Quot schemes and Ricci semipositivity

Abstract

Abstract. Let X be a compact connected Riemann surface of genus at least two, and let Q X (r, d) be the quot scheme that parametrizes all the torsion coherent quotients of O ⊕r X of degree d. This 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 Q X (r, d) is not nef. Equivalently, Q X (r, d) does not admit any Kähler metric whose Ricci curvature is semipositive. Résumé. Schéma quot et semi-positivité de RicciSoit X une surface de Riemann compacte et connexe de genre au moins deux, et soit Q X (r, d) le schéma quot qui paramétrise tous les quotients torsion cohérents de O) est aussi un espace de modules de vortex sur X. Nous démontrons que le fibré anticanonique de X n'a pas la propriété nef. De faonéquivalente, Q X (r, d) n'admet aucune métrique kahléienne dont la courbure de Ricci est semi-positive.

Chern scalar curvature and symmetric products of compact Riemann surfaces

“…Let X be a compact Riemann surface of genus g ≥ 0. For d ≥ 1, the d-fold symmetric product Sym d (X) of X admits a Hermitian metric with This result is a continuation of our earlier works on the existence of Kähler metrics with holomorphic bisectional curvature [BS1] and Ricci curvature [BS2] with fixed sign on symmetric products, and, more generally, Quot schemes associated to Riemann surfaces. Theorem 1.1 raises some interesting questions which may have to be tackled by methods different from those in this paper: The proof of Theorem 1.1 is based on a recent result of X. Yang, [Ya], giving criteria for the existence of positive or negative Chern scalar curvature Hermitian metrics on a compact complex manifold.…”