Steven Bradlow

Steven Bradlow
University of Illinois Urbana-Champaign | UIUC · Department of Mathematics

Ph.D. (Mathematics)

About

75
Publications
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2,031
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Additional affiliations
September 1992 - present
January 1990 - August 1991
University of California, San Diego

Publications

Publications (75)
Chapter
The moduli spaces for Higgs bundles associated to real Lie groups and a closed Riemann surface have multiple connected components. This survey provides a compendium of results concerning the counting of these components in cases where the Lie group is a real forms of a complex simple Lie group. In some cases the components can be described quite ex...
Article
Given a compact connected Riemann surface [Formula: see text] of genus [Formula: see text], and an effective divisor [Formula: see text] on [Formula: see text] with [Formula: see text], there is a unique cone metric on [Formula: see text] of constant negative curvature [Formula: see text] such that the cone angle at each point [Formula: see text] i...
Preprint
Given a compact Riemann surface $\Sigma$ of genus $g_\Sigma\, \geq\, 2$, and an effective divisor $D\, =\, \sum_i n_i x_i$ on $\Sigma$ with $\text{degree}(D)\, <\, 2(g_\Sigma -1)$, there is a unique cone metric on $\Sigma$ of constant negative curvature $-4$ such that the cone angle at each $x_i$ is $2\pi n_i$ (see McOwen and Troyanov [McO,Tr]). We...
Preprint
We examine Higgs bundles for non-compact real forms of SO(4,C) and the isogenous complex group SL(2,C)XSL(2,C). This involves a study of non-regular fibers in the corresponding Hitchin fibrations and provides interesting examples of non-abelian spectral data.
Article
Some connected components of a moduli space are mundane in the sense that they are distinguished only by obvious topological invariants or have no special characteristics. Others are more alluring and unusual either because they are not detected by primary invariants, or because they have special geometric significance, or both. In this paper we de...
Chapter
These volumes contain the proceedings of the conference held at Aarhus, Oxford and Madrid in September 2016 to mark the seventieth birthday of Nigel Hitchin, one of the world’s foremost geometers and Savilian Professor of Geometry at Oxford. The proceedings contain twenty-nine articles, including three by Fields medallists (Donaldson, Mori and Yau)...
Article
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Some connected components of a moduli space are mundane in the sense that they are distinguished only by obvious topological invariants or have no special characteristics. Others are more alluring and unusual either because they are not detected by primary invariants, or because they have special geometric significance, or both. In this paper we de...
Article
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For semisimple Lie groups, moduli spaces of Higgs bundles on a Riemann surface correspond to representation varieties for the surface fundamental group. In many cases, natural topological invariants label connected components of the moduli spaces. Hitchin representations into split real forms, and maximal representations into Hermitian Lie groups,...
Preprint
For semisimple Lie groups, moduli spaces of Higgs bundles on a Riemann surface correspond to representation varieties for the surface fundamental group. In many cases, natural topological invariants label connected components of the moduli spaces. Hitchin representations into split real forms, and maximal representations into Hermitian Lie groups,...
Article
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We propose a method for demonstrating sub community structure in scientific networks of relatively small size from analyzing databases of publications. Research relationships between the network members can be visualized as a graph with vertices corresponding to authors and with edges indicating joint authorship. Using a fast clustering algorithm c...
Article
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We give necessary and sufficient conditions for moduli spaces of semistable chains on a curve to be irreducible and non-empty. This gives information on the irreducible components of the nilpotent cone of GL_n-Higgs bundles and the irreducible components of moduli of systems of Hodge bundles on curves. As we do not impose coprimality restrictions,...
Article
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We explore relations between Higgs bundles that result from isogenies between low-dimensional Lie groups, with special attention to the spectral data for the Higgs bundles. We focus on isogenies onto $SO(4,C)$ and $SO(6,C)$ and their split real forms. Using fiber products of spectral curves, we obtain directly the desingularizations of the (necessa...
Article
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Higgs bundles over a closed orientable surface can be defined for any real reductive Lie group G. In this paper we examine the case G=SO*(2n). We describe a rigidity phenomenon encountered in the case of maximal Toledo invariant. Using this and Morse theory in the moduli space of Higgs bundles, we show that the moduli space is connected in this max...
Article
Let (E,φ)(E,φ) be a Higgs vector bundle over a compact connected Kähler manifold X. Fix any filtration of EE by coherent analytic subsheaves in which each sheaf is preserved by the Higgs field, and each successive quotient is a torsionfree and stable Higgs sheaf. Denote by GG the direct sum of these stable quotients, and let the singular set of GG...
Article
Let $X$ be a compact connected Riemann surface and $G$ a connected reductive complex affine algebraic group. Given a holomorphic principal $G$ -bundle $E_G$ over $X$ , we construct a $C^\infty $ Hermitian structure on $E_G$ together with a $1$ -parameter family of $C^\infty $ automorphisms $\{F_t\}_{t\in \mathbb R }$ of the principal $G$...
Article
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In this mostly expository paper we describe applications of Morse theory to moduli spaces of Higgs bundles. The moduli spaces are finite-dimensional analytic varieties but they arise as quotients of infinite-dimensional spaces. There are natural functions for Morse theory on both the infinite-dimensional spaces and the finite-dimensional quotients....
Article
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We use Higgs bundles to answer the following question: When can a maximal Sp(4,R)-representation of a surface group be deformed to a representation which factors through a proper reductive subgroup of Sp(4,R)?
Article
In this mostly expository paper we describe applications of Morse theory to moduli spaces of Higgs bundles. The moduli spaces are finite dimensional analytic varieties but they arise as quotients of infinite dimensional spaces. There are natural functions for Morse theory on both the infinite dimensional spaces and the finite dimensional quotients....
Article
We introduce the twisted coupled vortex equations defined over a closed Kähler manifold X. There is an associated notion of stability for certain triples of holomorphic data on X. We establish a Hitchin–Kobayashi correspondence which relates the existence of solutions to these equations and the stability of a corresponding triple.
Article
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Let C be an algebraic curve of genus g ≥ 2. A coherent system on C consists of a pair (E,V), where E is an algebraic vector bundle over C of rank n and degree d and V is a subspace of dimension k of the space of sections of E. The stability of the coherent system depends on a parameter α. We study the geometry of the moduli space of coherent system...
Article
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It this paper we study the space of gauge equivalence classes of pairs where represents a holomorphic structure on a complex bundle, E, over a closed Riemann Surface, and ϕ is a holomorphic section. We define a space of stable pairs and consider the moduli space problem for this space. The space of stable pairs, , is related to the space of solutio...
Article
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Let X be a curve of genus g. A coherent system on X consists of a pair (E,V), where E is an algebraic vector bundle over X of rank n and degree d and V is a subspace of dimension k of the space of sections of E. The stability of the coherent system depends on a parameter α. We study the variation of the moduli space of coherent systems when we move...
Article
We calculate certain homotopy groups of the moduli spaces for representations of a compact oriented surface in the Lie groups and . Our approach relies on the interpretation of these representations in terms of Higgs bundles and uses Bott–Morse theory on the corresponding moduli spaces.
Article
It is becoming routine for cryoEM single particle reconstructions to result in 3D electron density maps with resolutions of approximately 10A, but maps with resolutions of 5A or better are still celebrated events. The electron microscope has a resolving power to better than 2A, and thus should not be a limiting factor; instead the practical limitat...
Article
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A Higgs bundle is a holomorphic vector bundle together with a Higgs field. Such objects first emerged twenty years ago in Nigel Hitchin's study of the self-duality equations on a Riemann surface and in Carlos Simp-son's Ph.D. thesis and subsequent work on nonabelian Hodge theory. Hitchin introduced the term "Higgs field" because of similarities to...
Article
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Let $C$ be an algebraic curve of genus $g\ge2$. A coherent system on $C$ consists of a pair $(E,V)$, where $E$ is an algebraic vector bundle over $C$ of rank $n$ and degree $d$ and $V$ is a subspace of dimension $k$ of the space of sections of $E$. The stability of the coherent system depends on a parameter $\alpha$. We study the geometry of the mo...
Article
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Higgs bundles and non-abelian Hodge theory provide holomorphic methods with which to study the moduli spaces of surface group representations in a reductive Lie group G. In this paper we survey the case in which G is the isometry group of a classical Hermitian symmetric space of non-compact type. Using Morse theory on the moduli spaces of Higgs bun...
Preprint
Higgs bundles and non-abelian Hodge theory provide holomorphic methods with which to study the moduli spaces of surface group representations in a reductive Lie group G. In this paper we survey the case in which G is the isometry group of a classical Hermitian symmetric space of non-compact type. Using Morse theory on the moduli spaces of Higgs bun...
Preprint
We calculate certain homotopy groups of the moduli spaces for representations of a compact oriented surface in the Lie groups GL(n,C) and U(p,q). Our approach relies on the interpretation of these representations in terms of Higgs bundles and uses Bott--Morse theory on the corresponding moduli spaces.
Article
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A holomorphic triple over a compact Riemann surface consists of two holomorphic vector bundles and a holomorphic map between them. After fixing the topological types of the bundles and a real parameter, there exist moduli spaces of stable holomorphic triples. In this paper we study non-emptiness, irreducibility, smoothness, and birational descripti...
Article
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Given a Kaehlerian holomorphic fiber bundle whose fiber is a compact homogeneous Kaehler manifold, we describe the perturbed Hermitian-Einstein equations relative to certain holomorphic vector bundles. With respect to special metrics on the holomorphic bundles, there is a dimensional reduction procedure which reduces these equations to a system of...
Conference Paper
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We determine the number of connected components of the moduli space for representations of a surface group in the general linear group.
Article
A principal pair consists of a holomorphic principal G‐bundle together with a holomorphic section of an associated Kaehler fibration. Such objects support natural gauge theoretic equations coming from a moment map condition, and also admit a notion of stability based on Geometric Invariant Theory. The Hitchin–Kobayashi correspondence for principal...
Article
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Using the L<sup>2</sup> norm of the Higgs field as a Morse function, we study the moduli spaces of U(p, q)-Higgs bundles over a Riemann surface. We require that the genus of the surface be at least two, but place no constraints on (p, q). A key step is the identification of the function's local minima as moduli spaces of holomorphic triples. In a c...
Article
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Using the L^2 norm of the Higgs field as a Morse function, we study the moduli spaces of U(p,q)-Higgs bundles over a Riemann surface. We require that the genus of the surface be at least two, but place no constraints on (p,q). A key step is the identification of the function's local minima as moduli spaces of holomorphic triples. In a companion pap...
Article
Full-text available
A principal pair consists of a holomorphic principal $G$-bundle together with a holomorphic section of an associated Kaehler fibration. Such objects support natural gauge theoretic equations coming from a moment map condition, and also admit a notion of stability based on Geometric Invariant Theory. The Hitchin--Kobayashi correspondence for princip...
Article
Full-text available
Using the $L^2$ norm of the Higgs field as a Morse function, we study the moduli spaces of $U(p,q)$-Higgs bundles over a Riemann surface. We require that the genus of the surface be at least two, but place no constraints on $(p,q)$. A key step is the identification of the function's local minima as moduli spaces of holomorphic triples. We prove tha...
Article
We prove a Hitchin-Kobayashi correspondence for extensions of Higgs bundles. The results generalize known results for extensions of holomorphic bundles. Using Simpson's methods, we construct moduli spaces of stable objects. In an appendix we construct Bott-Chern forms for Higgs bundles.
Article
A coherent system (CS) consists of a holomorphic bundle together with a subspace of its space of holomorphic sections. There is a notion of stability for this type of 'augmented bundle', with resulting moduli spaces of stable objects. The definition of stability, and hence the moduli spaces, depend on a real parameter. At large values of the parame...
Article
We count the connected components in the moduli space of PU(p,q)-representations of the fundamental group for a closed oriented surface. The components are labelled by pairs of integers which arise as topological invariants of the flat bundles associated to the representations. Our results show that for each allowed value of these invariants, which...
Article
We count the connected components in the moduli space of PU(p,q)-representations of the fundamental group for a closed oriented surface. The components are labelled by pairs of integers which arise as topological invariants of the flat bundles associated to the representations. Our results show that for each allowed value of these invariants, which...
Article
Full-text available
We prove a Hitchin-Kobayashi correspondence for extensions of Higgs bundles. The results generalize known results for extensions of holomorphic bundles. Using Simpson's methods, we construct moduli spaces of stable objects. In an appendix we construct Bott-Chern forms for Higgs bundles
Article
We introduce the twisted coupled vortex equations defined over a closed Kähler manifold X. There is an associated notion of stability for certain triples of holomorphic data on X. We establish a Hitchin-Kobayashi correspondence which relates the existence of solutions to these equations and the stability of a corresponding triple.
Article
Full-text available
. The technique of dimensional reduction of an integrable system usually requires symmetry arising from a group action. In this paper we study a situation in which a dimensional reduction can be achieved despite the absence of any such global symmetry. We consider certain holomorphic vector bundles over a Kahler manifold which is itself the total s...
Article
Full-text available
In order to use the technique of dimensional reduction, it is usually necessary for there to be a symmetry coming from a group action. In this paper we consider a situation in which there is no such symmetry, but in which a type of dimensional reduction is nevertheless possible. We obtain a relation between the Coupled Vortex equations on a closed...
Article
Full-text available
The Seiberg-Witten equations are defined on certain complex line bundles over smooth oriented four manifolds. When the base manifold is a complex Kahler surface, the Seiberg-Witten equations are essentially the Abelian vortex equations. Using known non-abelian generalizations of the vortex equations as a guide, we explore some non-abelian versions...
Article
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A coherent system (E, V) consists of a holomorphic bundle plus a linear subspace of its space of holomorphic sections. Motivated by the usual notion in geometric invariant theory, a notion of slope stability can be defined for such objects. In the paper it is shown that stability in this sense is equivalent to the existence of solutions to a certai...
Article
Full-text available
We introduce equations for special metrics, and notions of stability for some new types of augmented holomorphic bundles. These new examples include holomorphic extensions, and in this case we prove a Hitchin-Kobayashi correspondence between a certain deformation of the Hermitian-Einstein equations and our definition of stability for an extension.
Article
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Unstable holomorphic bundles can be described algebraically by Harder-Narasimhan filtrations. In this note we show how such filtrations correspond to the existence of special metrics defined by Hermitian-Einstein inequalities.
Chapter
The study of vector bundles over algebraic varieties has been stimulated over the last few years by successive waves of migrant concepts, largely from mathematical physics, whilst retaining its roots in old questions concerning subvarieties of projective space. The 1993 Durham Symposium on Vector Bundles in Algebraic Geometry brought together some...
Article
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This is a resubmission of preprint 9401008 , which has some TeXnical errors introduced by the "reform" procedure (designed to avoid precisely these problems!). The original can be formatted by editing out the messages "%% following line cannot be broken..." at line 1674. This problem has been corrected in the current version. The content of the pap...
Article
We construct a finite-dimensional Kähler manifold with a holomorphic, symplectic circle action whose symplectic reduced spaces may be identified with the τ-vortex moduli spaces (or τ-stable pairs). The Morse theory of the circle action induces natural birational maps between the reduced spaces for different values of τ which in the case of rank two...
Preprint
We construct a finite dimensional Kaehler manifold with a holomorphic, symplectic circle action whose symplectic reduced spaces may be identified with the tau-vortex moduli spaces (or tau-stable pairs). The Morse theory of the circle action induces natural birational maps between the reduced spaces for different values of tau which in the case of r...
Article
In this paper we describe canonical metrics on holomorphic bundles in which there are global holomorphic sections. Such metrics are defined by a constraint on the curvature of the corresponding metric connection. The constraint is in the form of a P.D.E which looks like the Hermitian-Yang-Mills equation with an extra zeroth order term. We identify...
Article
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We apply a modified Yang-Mills-Higgs functional to unitary bundles over closed Khler manifolds and study the equations which govern the global minima. The solutions represent vortices in holomorphic bundles and are direct analogs of the vortices overR 2. We obtain a complete description of the moduli space of these new vortices where the bundle is...
Article
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Collins and Rice’s1 calculations are extended to the fifth excited state, and the effect of anharmonicity in the potential on mode shape and energy is examined. (AIP)
Article
The optical line shape of the GR1 zero phonon line centred around 1.673 eV is profoundly affected by random strain fields. Sometimes, an interaction with the lattice strain produces a finite stress at the defect removing the degeneracy of the electronic levels concerned in the GR1 transition and resulting in an asymmetric line shape. For Type IIA d...
Article
. The technique of dimensional reduction of an integrable system usually requires symmetry arising from a group action. In this paper we study a situation in which a dimensional reduction can be achieved despite the absence of any such global symmetry. We consider certain holmorphic vector bundles over a Kahler manifold which is itself the total sp...
Article
A coherent system is a pair (E;V ) where E is a holomorphic bundle and V is a linear subspace of its space of holomorphic sections. If E is a semistable bundle, then the existence of such objects is equivalent to the non-emptiness of a higher rank Brill- Noether locus. This connection to higher rank Brill-Noether theory provides one of the motivati...

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