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- W2056568055 abstract "Let f:X-*S be a morphism of complex spaces and 3 a coherent analytic sheaf on X. If is proper, then the higher direct image sheaves Rf#$ are coherent for all z'2^0 by Grauert [8]. Generalizations of this theorem to certain classes of non-proper morphisms have also been obtained by several authors, among which we shall mention the following results, being of particular interest to us here (cf. also [22]). Let q^>Q be an integer. Then: a) If is (/-concave, then Rf*5 is coherent for /<;codhff -dim 5-(?-l [17]. b) If / is g-convex, then R'ftf is coherent for z<Jcodh S —dim S— q [4], where f denotes the direct image with proper supports, c) If is (1, 1)-convex-concave, then Rfj.S is coherent for l<^/5jcodh 3 —dim S~2 [15]. (Note that here and in what follows the terms '^-convex' and 'g-concave' are used in such a way that when S reduces to a point, they coincide with the notion of 'fortement qpseudoconvexe' and 'fortement #-pseudoconcave' respectively of AndreottiGrauert [1]. Hence they should be called ' (q — 1)-convex and (q — 1)concave respectively in the terminology of [4], [17], [18] etc.) Though these results are best possible as they stand, we could expect to improve the bounds for i under the additional assumption that 3 is /-flat, and in fact in such a way that the condition is stable under base change. The latter fact would indeed be useful in certain applications (cf. e.g. § 5" @default.
- W2056568055 created "2016-06-24" @default.
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- W2056568055 date "1982-01-01" @default.
- W2056568055 modified "2023-09-24" @default.
- W2056568055 title "A coherency theorem for direct images with proper supports in the case of a 1-convex map" @default.
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- W2056568055 doi "https://doi.org/10.2977/prims/1195183568" @default.
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