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A pseudoconcave generalization of Grauert's direct image theorem : I

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A pseudoconcave generalization of Grauert's direct image theorem : I ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA Classe di Scienze YUM-TONG SIU Apseudoconcave generalization ofGrauert’s direct image theorem : I Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 3e série, tome 24, no 2 (1970), p. 279-330. <http://www.numdam.org/item?id=ASNSP_1970_3_24_2_279_0> © Scuola Normale Superiore, Pisa, 1970, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ A PSEUDOOONOAVE GENERALIZATION OF GRAUERT’S DIRECT IMAGE THEOREM: I by YUM.ToNG SIU (*) Table of Contents ~ o, Introduction. A. In [2] Grauert proves the following direct image theorem. THEOREM G. is a proper holomorphic map of (not necessarily reduced) complex spaces and F is a coherent analytic sheaf on X. Then the lth direct image of 7 under n is a coherent analytic sheaf on Y for all I &#x3E; 0. (A simplified treatment of a key point of the proof for a special case is given in [3] to illustrate the idea of the proof. In [5] Knorr gives an amplified version of Grauert’s original proof.) Pervenuto alla Redazione il 22 Set. 1969. (*) Partially supported by NSF Grant GP-7265. 280 When the dimension of the complex space Y in Theorem G is zero, Theorem G is reduced to the following finiteness theorem of Cartan-Serre. THEOREM C-S. Suppose X is a compact complex space and F is a coherent analytic sheaf on X. Then the dimension is finite for all I &#x3E; 0. Theorem G can be

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