Vortices and Jacobian varieties

Abstract

We investigate the geometry of the moduli space of N -vortices on line bundles over a closed Riemann surface Σ of genus g > 1, in the little explored situation where 1 ≤ N < g. In the regime where the area of the surface is just large enough to accommodate N vortices (which we call the dissolving limit), we describe the relation between the geometry of the moduli space and the complex geometry of the Jacobian variety of Σ. For N = 1, we show that the metric on the moduli space converges to a natural Bergman metric on Σ. When N > 1, the vortex metric typically degenerates as the dissolving limit is approached, the degeneration occurring precisely on the critical locus of the Abel-Jacobi map of Σ at degree N . We describe consequences of this phenomenon from the point of view of multivortex dynamics.

On the Kähler metrics over Symd(X)

“…The natural inner product on H 0 (X, K X ), where K X −→ X is the holomorphic cotangent bundle, produces a flat Kähler metric on Pic d (X). It is natural to construct a metric on Sym d (X) by pulling back the flat metric using the embedding ϕ; see [15], [12] (especially [12, p. 1137, (1.2)], [12, § 7]). Our aim here is to study this metric on Sym d (X).…”

Singularities of Abel-Jacobi maps and geometry of dissolving vortices

“…It is not hard to see that this induced metric coincides with the natural L 2 -geometry on the space of dissolved vortices (see Section 3 of [22] for the explicit argument). This geometry on the dual Jacobian is independent of the first Chern class k, the vortex number of the dissolved vortex.…”

“…In [22], the following result is proven: Theorem 2.3. In the dissolving limit (1.7), the L 2 -metric on M 1 converges to a natural Bergman metric on Σ, regarded as the moduli space of one dissolving vortex.…”

“…which is associated to any basis [22]. Note that (2.8) is constant on U(g)-orbits of the space of bases.…”

“…This was considered in [32] and [3] when Σ has genus g = 0; and for genus g ≥ 1 first by Nasir [26], and then by Manton and Romão [22]. In the following, we shall give an account of the results in [22] -we will add little to the presentation in the original paper, and shall focus on the k > 1 (multivortex) case, which brings in some interesting issues that relate to the theory of singularities.…”