Monge-Ampère type equation for the Nakano positive curvature tensor of holomorphic vector bundles

TL;DR

This work extends the Calabi–Yau-type Monge–Ampère framework to Nakano-positive Hermitian holomorphic vector bundles by solving a determinant-type equation in the conformal class of a Nakano-positive metric. Employing a continuity method, it establishes existence and uniqueness for $\lambda>0$ and a normalized solution for $\lambda=0$, underpinned by a sequence of sharp a priori estimates: $C^{0}$, $C^{1}$, $C^{2}$, and finally $C^{2,\alpha}$ via the complex Evans–Krylov theory. The results demonstrate strong regularity and control of solutions, closely mirroring the classical Yau–Aubin theory but in a higher-rank, curvature-driven setting. These findings contribute to the broader program toward Griffiths positivity and offer a foundational PDE toolkit for Nakano-positive curvature tensors in holomorphic vector bundles.

Abstract

For any Hermitian holomorphic vector bundle with Nakano positive curvature tensor, Demailly introduced a Monge-Ampère type equation. When the rank of the bundle is $1$, it becomes the usual Monge-Ampère equation. In this paper, we solve this equation in the conformal class of a Nakano positive Hermitian metric.

Theorems & Definitions (14)

• Theorem 1.1
• Lemma 2.1
• Lemma 2.2
• Lemma 3.1
• Proposition 3.2
• Lemma 3.3
• proof
• Proposition 3.4
• proof
• Lemma 4.1