By Robert Friedman, John W. Morgan

This booklet applies the hot thoughts of gauge conception to review the sleek category of compact complicated surfaces. The research is split into 4 major components: Classical complicated floor conception, gauge conception and Donaldson invariants, deformations of holomorphic vector bundles, and particular calculations for elliptic sur§ faces. The e-book represents a wedding of the thoughts of algebraic geometry and 4-manifold topology and offers an in depth exposition of a few of the most topics during this very energetic quarter of present study.

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47) The main point now is that JRn c/J dJL is a good approximation for JL(K) and JRn c/JdJLj is a good approximation for JLj(Kj) when r is small and j is large. Indeed, IJL(K)-lnc/JdJL ! 18 in [R]). Let us check that IJLj (I 0 and all sufficiently large j. How large j has to be depends on r, but the constants C, a do not depend on either j or r. 50) for all sufficiently large j, because K j ~ K(r) when j is large enough (since Kj converges to K).

One can make the same kind of observations about the dependence of the limits of mapping packages on the embeddings, etc. 15. In other words, there is practically no dependence, if we are willing always to pass to subsequence, and if we work up to isometric equivalence. 6 Convergence of measures As usual we begin with convergence on Euclidean spaces before proceeding to the general case. 23 (Convergence of measures on Euclidean spaces) Let {rj} be a sequence of nonnegative Borel measures on some R n which are finite on compact sets, and let r be another measure on Rn with the same properties.

We may as well assume that all of our metric spaces Mj lie in R n already, since the effect of our definitions is to reduce everything to that case. l already live on R n , and that the Kj's and K are compact subsets of R n . 25. 18 in [R). L(K) + f. 32) Let ¢ be a continuous function with compact support on R n such that 0 ::; ¢ ::; 1 everywhere, ¢ == 1 on a neighborhood of K, ¢ == 0 on the complement of U. 28) be a sequence of mapping packages which converges to the mapping package {(M, d(x, y),p), (N,p(u,v),q),h}.

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