Y. Yang observed 20 years ago that the 2-sphere is the only compact orientable surface admitting solutions of the Einstein-Bogomol'nyi equations, coupling vortices with gravity, and obtained sufficient conditions for the existence of cosmic strings in this situation. In this talk, we will give an algebro-geometric interpretation of Yang's conditions, and explain why they are in fact necessary for the existence of solutions. The key ideas, coming from the theory of quotient spaces in symplectic geometry and algebraic geometry, will also enable us to find suitable generalizations of these equations to higher-genus compact orientable surfaces. This is joint work with Mario García-Fernández and Óscar García-Prada.