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18
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
The Hodge filtration in nonabelian cohomology. alggeom preprint
, 1996
"... Whereas usual Hodge theory concerns mainly the usual or abelian cohomology of an algebraic variety—or eventually the rational homotopy theory or nilpotent completion of π1 which are in some sense obtained by extensions—nonabelian Hodge theory concerns the cohomology of a variety with nonabelian coef ..."
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Cited by 53 (5 self)
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Whereas usual Hodge theory concerns mainly the usual or abelian cohomology of an algebraic variety—or eventually the rational homotopy theory or nilpotent completion of π1 which are in some sense obtained by extensions—nonabelian Hodge theory concerns the cohomology of a variety with nonabelian coefficients. Because of the basic fact
Moduli spaces of parabolic Higgs bundles and parabolic K(D) pairs over smooth curves: I
, 1998
"... This paper concerns the moduli spaces of rank two parabolic Higgs bundles and parabolic K(D) pairs over a smooth curve. Precisely which parabolic bundles occur in stable K(D) pairs and stable Higgs bundles is determined. Using Morse theory, the moduli space of parabolic Higgs bundles is shown to b ..."
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Cited by 19 (2 self)
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This paper concerns the moduli spaces of rank two parabolic Higgs bundles and parabolic K(D) pairs over a smooth curve. Precisely which parabolic bundles occur in stable K(D) pairs and stable Higgs bundles is determined. Using Morse theory, the moduli space of parabolic Higgs bundles is shown to be a noncompact, connected, simply connected manifold, and a computation of its Poincaré polynomial is given.
BETTI NUMBERS OF THE MODULI SPACE OF RANK 3 PARABOLIC HIGGS BUNDLES
, 2004
"... Abstract. Parabolic Higgs bundles on a Riemann surface are of interest for many reasons, one of them being their importance in the study of representations of the fundamental group of the punctured surface in the complex general linear group. In this paper we calculate the Betti numbers of the modul ..."
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Cited by 12 (5 self)
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Abstract. Parabolic Higgs bundles on a Riemann surface are of interest for many reasons, one of them being their importance in the study of representations of the fundamental group of the punctured surface in the complex general linear group. In this paper we calculate the Betti numbers of the moduli space of rank 3 parabolic Higgs bundles, using Morse theory. A key point is that certain critical submanifolds of the Morse function can be identified with moduli spaces of parabolic triples. These moduli spaces come in families depending on a real parameter and we carry out a careful analysis of them by studying their variation with this parameter. Thus we obtain in particular information about the topology of the moduli spaces of parabolic triples for the value of the parameter relevant to the study of parabolic Higgs bundles. The remaining critical submanifolds are also described: one of them is the moduli space of parabolic bundles, while the remaining ones have a description in terms of symmetric products of the Riemann surface. As another consequence of our Morse theoretic analysis, we obtain a proof of the parabolic version of a theorem of Laumon, which states that the nilpotent cone (the preimage of zero under the Hitchin map) is a Lagrangian subvariety of the moduli space of parabolic Higgs bundles. 1.
Nonminimal scalarflat Kähler surfaces and parabolic stability
 Invent. Math
"... Abstract. A new construction is presented of scalarflat Kähler metrics on nonminimal ruled surfaces. The method is based on the resolution of singularities of orbifold ruled surfaces which are closely related to rank2 parabolically stable holomorphic bundles. This rather general construction is sh ..."
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Cited by 10 (3 self)
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Abstract. A new construction is presented of scalarflat Kähler metrics on nonminimal ruled surfaces. The method is based on the resolution of singularities of orbifold ruled surfaces which are closely related to rank2 parabolically stable holomorphic bundles. This rather general construction is shown also to give new examples of low genus: in particular, it is shown that CP 2 blown up at 10 suitably chosen points, admits a scalarflat Kähler metric; this answers a question raised by Claude LeBrun in 1986 in connection with the classification of compact selfdual 4manifolds. 1.
Gluing theorems for complete antiselfdual spaces
 Geom. Func. Analysis
"... 1.1. Summary. One of the special features of 4dimensional differential geometry is the existence of objects with selfdual (SD) or antiselfdual (ASD) curvature. The objects in question can be connections in an auxiliary bundle over a 4manifold, leading to the study of instantons in Yang–Mills th ..."
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Cited by 9 (3 self)
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1.1. Summary. One of the special features of 4dimensional differential geometry is the existence of objects with selfdual (SD) or antiselfdual (ASD) curvature. The objects in question can be connections in an auxiliary bundle over a 4manifold, leading to the study of instantons in Yang–Mills theory [DK91], or as in this paper, Riemannian metrics or conformal
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.
Periodic Instantons and the Loop Group
"... We construct a large class of periodic instantons. Conjecturally we produce all periodic instantons. This confirms a conjecture of Garland and Murray that relates periodic instantons to orbits of the loop group acting on an extension of its Lie algebra. AMS classification: 81T13, 53C07, 55P10 1 Int ..."
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Cited by 4 (1 self)
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We construct a large class of periodic instantons. Conjecturally we produce all periodic instantons. This confirms a conjecture of Garland and Murray that relates periodic instantons to orbits of the loop group acting on an extension of its Lie algebra. AMS classification: 81T13, 53C07, 55P10 1 Introduction Periodic instantons are solutions of the antiselfdual equations FB = \Gamma FB for a connection B on a trivial vector bundle with structure group G over S 1 \Theta R 3 . In this paper, G is a compact Lie group with complexification G c equipped with a representation acting on C n that is unitary on G. Put B = A+ \Phid` so F A = dA \Phi \Gamma @ ` A (1) where we use the threedimensional Hodge star operator and is the reciprocal of the radius of the circle. One can think of the connection and Higgs field as defined over R 3 and dependent on the circlevalued `. Nahm studied periodic instantons, calling them calorons [17]. Later, Garland and Murray studied perio...
Coassociative K3 Fibrations of Compact G2Manifolds
, 2005
"... Abstract. A class of examples of Riemannian metrics with holonomy G2 on compact 7manifolds was constructed by the author in [Ko] using a certain ‘generalized connected sum ’ of two asymptotically cylindrical manifolds with holonomy SU(3). We consider, on each of the two initial SU(3)manifolds, a f ..."
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Cited by 3 (2 self)
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Abstract. A class of examples of Riemannian metrics with holonomy G2 on compact 7manifolds was constructed by the author in [Ko] using a certain ‘generalized connected sum ’ of two asymptotically cylindrical manifolds with holonomy SU(3). We consider, on each of the two initial SU(3)manifolds, a fibration arising from a Lefschetz pencil of K3 surfaces. The gluing of the two K3 fibrations yields a coassociative fibration of the connected sum G2manifold over a 3dimensional sphere. The singular fibres of this fibration are diffeomorphic to K3 orbifolds with ordinary double points and are parameterized by a Hopftype link. We believe that these are the first examples of fibrations of compact manifolds of holonomy G2 by coassociative minimal submanifolds.