Demaiily, Jean-Pierre o 32003 Constnictibilite des faisceaux de solutions des systemes differentiels holononws, (D'apres Masaki Kashiwara). Semin. d* Analyse P. Ulong - P. Dolbeault - H. Skoda, Annces 1981/83, Lect, Notes Math. 1028,83 - 95(1983). [This article was published in this book announced in this Zbl. 511.00025.] This paper is a written account of the second part of the author's doctoral thesis; it contains an elementary introduction to a few basic ideas in the theory of partial differential equations developed by Sato- Kashiwara - Kawai. The sheaves fi*, S of differential and microdiffcrential operators on a complex manifold X arc introduced s and coherent ^-modules M are shown to correspond to the usual notion of 132 529.32004 differential systems* The sheaf of solutions of Jl in Ox can then be interpreted as Jfomfl {M > (5), and the higher £xt£(^>0) sheaves express "obstructions" to solvability. To each coherent @* module Jt is attached its characteristic variety SS( Jt) c TX defined as the common zero set of symbols of operators P defining Jt. SS(v#) is always involutive with respect to the natural symplcctic structure of TXy and Jt is said to be holonomic when SS(Jt) is lagrangian. According to Kashiwara, one proves an extension theorem for solutions across non characteristic boundaries. If Jt is holonomic> it follows that the sheaves £xl\}(Jt %G) are constructible,i.e. locally constant of finite rank along the strata of a suitable stratification of X. Autorreferat. Demailly, J. P. 32006 Sur la structure des courants positifs fermes- Inst, Elie Cartan, Univ. Nancy I 8, 52 - 62 (1983). This article is a brief account of three detailed papers of the author: Ann, Inst- Fourier 32, No, 2, 37 — 66 (1982; Zbi.457,32005), Bull. Soc. Math. Fr, 110,75 -102 (1982; Zbl, 493.32003), and Invent, Math. 69, 347- 374 (1982; Zbl, 476.58001), The first two deal with Lclong-Jcnsen formulas and define generalized Lelong numbers of currents with respect to a plurisubharmonic weight: v(T, on X such that ic(M) + i53tp + iRicci(uj} £kc(detQ), where k>inf(n,q) + lnf(n,s), n<=dimX, q«=rarikQ, s = rankS, E being semi-positive in the sense of Griffiths* Using H. Skoda's results, we find for every f€r(X,Hora(Q,Q&M)) a section h of Hom{Q,E»M) such thatg*h=f and [Ih| e^dV^cflfj e^dV, provided that the second integral is finite. Consider­ ing the exact sequence O-TX-TC* |x -NX-0 where X is an n-subroanifold of ©* and NX is normal bundle, wo use the above splitting to construct a tubular neigh­ borhood U of X and a holoaorphic retraction o:U-X* vfe also prove an extension theorem with estimates, inspired from B* Jennane's work, and which generalizes the HKraandcr-Borobieri-Skoda theorem in an optimal form. Combining the above results, we find explicit estimates for U, p and for the extension of functions froo X to &, Involving only geo&etric invariants of X, Autorreferat, DcaaiIty* 3ean~Plerr«: Un exempte de f\br4 hotomorphe non d* Stein £ fibre c2 ayant poup base le disque ou te Plan* Inventiones math, (4 parattre) CNorlu, F-BQ240 Roiset). We give an example of a non Stein analytic fibre bundle over the unit disc or the complex plane, with fibre G* and transition automorphisms of expo­nential type* The main argument we use is an inequality due to P. Lelong, according to which plurisubharaonie functions grow "uniformly*1 on the fibres. Autorreferat. Demat Uy, 3.-P.: j ^ o 32010 Relations entre ies dltferentes notions de fibres et de ecurants positlfs. Semln* P, Lelons - H. Skoda/ Analyser ftnnees 1980/81/ et; Les fonctions plurisousharaonfqyes en dimension finie ou inftnle/ Colloq. Ui*ereu* 1981, Lect. Motes Hath. 919/ 56-76 (1982). fThis article was published in the book announced In this Zbl* 4?l,0CO12.j To every herraitian holorcori&ic vector bundle E over an analytic c&nifold X is attached a curvature tensor ctE)*Ef, which is a (l,l)-forai with values in Henn(E,E), One says that E is*cO (Griffiths seici-posUivc) if for every z € X, SeTiXareieeSj one has ic(E)(S®e,£Se)d='(ie(E) (£,15) .cle) 2 0. We say that E is *H0 (Nakano), respec­tively *sO (strongly iO) If for every s€X and C€ (TX©E)£ we have ic{E)(C,0 *0> respectively ic(E)(C,C) = 25?»ef ®f?®e£(C,C) for some elects Sf€T?X, ef€E?. Then it Is clear that E*s0=*E ^O^E^O, and conversely we prove: Theorem* (1) E^G0=E®detEss0; (2) E*c0=>E® (detE)*a *0, where q« inf(rankE,dirtX). -Similarly, we study relationships between the notions of weakly and strongly posi­tive differential forms, introduced by P. Lelong. Let iu be a herroitian metric and or a weakly 2 0 (p,p)-form on t* . We coajpute explicit constants C=*C(n,p), C1 -C'tn^p) such that -C (Tr«) —£s<*£sC(Trff)™ . Autorref erat, Deaailly/ D.-P-: ^ o 32011 Sdndaae holomorphe d'un atorphisme de fibres vectorlels semi- posltlfs avec estimations L*. Seatn. P. Lelonfl - H. Skoda/ Analyse/ Annees 1980/81/ et; Les fonctions plurlsousharaon1*ues en dimension finie oy Inf1n1e/ Colloq- Uinereux 1981/ Lect. Notes Math. 919/ 77-107 C1982)« [This article was published in the book announced in this Zbl. 471.00012.] Let 0-*S-*E-Q-0 an exact sequence of holomorphic heraitian vector bundles over a weakly pseudoconvex kShlerian manifold (X,w)* Assume given a hermitian line bundle M and a psh function on X such that ic(M) + i53tp + iRicci(uj} £kc(detQ), where k>inf(n,q) + lnf(n,s), n<=dimX, q«=rarikQ, s = rankS, E being semi-positive in the sense of Griffiths* Using H. Skoda's results, we find for every f€r(X,Hora(Q,Q&M)) a section h of Hom{Q,E»M) such thatg*h=f and [Ih| e^dV^cflfj e^dV, provided that the second integral is finite. Consider­ ing the exact sequence O-TX-TC* |x -NX-0 where X is an n-subroanifold of ©* and NX is normal bundle, wo use the above splitting to construct a tubular neigh­ borhood U of X and a holoaorphic retraction o:U-X* vfe also prove an extension theorem with estimates, inspired from B* Jennane's work, and which generalizes the HKraandcr-Borobieri-Skoda theorem in an optimal form. Combining the above results, we find explicit estimates for U, p and for the extension of functions froo X to &, Involving only geo&etric invariants of X, Autorreferat, DemqUlx/ Dean-Pierre: _ 32021 Estimations L pour I'operateur & d'un flbre^ vectorial holomorphe sewt-posi tif ao-dessus dJune variete Kahlerienne complete. Ann- Sci. Ec. Norm. Super./ IV. Ser. 15^ 457-511 (1982), Let E be a hermitian vector bundle of rank r over a n-dimensional KShler m&ni- fold X, The bundle E is said to be s-positive if its curvature tensor ictE) identified with a hermitian form onTX®E takes >0 values on tensors of rankss and ^0. For example* if E is Griffiths >0 (i.e. L-positive) of rank ra8, one shows that E*®(detE)* is 5-positive and that E®detE is Nakano >0 (i.e. n-posi- tive). In connection with these results, one proves the following vanishing theorem; if E is 5-positive and X is weakly pseudocode*, then H* (X,AnT*X®E) *0 for q ssup(l,n- s + 1). Given a surjective rcorphisn: E-+Q-0 of herraltian bundles, one also obtains curvature conditions which imply the surjectivity of the taap K* (X>E {