By Friedrich Hirzebruch
In recent times new topological tools, specially the idea of sheaves based by way of J. LERAY, were utilized effectively to algebraic geometry and to the idea of services of a number of complicated variables. H. CARTAN and J. -P. SERRE have proven how basic theorems on holomorphically entire manifolds (STEIN manifolds) could be for mulated when it comes to sheaf conception. those theorems indicate many evidence of functionality concept as the domain names of holomorphy are holomorphically whole. they could even be utilized to algebraic geometry as the supplement of a hyperplane element of an algebraic manifold is holo morphically whole. J. -P. SERRE has bought very important effects on algebraic manifolds by means of those and different equipment. lately lots of his effects were proved for algebraic types outlined over a box of arbitrary attribute. ok. KODAIRA and D. C. SPENCER have additionally utilized sheaf idea to algebraic geometry with nice good fortune. Their equipment vary from these of SERRE in that they use concepts from differential geometry (harmonic integrals and so forth. ) yet don't make any use of the speculation of STEIN manifolds. M. F. ATIYAH and W. V. D. HODGE have dealt effectively with difficulties on integrals of the second one type on algebraic manifolds with assistance from sheaf thought. i used to be capable of interact with okay. KODAIRA and D. C. SPENCER in the course of a remain on the Institute for complicated research at Princeton from 1952 to 1954.
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