Quot schemes and Ricci semipositivity

Let $X$ be a compact connected Riemann surface of genus at least two, and let ${\mathcal{Q}}_X(r,d)$ be the quot scheme that parametrizes all the torsion coherent quotients of ${\mathcal{O}}_X^{\oplus r}$ of degree $d$. This ${\mathcal{Q}}_X(r,d)$ is also a moduli space of vortices on $X$. Its geometric properties have been extensively studied. Here we prove that the anticanonical line bundle of ${\mathcal{Q}}_X(r,d)$ is not nef. Equivalently, ${\mathcal Q}_X(r,d)$ does not admit any K"ahler metric whose Ricci curvature is semipositive.