On the Kähler structures over Quot schemes, II

Abstract

Let X be a compact connected Riemann surface of genus g, with g ≥ 2, and let O X denote the sheaf of holomorphic functions on X. Fix positive integers r and d and let Q(r, d) be the Quot scheme parametrizing all torsion coherent quotients of O ⊕r X of degree d. We prove that Q(r, d) does not admit a Kähler metric whose holomorphic bisectional curvatures are all nonnegative.

Quot schemes and Ricci semipositivity

“…Since semipositive holomorphic bisectional curvature implies semipositive Ricci curvature for a Kähler metric, Theorem 1.1 generalizes the main result of [11].…”

“…Bökstedt and Romão proved some interesting differential geometric properties of Q (see [12]). In [10] and [11] we proved that Q does not admit Kähler metrics with semipositive or seminegative holomorphic bisectional curvature. In this note, we continue the study the question of existence of metrics on Q whose curvature has a sign.…”

“…The projection q in (2.6) is clearly the Albanese morphism for Q, because C, H and F are all simply connected. On the other hand, p • ϕ is the Albanese morphism for Q [11,Corollary 2.2]. Therefore, its pullback, namely, λ, is the Albanese morphism for Q. Consequently, we have A = J with λ coinciding with the projection q in (2.6).…”

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.…”