3. The proof of Theorem

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1 3. The proof of Theorem Objekttyp: Chapter Zeitschrift: L'Enseignement Mathématique Band (Jahr): 31 (195) Heft 1-2: L'ENSEIGNEMENT MATHÉMATIQUE PDF erstellt am: Nutzungsbedingungen Die ETH-Bibliothek ist Anbieterin der digitalisierten Zeitschriften. Sie besitzt keine Urheberrechte an den Inhalten der Zeitschriften. Die Rechte liegen in der Regel bei den Herausgebern. Die auf der Plattform e-periodica veröffentlichten Dokumente stehen für nicht-kommerzielle Zwecke in Lehre und Forschung sowie für die private Nutzung frei zur Verfügung. Einzelne Dateien oder Ausdrucke aus diesem Angebot können zusammen mit diesen Nutzungsbedingungen und den korrekten Herkunftsbezeichnungen weitergegeben werden. Das Veröffentlichen von Bildern in Print- und Online-Publikationen ist nur mit vorheriger Genehmigung der Rechteinhaber erlaubt. Die systematische Speicherung von Teilen des elektronischen Angebots auf anderen Servern bedarf ebenfalls des schriftlichen Einverständnisses der Rechteinhaber. Haftungsausschluss Alle Angaben erfolgen ohne Gewähr für Vollständigkeit oder Richtigkeit. Es wird keine Haftung übernommen für Schäden durch die Verwendung von Informationen aus diesem Online-Angebot oder durch das Fehlen von Informationen. Dies gilt auch für Inhalte Dritter, die über dieses Angebot zugänglich sind. Ein Dienst der ETH-Bibliothek ETH Zürich, Rämistrasse 11, 92 Zürich, Schweiz,

2 to be strongly plurisubharmonic if for every C real-valued fonction 6 with compact support there exists an e > such that (p + is plurisubharmonic for. A main resuit in [3] tells us that the above définition agrées with the usual one as given in [6]. Let us also recall that a complex space X is said to be 1-convex if there exist : i) a compact analytic set a X with dim x S > S for any x e S, Stein space Y, a finite set A <= Y and a proper holomorphic map ii) : X -? 7 inducing a bilholomorphism p X\5 = 7\^4 and which satisfies P*Ox = <!)y. a called the exceptional set of S is X and Y the Remmert réduction of X. Using the analytic version of Chow's lemma (Hironaka [5]) [2] that any 1-convex space X carries a strongly pluri subharmonicexhaustion function cp:x-> [?,), i.e. the converse of Theorem 1 holds too. Remark. it was proved in 3. The proof of Theorem We shall apply Andreotti-Grauert's technique [1] with suitable modifica tionsrequired by the upper semicontinuity. Throughout this section!f will dénote a cohérent sheaf on X and X c = {x e X cp(x) < c}. To prove Theorem 1 we need some lemmas. Lemma H\X map For any 1. C +Z c e R there exists,&) -> H\X C+E >,^) > such that is restriction is surjective for any. Proof. We may assume c =. Set K = {(p < 1} and let {L^,..., U m } be a covering of K with Stein open sets, Ui c a X and /i i GC^((7 i ), fo f r such that 9? X» = i s strongly plurisubharmonic for r = 5] /z; i=l 1,..., m and wj m < > on K. Choose a > < min(a, 1). such that i= We shall prove that this a for any /î;(*) xe K and take 1 in Lemma e we set X\. = {xex\ For any for r =,..., m (by définition X% = X e >). satisfies the conditions required 1. <p(x) < + h^x) K(x)}

3 We make the following remark: for any we hâve X z c X?. Indeed, let xex such that cp(x) <. In particular cp(x) < 1, hence xek. m From the définition of a it follows that /i (x) > a and from the inequalities ; = i cp(x) < < a < /i f (x) < s 7 + /i f(x) we get x e X%. i = 1 i=l Due to this remark Lemma 1 will be proved if we prove that the restriction map H l (X?., -? H 1 (X e^) is surjective for any s. The inclusions X z. = Xj c Ij c... cz X? show that it suffices to prove that the restrictions H 1 (X r/1 If we set, &) -> H\X r E >, &) are surjective for r =,..., m - 1. then V r/1 and X^nV^ 1 are Stein open X'/^XÏ c supp(/i r+1 ) c= U r + 1 and so X^+1 = X\. u K^+1. From the Mayer-Vietoris exact séquence : sets. On the other hand it follows that the restriction map H 1 (X r/1 and so Lemma 1 is proved.,^r) -> H^X^,^) is surjective Lemma 2. For any oc p the restriction map H\X^&) -? H\X a, &) is surjective. Proof. Set M(ot) = {S a for any a Ô y the restriction map From Lemma 1 and Lemma [1, p. 241] we deduce that M(oc) = [a, oo) which proves Lemma 2. Lemma 3. For any oe e R H 1(X a, &) has finite dimension. Proof. Choose p > oc such that X a o X p. From Lemma 2 the restriction map HH^p, &) -+ H\X a, &) is surjective and from [1, p. 24] Lemma 4. For any cer there exists > such that the restriction map r(x c + E,^) -s. T(X c+e,,^) has dense image for any s'.

4 Proof. We may assume c = and choose > as in Lemma 1. Exactly as in the proof of Lemma 1 it suffices to prove that the restriction,^) -? T{X r t.,&) has dense image for r =,..., m - 1. map TiX 1^ 1 Consider the Mayer-Vietoris exact séquence : Since (X r t. n V/ 1, K^+1 ) is a Runge pair it follows that a has dense image. On the other hand, applying Lemma 3 to the function we deduce that H^X^ 1, 3F) has finite dimension, in particular it is separated, hence a has closed image. Consequently a is surjective. From the open mapping theorem it follows easily that the restriction map has dense image and so Lemma 4 is proved. Lemma 5. For any oc p the restriction map F(Z p, J^) -? T(X ai $F) has dense image. Proof. Lemma 5 is an immédiate conséquence of Lemma 4 and of Lemma [1, p. 246]. Lemma 6. For any cer there exists s > such that the restriction map H\XC+Z,&) -? H\XC + Z.,&) is bijective for any s'. Proof. We may assume c = and choose > as in Lemma 1. Due to the inclusions X. z a X e c X? and using Lemma 2 it follows that it suffices to show that the restriction map H\X? 9 &) -? H\X Z., &) is bijective. The inclusions X z. = X ci X\. ci... ci X^îshow thatitis enoughto prove that the restrictions H 1 (X r /1, H 1 (X r r, are bijective for r =,..., m? 1. Consider the Mayer-Vietoris exact séquence : As remarked in the proof of Lemma 4 the map

5 is surjective. Since it follows that the restriction map is bijective and so Lemma 6 is proved. Lemma 7. For any oc < p the restriction map H\X^&) -? H 1 {X a, &) is bijective. Proof. Set M(a) = { > for oc any a y the restriction map and let a = sup M(oc). From Lemma 2 it follows that if e M(a) then [a, ] cz M(a), consequently [a, a ) cz M(a). To prove Lemma 7 we hâve to show that oc = oo. Suppose that a < oo. From Lemma 5 and Lemma [1, p. 25] we deduce that a g M(a). From Lemma 6 there exists s > such that oc + e e M(a). This contradicts the définition of oc, and so Lemma 7 is proved. We are now in a position to prove Theorem 1. Choose oc e R and take oc = oc < (*! <... < oi n <... an increasing séquence of real numbers tending to oo. By Lemma 7 the restriction map H 1 (X aln + 1, J* 7 ) -? H\X nn, &) is bijective and by Lemma 5 the restriction map T{X an + 1, $F) -»? r(x?n, J^) has dense image. It follows then from Lemma [1, p. 25] that the restriction map H\X, &) -? H\X^&) is also bijective and from Lemma 3 H\X, &) has finite dimension. Theorem V. in [6] tells us that X is 1-convex, as required.

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