In recent years new topological methods, especially the theory of
sheaves founded by J. LERAY, have been applied successfully to
algebraic geometry and to the theory of functions of several
complex variables. H. CARTAN and J. -P. SERRE have shown how
fundamental theorems on holomorphically complete manifolds (STEIN
manifolds) can be for mulated in terms of sheaf theory. These
theorems imply many facts of function theory because the domains of
holomorphy are holomorphically complete. They can also be applied
to algebraic geometry because the complement of a hyperplane
section of an algebraic manifold is holo morphically complete. J.
-P. SERRE has obtained important results on algebraic manifolds by
these and other methods. Recently many of his results have been
proved for algebraic varieties defined over a field of arbitrary
characteristic. K. KODAIRA and D. C. SPENCER have also applied
sheaf theory to algebraic geometry with great success. Their
methods differ from those of SERRE in that they use techniques from
differential geometry (harmonic integrals etc. ) but do not make
any use of the theory of STEIN manifolds. M. F. ATIYAH and W. V. D.
HODGE have dealt successfully with problems on integrals of the
second kind on algebraic manifolds with the help of sheaf theory. I
was able to work together with K. KODAIRA and D. C. SPENCER during
a stay at the Institute for Advanced Study at Princeton from 1952
to 1954."
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