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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 68 (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
Nahm transform for doublyperiodic instantons
"... This work concerns the study of certain finiteenergy solutions of the antiselfdual YangMills equations on Euclidean 4dimensional space which are periodic in two directions, socalled doublyperiodic instantons. We establish a circle of ideas involving equivalent analytical and algebraicgeometri ..."
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Cited by 11 (6 self)
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This work concerns the study of certain finiteenergy solutions of the antiselfdual YangMills equations on Euclidean 4dimensional space which are periodic in two directions, socalled doublyperiodic instantons. We establish a circle of ideas involving equivalent analytical and algebraicgeometric descriptions of these objects. In the first introductory chapter we provide an overview of the problem and state the main results to be proven in the thesis. In chapter 2, we study the asymptotic behaviour of the connections we are concerned with, and show that the coupled Dirac operator is Fredholm. After laying these foundations, we are ready to address the main topic of the thesis, the construction of a Nahm transform of doublyperiodic instantons. By combining differentialgeometric and holomorphic methods, we show in chapters 3 through 5 that doublyperiodic instantons correspond bijectively to certain singular Higgs pairs, i.e. meromorphic solutions of
Boalch – Wild nonabelian Hodge theory on curves
 Compositio Math
"... On a complex curve, we establish a correspondence between integrable connections with irregular singularities, and Higgs bundles such that the Higgs field is meromorphic with poles of any order. The moduli spaces of these objects are obtained by fixing at each singularity the polar part of the conne ..."
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Cited by 5 (0 self)
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On a complex curve, we establish a correspondence between integrable connections with irregular singularities, and Higgs bundles such that the Higgs field is meromorphic with poles of any order. The moduli spaces of these objects are obtained by fixing at each singularity the polar part of the connection, which amounts to fixing a coadjoint orbit of the group GLr(C[z]/z n). We prove that they carry hyperKähler metrics, which are complete when the residues of the connection are semisimple.
PARABOLIC BUNDLES ON ALGEBRAIC SURFACES I THE DONALDSON–UHLENBECK COMPACTIFICATION
, 2006
"... Abstract. The aim of this paper is to construct the parabolic version of the Donaldson–Uhlenbeck compactification for the moduli space of parabolic stable bundles on an algenraic surface with parabolic structures along a divisor with normal crossing singularities. We prove the non–emptiness of the m ..."
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Abstract. The aim of this paper is to construct the parabolic version of the Donaldson–Uhlenbeck compactification for the moduli space of parabolic stable bundles on an algenraic surface with parabolic structures along a divisor with normal crossing singularities. We prove the non–emptiness of the moduli space of parabolic stable bundles of rank 2 and also prove the existence of components with smooth points. 1.
and
, 2008
"... We study doublyperiodic instantons, i.e. instantons on the product of a 1dimensional complex torus T with a complex line C, with quadratic curvature decay. We determine the asymptotic behaviour of these instantons, constructing new asymptotic invariants. We show that the underlying holomorphic bun ..."
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We study doublyperiodic instantons, i.e. instantons on the product of a 1dimensional complex torus T with a complex line C, with quadratic curvature decay. We determine the asymptotic behaviour of these instantons, constructing new asymptotic invariants. We show that the underlying holomorphic bundle extends to T × P 1. The converse statement is also true, namely a holomorphic bundle on T × P 1 which is flat on the torus at infinity, and satisfies a stability condition, comes from a doublyperiodic instanton. Finally, we study the hyperkähler geometry of the moduli space of doublyperiodic instantons, and prove that the Nahm transform previously defined by the second author is a hyperkähler isometry with the moduli space of
Stable Parabolic Bundles over Elliptic Surfaces and over Orbifold Riemann Surfaces
, 2008
"... If q: Y → Σ is an elliptic surface (to be made precise) then the induced map of fundamental groups is an isomorphism if we consider Σ as an orbifold, [U], [Dol]. Hence, we obtain a correspondence of flat bundles (by bundles we always mean complex vector bundles). Donaldson showed that each ..."
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If q: Y → Σ is an elliptic surface (to be made precise) then the induced map of fundamental groups is an isomorphism if we consider Σ as an orbifold, [U], [Dol]. Hence, we obtain a correspondence of flat bundles (by bundles we always mean complex vector bundles). Donaldson showed that each