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The genus of embedded surfaces in the projective plane
 Math. Research Letters
, 1994
"... Abstract. We show how the new invariants of 4manifolds resulting from the SeibergWitten monopole equation lead quickly to a proof of the ‘Thom conjecture’. 1. Statement of the result The genus of a smooth algebraic curve of degree d in CP 2 is given by the formula g =(d−1)(d − 2)/2. A conjecture s ..."
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Abstract. We show how the new invariants of 4manifolds resulting from the SeibergWitten monopole equation lead quickly to a proof of the ‘Thom conjecture’. 1. Statement of the result The genus of a smooth algebraic curve of degree d in CP 2 is given by the formula g =(d−1)(d − 2)/2. A conjecture sometimes attributed to Thom states that the genus of the algebraic curve is a lower bound for the genus of any smooth 2manifold representing the same homology class. The conjecture has previously been proved for d ≤ 4andford = 6, and less sharp lower bounds for the genus are known for all degrees [5,6,7,8,10]. In this note we confirm the conjecture. Theorem 1. Let Σ be an oriented 2manifold smoothly embedded in CP 2 so as to represent the same homology class as an algebraic curve of degree d. Then the genus g of Σ satisfies g ≥ (d − 1)(d − 2)/2. Very recently, Seiberg and Witten [11,12,13] introduced new invariants of 4manifolds, closely related to Donaldson’s polynomial invariants [2], but in many respects much simpler to work with. The new techniques have led to more elementary proofs of many theorems in the area. Given the monopole equation and the vanishing theorem which holds when the scalar curvature is positive (something which was pointed out by Witten), the rest of the argument presented here is not hard to come by. A slightly different proof of the Theorem, based on the same techniques, has been found by Morgan, Szabo and Taubes. It is also possible to prove a version of Theorem 1 for other complex surfaces, without much additional work. This and various other applications will be treated in a later paper, with joint authors. 2. The monopole equation and the SeibergWitten invariants Let X be an oriented, closed Riemannian 4manifold. Let a spin c structure on X be given. We write c for the spin c structure and write W + = W + c and W − = W − c for the associated spin c bundles. Thus W + is a U(2) bundle and Clifford multiplication
The SeibergWitten equations and 4–manifold topology
 Bull. Amer. Math. Soc
, 1996
"... Since 1982 the use of gauge theory, in the shape of the YangMills instanton equations, has permeated research in 4manifold topology. At first this use of differential geometry and differential equations had an unexpected and unorthodox flavour, but over the years the ideas have become more familia ..."
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Cited by 79 (0 self)
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Since 1982 the use of gauge theory, in the shape of the YangMills instanton equations, has permeated research in 4manifold topology. At first this use of differential geometry and differential equations had an unexpected and unorthodox flavour, but over the years the ideas have become more familiar; a body of techniques has built up through the efforts of many mathematicians, producing results which have uncovered some of the mysteries of 4manifold theory, and leading to substantial internal conundrums within the field itself. In the last three months of 1994 a remarkable thing happened: this research area was turned on its head by the introduction of a new kind of differentialgeometric equation by Seiberg and Witten: in the space of a few weeks longstanding problems were solved, new and unexpected results were found, along with simpler new proofs of existing ones, and new vistas for research opened up. This article is a report on some of these developments, which are due to various mathematicians, notably Kronheimer, Mrowka, Morgan, Stern and Taubes, building on the seminal work of Seiberg [S] and Seiberg and Witten [SW]. It is written as an attempt to take stock of the progress stemming
Gauge theory for embedded surfaces
 I, Topology
, 1993
"... (i) Topology of embedded surfaces. Let X be a smooth, simplyconnected 4manifold, and ξ a 2dimensional homology class in X. One of the features of topology in dimension 4 is the fact that, although one may always represent ξ as the fundamental class of some smoothly ..."
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Cited by 67 (6 self)
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(i) Topology of embedded surfaces. Let X be a smooth, simplyconnected 4manifold, and ξ a 2dimensional homology class in X. One of the features of topology in dimension 4 is the fact that, although one may always represent ξ as the fundamental class of some smoothly
Symplectic Fillings and Positive Scalar Curvature
 GEOM. AND TOPOLOGY
, 1998
"... Let X be a 4manifold with contact boundary. We prove that the monopole invariants of X introduced by Kronheimer and Mrowka vanish under the following assumptions: (i) a connected component of the boundary of X carries a metric with positive scalar curvature and (ii) either b^+_ 2 (X) > 0 or the bou ..."
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Cited by 26 (9 self)
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Let X be a 4manifold with contact boundary. We prove that the monopole invariants of X introduced by Kronheimer and Mrowka vanish under the following assumptions: (i) a connected component of the boundary of X carries a metric with positive scalar curvature and (ii) either b^+_ 2 (X) > 0 or the boundary of X is disconnected. As an application we show that the Poincaré homology 3sphere, oriented as the boundary of the positive E_8 plumbing, does not carry symplectically semillable contact structures. This proves, in particular, a conjecture of Gompf, and provides the first example of a 3manifold which is not symplectically semifillable. Using work of Fr yshov, we also prove a result constraining the topology of symplectic fillings of rational homology 3spheres having positive scalar curvature metrics.
Uhlenbeck spaces via affine Lie algebras, math.AG/0301176; to appear in ”Unity of Mathematics” (proceedings of a conference dedicated to I. M. Gelfand’s 90th birthday
, 2003
"... Abstract. Let G be an almost simple simply connected group over C, and let Buna G (P2, P1) be the moduli scheme of principal Gbundles on the projective plave P2, of second Chern class a, trivialized along a line P1 ⊂ P2. We define the Uhlenbeck compactification Ua G of Buna G (P2, P1), which classi ..."
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Abstract. Let G be an almost simple simply connected group over C, and let Buna G (P2, P1) be the moduli scheme of principal Gbundles on the projective plave P2, of second Chern class a, trivialized along a line P1 ⊂ P2. We define the Uhlenbeck compactification Ua G of Buna G (P2, P1), which classifies, roughly, pairs (FG, D), where D is a 0cycle on A2 = P2 − P1 of degree b, and FG is a point of Bun a−b G (P2, P1), for varying b. In addition, we calculate the stalks of the Intersection Cohomology sheaf of Ua G. To do that we give a geometric realization of Kashiwara’s crystals for affine KacMoody algebras.
Generic metrics, irreducible rankone PU(2) monopoles, and transversality
 Comm. Anal. Geom
"... Our main purpose in this article is to prove that the moduli space of solutions to the PU(2) monopole equations is a smooth manifold of the expected dimension for simple, generic parameters such as (and including) the Riemannian metric on the given fourmanifold: see Theorem 1.3. In [16] we proved t ..."
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Cited by 18 (7 self)
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Our main purpose in this article is to prove that the moduli space of solutions to the PU(2) monopole equations is a smooth manifold of the expected dimension for simple, generic parameters such as (and including) the Riemannian metric on the given fourmanifold: see Theorem 1.3. In [16] we proved transversality using an
Donaldson invariants and wallcrossing formulas. I: Continuity of gluing and obstruction maps
, 1999
"... The present article is the first in a series whose ultimate goal is to prove the KotschickMorgan conjecture concerning the wallcrossing formula for the Donaldson invariants of a fourmanifold with b + = 1. The conjecture asserts that, in essence, the wallcrossing terms due to changes in the metri ..."
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Cited by 11 (9 self)
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The present article is the first in a series whose ultimate goal is to prove the KotschickMorgan conjecture concerning the wallcrossing formula for the Donaldson invariants of a fourmanifold with b + = 1. The conjecture asserts that, in essence, the wallcrossing terms due to changes in the metric depend at most on the homotopy type of the fourmanifold and the degree of the invariant. We prove the first half of a general gluing theorem endowed with several important enhancements not present in previous treatments due to Taubes, Donaldson, and Mrowka. In particular, we show that the gluing maps, with obstructions to deformation permitted, extend to maps on the closure of the gluing data which are continuous with respect to the Uhlenbeck topology. The second half of the proof of our general gluing theorem is proved in the sequel [13].
PU(2) monopoles. II: Toplevel SeibergWitten moduli spaces and Witten's conjecture in low degrees
 J. REINE ANGEW. MATH.
, 2001
"... In this article, a continuation of [10], we complete the proof  for a broad class of fourmanifolds  of Witten's conjecture that the Donaldson and SeibergWitten series coincide, at least through terms of degree less than or equal to c 2, where c ˆ 1 …7w ‡ 11s † and w and s are the Euler charac ..."
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In this article, a continuation of [10], we complete the proof  for a broad class of fourmanifolds  of Witten's conjecture that the Donaldson and SeibergWitten series coincide, at least through terms of degree less than or equal to c 2, where c ˆ 1 …7w ‡ 11s † and w and s are the Euler characteristic and signature of the four4 manifold. We use our computations of Chern classes for the virtual normal bundles for the SeibergWitten strata from the companion article [10], a comparison of all the orientations, and the PU…2 † monopole cobordism to compute pairings with the links of levelzero SeibergWitten moduli subspaces of the moduli space of PU…2 † monopoles. These calculations then allow us to compute lowdegree Donaldson invariants in terms of SeibergWitten invariants and provide a partial veri®cation of Witten's conjecture.
A SIMPLE GEOMETRIC REPRESENTATIVE FOR µ OF A POINT
, 1995
"... Abstract. For SU(2) (or SO(3)) Donaldson theory on a 4manifold X, we construct a simple geometric representative for µ of a point. Let p be a generic point in X. Then the set {[A]F − A (p) is reducible}, with coefficient1/4 and appropriate orientation, is our desired geometric representative. 1. ..."
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Abstract. For SU(2) (or SO(3)) Donaldson theory on a 4manifold X, we construct a simple geometric representative for µ of a point. Let p be a generic point in X. Then the set {[A]F − A (p) is reducible}, with coefficient1/4 and appropriate orientation, is our desired geometric representative. 1.
CRITICALEXPONENT SOBOLEV NORMS AND THE SLICE THEOREM FOR THE QUOTIENT SPACE OF CONNECTIONS
, 2001
"... Following Taubes, we describe a collection of criticalexponent Sobolev norms, discuss their embedding and multiplication properties, and describe optimal Green’s operator estimates where the constants depend at most on the first positive eigenvalue of the covariant Laplacian of a G connection and t ..."
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Following Taubes, we describe a collection of criticalexponent Sobolev norms, discuss their embedding and multiplication properties, and describe optimal Green’s operator estimates where the constants depend at most on the first positive eigenvalue of the covariant Laplacian of a G connection and the L2 norm of the connection’scurvature, for arbitrary compact Lie groups G. Using these criticalexponent norms, we prove a sharp, global analogue of Uhlenbeck’s Coulomb gaugefixing theorem, where the usual product connection over a ball isreplaced by an arbitrary reference connection over the entire manifold. We also prove a quantitative version of the conventional slice theorem for the quotient space of G connections, with an invariant and sharp characterization of those pointsin the quotient space which are contained in the image of an L4 ball in the Coulombgauge slice. Our gaugefixing