By D. E. Lerner, P. D. Sommers

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Phys. [10] R. Penrose, Gen. Rel. Grav. S. Ward, A class of self-dual solutions of Einstein's equations ~ (1978), 1-15. ~2~ (1977), 81-83. ~ (1967), 345-366. l (1976), 31-52. (preprint, Oxford, 1978; to appear in Proc. Roy. ). S. Ward Merton College Oxford England R Hartshorne Algebraic vector bundles on projective spaces, with applications to the Yang-Mills equation My purpose in these lectures is to give some idea of recent work in algebraic geometry concerning algebraic vector bundles on projective spaces, and how it is related to the solutions of the Yang-Mills equation.

Remarks (1) Various bits of the proof may be found spread over a number of references, for example [10,5,6]; the deformation theory of Kodaira is used extensively. An outline of the correspondence will be given below. (2) The above theorem is a local version: in this context "local" means virtually the same thing as "sufficiently small". Conversely, a global theorem would have to do without the "sufficient smallness". global positive definite theorem. 1, this extra structure is guaranteed by Kodaira's deformation theorems and so it doesn't have to be mentioned explicitly.

F. Atiyah, R, S, Ward, Instantons and algebraic geometry, Comm. Math. Phys. 55 (1977) 117-124. 4 N. H. Christ, E. J. Weinberg, N. K. Stanton, General self-dual YangMills solutions (preprint) 5 V. G. Drinfeld, Ju. I. Manin, Instantons and sheaves on ,F 3 (preprint) 6 R. Hartshorne, Algebraic Geometry, Graduate Texts in Math 52, Springer-Verlag, New York (1977r-xvi + 496 pp. 7 R. Hartshorne, Stable vector bundles and instantons, comm: Math. Phys. 59 (1978) 1-15. 8 R. Hartshorne, Stable vector bundles of rank 2 on P 3 , Math.