On the curvature of vortex moduli spaces

Bökstedt, Marcel; Romão, Nuno M.

doi:10.1007/s00209-013-1265-3

Cited by 11 publications

(14 citation statements)

References 49 publications

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“…Bökstedt and Romão proved some interesting differential geometric properties of Q (see [12]). In [10] and [11] we proved that Q does not admit Kähler metrics with semipositive or seminegative holomorphic bisectional curvature.…”

Section: Introductionmentioning

confidence: 99%

Quot schemes and Ricci semipositivity

Biswas

Seshadri

2017

Comptes Rendus. Mathématique

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Abstract. Let X be a compact connected Riemann surface of genus at least two, and let Q X (r, d) be the quot scheme that parametrizes all the torsion coherent quotients of O ⊕r X of degree d. This Q X (r, d) is also a moduli space of vortices on X. Its geometric properties have been extensively studied. Here we prove that the anticanonical line bundle of Q X (r, d) is not nef. Equivalently, Q X (r, d) does not admit any Kähler metric whose Ricci curvature is semipositive. Résumé. Schéma quot et semi-positivité de RicciSoit X une surface de Riemann compacte et connexe de genre au moins deux, et soit Q X (r, d) le schéma quot qui paramétrise tous les quotients torsion cohérents de O) est aussi un espace de modules de vortex sur X. Nous démontrons que le fibré anticanonique de X n'a pas la propriété nef. De faonéquivalente, Q X (r, d) n'admet aucune métrique kahléienne dont la courbure de Ricci est semi-positive.

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Section: Introductionmentioning

confidence: 99%

Quot schemes and Ricci semipositivity

Biswas

Seshadri

2017

Comptes Rendus. Mathématique

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“…For any positive integer d, let S d (X) denote the d-fold symmetric product of X. The main theorem of [BoR] says the following (see [BoR,Theorem 1.1]): If d ≤ 2(g − 1), then S d (X) does not admit any Kähler metric for which all the holomorphic bisectional curvatures are nonnegative.…”

Section: Introductionmentioning

confidence: 99%

On the Kähler structures over Quot schemes, II

Biswas¹,

Seshadri²

2014

Illinois J. Math.

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“…We stress that Theorem 2 holds in general, not only near the Bradlow limit. The fact that the fibre PH 0 (Σ, O(π −1 (p))) is a geodesic submanifold of the vortex moduli space is also a consequence of Proposition 7.1 in [13], for which an isometry on the moduli space is required.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Vortices and the Abel–Jacobi map

Rink

2014

Journal of Geometry and Physics

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The abelian Higgs model on a compact Riemann surface Σ supports vortex solutions for any positive vortex number d ∈ Z. Moreover, the vortex moduli space for fixed d has long been known to be the symmetrized d-th power of Σ, in symbols, Sym d (Σ). This moduli space is Kähler with respect to the physically motivated metric whose geodesics describe slow vortex motion.In this paper we appeal to classical properties of Sym d (Σ) to obtain new results for the moduli space metric. Our main tool is the AbelJacobi map, which maps Sym d (Σ) into the Jacobian of Σ. Fibres of the Abel-Jacobi map are complex projective spaces, and the first theorem we prove states that near the Bradlow limit the moduli space metric restricted to these fibres is a multiple of the Fubini-Study metric. Additional significance is given to the fibres of the Abel-Jacobi map by our second result: we show that if Σ is a hyperelliptic surface, there exist two special fibres which are geodesic submanifolds of the moduli space. Even more is true: the Abel-Jacobi map has a number of fibres which contain complex projective subspaces that are geodesic. * nar43@cantab.net

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On the curvature of vortex moduli spaces

Cited by 11 publications

References 49 publications

Quot schemes and Ricci semipositivity

Quot schemes and Ricci semipositivity

On the Kähler structures over Quot schemes, II

Vortices and the Abel–Jacobi map

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