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We determine the second-order approximation of the anti-selfduality equation around a reducible Yang-Mills connection on a compact 1connected Kahler surface. The goal of this note is to explain how one can derive an approximation of the anti-self-duality (ASD) equation around a U( 1)-reduction on a compact simply connected Kahler surface. The idea is to study two different moment map models associated to such a reduction. This observation is due to Donaldson as he explained to the author his understanding of a result in [M]. To begin with, we recall first the following general facts about a reducible ASD connection A on a smooth compact simply connected oriented 4-manifold X. We work with a fixed metric mo on X throughout this discussion. Suppose A is a reducible ASD connection on an SU(2)-bundle P > X preserving a splitting L ?D L-1 for some line bundle L -+ X, where c2(P) = -L.L = k > 0. It is a well-known fact that a neighbourhood of [A] E Mk(mO) ,the moduli space of equivalence classes of mo-anti-self-dual connections on P, can be modelled as an SI-quotient q-1(0)/S1 for some finite-dimensional equivariant map (1.1) ?> HA A defined on a small neighbourhood of 0 E HA . (See for instance [L].) Here we write HA, i = 0, 1, 2, for the cohomology groups of the Atiyah-Hitchin-Singer deformation complex 0 -* QO(adP) ?4 nl(adP) A ? +(adP) -> 0 associated to the ASD connection A. More precisely in (1.1), we find a smooth map V E HA e kerd; C Q W(adP) modelled on suitable Hilbert spaces so that for Iv ? 1 the map v) solves q(v) = F+(A +v) e H2 Received by the editors July 25, 1990 and, in revised form, May 9, 1991. 1991 Mathematics Subject Classification. Primary 53C07. ? 1993 American Mathematical Society 0002-9939/93 $1.00+ $.25 perpage

Reducible Yang-Mills connections on Kähler surfaces and moment maps