  We consider the magnetic Ginzburg-Landau equations in a compact manifold $N$
$$ \begin{cases} -\varepsilon^2 \Delta^{A} u=\frac{1}{2}(1-|u|^2)u,\\
\varepsilon^2 d^*dA=\langle\nabla^A u,iu\rangle \end{cases} $$ formally
corresponding to the Euler-Lagrange equations for the energy functional $$
  E(u,A)=\frac{1}{2}\int_{N}\varepsilon^2|\nabla^Au|^{2}+\varepsilon^4|dA|^{2}+\frac{1}{4}(1-|u|^{2})^{2}.
$$ Here $u:N\to \mathbb{C}$ and $A$ is a 1-form on $N$. Given a codimension-2
minimal submanifold $M\subset N$ which is also oriented and non-degenerate, we
construct a solution $(u_\varepsilon,A_\varepsilon)$ such that $u_\varepsilon$
has a zero set consisting of a smooth surface close to $M$. Away from $M$ we
have $$
  u_\varepsilon(x)\to\frac{z}{|z|},\quad
A_\varepsilon(x)\to\frac{1}{|z|^2}(-z^2dz^1+z^1dz^2),\quad
x=\exp_y(z^\beta\nu_\beta(y)).
  $$ as $\varepsilon\to 0$, for all sufficiently small $z\ne 0$ and $y\in M$.
Here, $\{\nu_1,\nu_2\}$ is a normal frame for $M$ in $N$. This improves a
recent result by De Philippis and Pigati who built a solution for which the
concentration phenomenon holds in an energy, measure-theoretical sense.
