By D. E. Lerner, P. D. Sommers
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Topology (to appear). An algebra-geometrical construction of commuting operators and of solutions to the Toda lattice equation, KortewegD~ Vries equation and related nonlinear equations, Kyoto Conference (to appear) ROBIN HARSHORNE Department of Mathematics University of California Berkeley, CA 94720 N H Christ Self-dual Yang-Mills solutions Let us consider the application of the Horrocks-Barth construction to the problem of finding self-dual Euclidean Yang-Mills solutions, recently developed by Atiyah, Hitch~n, Drinfeld and Manin .
Then M(k) is a real algebraic variety of dimension Bk-3. Let M(k,q) be the subspace of M(k) consisting of those instantons which can be derived from the functional Aq. For each fixed value of k, there exists a value qk of q such that  But qk + ~ as k instantons. in other words, no finite value of q gives aU The following table illustrates these comments. 1 2 3 4 5 >5 dim M(k,l) 5 13 19 24 29 5k + 4 dim M(k,2) 5 13 21 29 36 4k + 16 dim M(k) 5 13 21 29 37 8k - 3 k 28 + ~; Much work remains to be done on investigating the spaces M(k); most recent investigations have concentrated on the Horrocks/Barth approach mentioned above.
So Mumford The stable bundles with given c 1 , c 2 are parameterized by a finite union of quasi-projective varieties, which we call their moduli space. Also it happens that the bundles arising from instantons are all stable, so that we have good moduli spaces for them. Fixing c 1 = 0, c 2 > 0 for simplicity (which includes in particular all the bundles coming from instantons), we can then speak of the moduli space M(c 2 ) of stable algebraic rank 2 vector bundles on P~ with those Chern classes.
Complex Manifold Techniques in Theoretical Physics by D. E. Lerner, P. D. Sommers