By S. K. Donaldson, C. B. Thomas

Those volumes are in line with lecture classes and seminars given on the LMS Durham Symposium at the geometry of low-dimensional manifolds. This quarter has been one among excessive examine lately, with significant breakthroughs that experience illuminated the way in which a few assorted topics (topology, differential and algebraic geometry and mathematical physics) engage.

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L4'loer, An instanton invariant for 3-manijolds, Commun. Math. Phys. 118 (1989),215-240. Uhlenbeck, "Instantons and Four-Manifolds," Springer, New York, 1984. 14)-iedman, Rank two vector bundles over regular elliptic surfaces, Invent. Math. 96 (1989), ~M:J-332. Morgan, Complez 1Jersus differentiable classification of algebraic surI(~ces, Columbia University Preprint. Harris, "Principles of Algebraic Geometry," John Wiley, New York, 1978. romov, Pseudo-holomorphic curves in Symplectic manifolds, Invent.

Gromov's argument reduces in the integrabl' case to classical geometry, effectively a step in the Enriques-Castuelnuevo theore [14]. For many more general results on these lines see the contribution of Mac D , to the Proceedings. e. if ' restricts to a symplectic form on E. So Gromov's Theorem asserts that a symplec' tic homotopy Cp2 in which a generator of the homology can be represented by symplectic 2-sphere is standard. With this in mind we consider what can be said about moduli spaces of holo morphic and pseudo-holomoxphic curves.

37 Donaldson: Yang-Mills invariants of four-manifolds Nc)w it is easy to see, in various ways, that when k is large the operator D = 8E+a~ lAking no,evenCek) to nO,odd(e k ) has a large kernel. y index D =< ch (el:) Td (V), [V] >, wlac'''(~ the Todd class is defined in the usual way by the almost complex structure .... V. This formula represents a polynomial of degree n in k and the leading term I" A~n Yolume(V), thus we have: dim ker D ~ indexD = kRYolume(V) + O(k n - 1 ). Ia(~rc~ s is a section of {k and u contains the terms in nO,2T(e k) for r ~ 1.