Curvature operator of holomorphic vector bundles and $L^2$-estimate condition for $(n,q)$ and $(p,n)$-forms

Abstract

We study the positivity properties of the curvature operator for holomorphic Hermitian vector bundles. The characterization of Nakano semi-positivity by L 2 -estimate is already known. Applying our results, we give new characterizations of Nakano semi-negativity. Contents 1. Introduction 1 2. Properties of the curvature operator 3 3. The (p, q)-L 2 -estimate condition and semi-positivity of the curvature operator 15 4. Characterizations of Nakano semi-negativity 19 5. Applications 21 References 23

Bogomolov–Sommese type vanishing theorem for holomorphic vector bundles equipped with positive singular Hermitian metrics

“…A n,1 E,h ≥ 0, for holomorphic vector bundles (E, h). Then we introduced the following positive notion of Hörmander type in [29], which is an extension of the optimal L 2 -estimate condition from (n, 1)-forms to ( p, n)-forms and which characterizes the condition A p,n E,h ≥ 0 (see Theorem 3.5). Definition 3.4 (cf.…”

“…This definition is based on characterization of Nakano positivity using the so called "optimal L 2estimate condition" for (n, 1)-forms by Deng-Ning-Wang-Zhou [13], and does not require the use of curvature currents. In [29], these characterizations of positivity using L 2 -estimates for (n, 1)-forms are extended to (n, q) and ( p, n)-forms.…”

Bogomolov-Sommese type vanishing theorem for holomorphic vector bundles equipped with positive singular Hermitian metrics

“…A n,1 E,h ≥ 0, for holomorphic vector bundles (E, h). Then we introduced the following positive notion of Hörmander type in [Wat21], which is an extension of the optimal L 2 -estimate condition from (n, 1)-forms to (p, q)-forms with p + q > n and which characterizes the condition A p,q E,h ≥ 0 (see Theorem 3.5).…”

“…This phenomenon is first mentioned in [Ina21] as an extension of the properties seen in plurisubharmonic functions. After that, it is extended to the case of Nakano semipositivity in [Ina22] and then to the case of the (p, q)-L 2 -estimate condition with p+q > n in [Wat21].…”

“…[DWZZ18], [DNWZ20], [HI20]) and does not require the use of curvature currents. In [Wat21], these characterizations of positivity using L 2 -estimates for (n, 1)-forms are extended to (p, q)-forms with p + q > n.…”