  In a recent paper, Selberg-Tesfahun proved that the abelian
Chern-Simons-Higgs system (CSH) is globally well-posed for finite energy
initial data under the Lorenz gauge condition. It has been suspected by Huh,
however, that such a result should hold in the Coulomb gauge as well. In this
note, we give an affirmative answer to this question by first establishing low
regularity local well-posededness of (CSH) in the Coulomb gauge for initial
data set $(f, g) \in H^{\gamma}_{x} \times H^{\gamma-1}_{x}$ for any $\gamma >
3/4$. Then by conservation of energy, global well-posedness for (CSH) in the
energy space $(f, g) \in H^{1}_{x} \times L^{2}_{x}$ follows rather
immediately.
