Abstract
We use algebraic topology to investigate local curvature properties of the moduli spaces of gauged vortices on a closed Riemann surface. After computing the homotopy type of the universal cover of the moduli spaces (which are symmetric powers of the surface), we prove that, for genus g>1, the holomorphic bisectional curvature of the vortex metrics cannot always be nonnegative in the multivortex case, and this property extends to all Kaehler metrics on certain symmetric powers. Our result rules out an established and natural conjecture on the geometry of the moduli spaces.
Highlights
Gauged vortices [28,39] are of interest as static, stable configurations in various classical gauge field theories with topological solitons
We address curvature properties of the moduli spaces that are consistent with the topological consequences of the result (5), and which are in a sense stronger than the sign of the Ricci curvature
It was pointed out to us that an alternative proof of Theorem 1.1 can be presented, based on results obtained by Fang [17] on the characterisation of compact Kähler manifolds whose holomorphic bisectional curvature satisfies a condition that is weaker than the notion of nonnegativity used in the present paper
Summary
Gauged vortices [28,39] are of interest as static, stable configurations in various classical gauge field theories with topological solitons. It was pointed out to us that an alternative proof of Theorem 1.1 can be presented, based on results obtained by Fang [17] on the characterisation of compact Kähler manifolds whose holomorphic bisectional curvature satisfies a condition that is weaker than the notion of nonnegativity used in the present paper. This argument explores the fact that the Albanese variety [23] of Symk( ) coincides with the Jacobian variety of. Remark 1.3 After this work was completed, Indranil Biswas wrote a paper [5] proving that the hypothesis k ≤ 2g − 2 in our Theorem 1.1 is unnecessary, using arguments in algebraic geometry which apply strictly to the case k > 2g − 2
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