Geometry of Low-Dimensional Manifolds, Vol. 1: Gauge Theory by S. K. Donaldson, C. B. Thomas

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.

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