By Friedrich Hirzebruch

Lately new topological tools, particularly the idea of sheaves based via J. LERAY, were utilized effectively to algebraic geometry and to the idea of features of a number of complicated variables. H. CARTAN and J. -P. SERRE have proven how primary theorems on holomorphically entire manifolds (STEIN manifolds) should be for mulated by way of sheaf thought. those theorems indicate many proof of functionality conception as the domain names of holomorphy are holomorphically entire. they could even be utilized to algebraic geometry as the supplement of a hyperplane component of an algebraic manifold is holo morphically entire. J. -P. SERRE has got very important effects on algebraic manifolds through those and different equipment. lately a lot of his effects were proved for algebraic forms outlined over a box of arbitrary attribute. ok. KODAIRA and D. C. SPENCER have additionally utilized sheaf thought to algebraic geometry with nice luck. Their equipment vary from these of SERRE in that they use suggestions from differential geometry (harmonic integrals and so forth. ) yet don't make any use of the idea of STEIN manifolds. M. F. ATIYAH and W. V. D. HODGE have dealt effectively with difficulties on integrals of the second one style on algebraic manifolds with the aid of sheaf idea. i used to be capable of interact with ok. KODAIRA and D. C. SPENCER in the course of a remain on the Institute for complex research at Princeton from 1952 to 1954.

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