By Friedrich Hirzebruch

In recent times new topological equipment, specially the idea 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 basic theorems on holomorphically whole manifolds (STEIN manifolds) will be for mulated by way of sheaf conception. those theorems indicate many proof of functionality thought 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 received very important effects on algebraic manifolds by way of those and different equipment. lately a lot of his effects were proved for algebraic kinds 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 on. ) yet are not making 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 assistance from sheaf thought. i used to be in a position to interact with ok. KODAIRA and D. C. SPENCER in the course of a remain on the Institute for complex examine at Princeton from 1952 to 1954.

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