Quot schemes and Ricci semipositivity

Abstract

Let X be a compact connected Riemann surface of genus at least two, and let QX(r,d) be the quot scheme that parameterizes all the torsion coherent quotients of OX⊕r of degree d. This QX(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 QX(r,d) is not nef. Equivalently, QX(r,d) does not admit any Kähler metric whose Ricci curvature is semipositive.