Symplectic vortex equations for Kahler cones over Sasakian manifolds

TL;DR

This paper establishes a Hitchin-Kobayashi-type correspondence for symplectic vortex equations targeting Kahler cones over Sasakian manifolds, linking solutions to geometric structures and effective divisors.

Contribution

It introduces a novel correspondence connecting symplectic vortex equations with geometric stability conditions on Kahler cones over Sasakian manifolds.

Findings

Reduced the problem to Kazdan-Warner equations

Constructed a map between vortex moduli space and effective divisors

Provided existence and uniqueness results for solutions

Abstract

We obtain a Hitchin-Kobayashi-type correspondence for symplectic vortex equations, with the target a Kahler cone over a compact Sasakian manifold. We show that the correspondence reduces to studying the existence and uniqueness of Kazdan-Warner equations. Using this, we construct a map between the moduli space of solutions to the symplectic vortex equations and effective divisors.