  We establish a new estimate for the Ginzburg-Landau energies
$E_{\epsilon}(u)=\int_M\frac{1}{2}|du|^2+\frac{1}{4\epsilon^2}(1-|u|^2)^2$ of
complex-valued maps $u$ on a compact, oriented manifold $M$ with $b_1(M)\neq
0$, obtained by decomposing the harmonic component $h_u$ of the one-form
$ju:=u^1du^2-u^2du^1$ into an integral and fractional part. We employ this
estimate to show that, for critical points $u_{\epsilon}$ of $E_{\epsilon}$
arising from the two-parameter min-max construction considered by the author in
previous work, a nontrivial portion of the energy must concentrate on a
stationary, rectifiable $(n-2)$-varifold as $\epsilon\to 0$.
