H. CARTAN and J.-P. SERRE have shown how fundamental theoremson holomorphically complete manifolds (STEIN manifolds) can be for-mulated in terms of sheaf theory. These theorems imply many facts offunction theory because the domains of holomorphy are holomorphicallycomplete. They can also be applied to algebraic geometry because thecomplement of a hyperplane section of an algebraic manifold is holo-morphically complete. J.-P. SERRE has obtained important results onalgebraic manifolds by these and other methods. Recently many of hisresults have been proved for algebraic varieties defined over a field ofarbitrary characteristic. K. KODAIRA and D. C. SPENCER have alsoapplied sheaf theory to algebraic geometry with great success. Theirmethods differ from those of SERRE in that they use techniques fromdifferential geometry (harmonic integrals etc.) but do not make any useof the theory of STEIN manifolds. M. F. ATIVAH and W. V. D. HODGE have dealt successfully with problems on integrals of the second kind onalgebraic manifolds with the help of sheaf theory.
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