |A∩Ω , then thef ν can be chosen so thatf ν|A∩Ω 0 =f |A∩Ω 0 .(if we are given a positive integerIn particular, kand a finite set of points(t j )inΩ 0 , then thef ν can be taken to have the same k-jets asfat each of the pointst j ) An algebraic subvarietyAofSis given by a coherent sheafO A onSof the formO A =O S I A , whereI A is an ideal sheaf inO S generated by a (finite) set 2

of polynomial functions onS. (IfI A the ideal sheaf of the algebraic subsetequals SuppO A ⊂S, thenAis reduced and we can identifyA Thewith this subset.) restriction of a holomorphic mapf: Ω→Xto an algebraic subvarietyAis given by f |A∩Ω =f◦ι A∩Ω :A∩Ω→X , whereι A∩Ω :A∩Ω→ note that the WeΩ is the inclusion morphism.k-jet of a holomorphic mapf: Ω→Xat a pointa∈Ω can be described as the restrictionf |{a} k :{a} k →Xoffto the nonreduced point{a} k with structure ring O Ωa M k Ω + a 1 (whereM Ωa denotes the maximal ideal inO Ωa  parenthetical). The statement in Theorem 1.1 then follows by settingAequal to the union of the nonreduced points{t j } k . If in Theorem 1.1 we are given an exhausting sequence (Ω ν ) of relatively compact open sets in Ω, then we can construct by the Cantor diagonal process a sequence of Nash algebraic mapsf ν : Ω ν →Xconverging tofuniformly on every compact subset of Ω. We are unable to extend Theorem 1.1 to the case whereXis singular. course, for the case dim OfS= 1, Theorem 1.1 extends to singularX, since thenfcan be lifted to a desingularization ofX case. TheS= C of Theorem 1.1 was obtained earlier by L. van den Dries [Dr82] using different methods. In the case whereXis an affine algebraic manifold, Theorem 1.1 is easily proved as follows: One approximatesf: Ω→X⊂C m with an algebraic map g: Ω 0 →C m and then applies a Nash algebraic retraction ontoXto obtain the desired approximations. (The existence of Nash algebraic retractions is standard and is given in Lemma 2.1.) In the case wherefis a map into a projective manifold X⊂IP m−1 , one can reduce to the case whereflifts to a mapginto the cone Y⊂C m overXgeneral one cannot find a global Nash algebraic retraction, but in ontoY our proof proceeds in this case by pulling back to Ω the normal. Instead bundle toY  {0}(after making some reductions) and then using this pull-back bundle to extendgto a mapG: Ω×C p →C m  then approximate. WeGby algebraic mapsG ν , which we compose (on the right) with Nash algebraic maps from Ω 0 × {0}intoG ν −1 (Y) to obtain the approximationsf ν . In the case where the mapfin Theorem 1.1 is an embedding of a smooth domain Ω into a projective manifoldXwith the property thatf ⋆ Lis trivial for some ample line bundleLonX, we can approximatefby Nash algebraic embeddingsf ν with images contained in affine Zariski open sets of the formX  D ν , where theD ν are divisors of powers ofL. This result (Theorem 4.1) is obtained by first modifyingfon a Runge domain Ω 1 ⊂⊂Ω in such a way as to create essential singularities on the boundary off(Ω 1 ) so that the closuref(Ω 1 ) becomes complete pluripolar (in the sense of plurisubharmonic function theory). The existence of an ample divisor avoidingf(Ω 0 then obtained by means of an approximation) is theorem(Proposition4.8)basedonHo¨rmander’sL 2 estimates for∂. One of our main applications is the study of the Kobayashi pseudodistance and the Kobayashi-Royden infinitesimal pseudometric on algebraic manifolds 3