By Friedrich Hirzebruch
In recent times new topological equipment, specially the speculation of sheaves based via J. LERAY, were utilized effectively to algebraic geometry and to the speculation of features of a number of complicated variables. H. CARTAN and J. -P. SERRE have proven how primary theorems on holomorphically whole manifolds (STEIN manifolds) will be for mulated when it comes to sheaf concept. those theorems suggest many evidence of functionality concept as the domain names of holomorphy are holomorphically entire. they could even be utilized to algebraic geometry as the supplement of a hyperplane portion of an algebraic manifold is holo morphically whole. J. -P. SERRE has got very important effects on algebraic manifolds through those and different tools. lately a lot of his effects were proved for algebraic kinds outlined over a box of arbitrary attribute. okay. KODAIRA and D. C. SPENCER have additionally utilized sheaf concept 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 on. ) yet are not making 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 style on algebraic manifolds with assistance from sheaf idea. 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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