Uniform RC-positivity of direct image bundles

2025/12/11 by Kuang-Ru Wu, Wu, Kuang-Ru
Mathematics · #Geometry and complex manifolds #Algebraic Geometry and Number Theory #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2512.10845

Abstract

The concept of RC-positivity and uniform RC-positivity is introduced by Xiaokui Yang to solve a conjecture of Yau on projectivity and rational connectedness of a compact Kähler manifold with positive holomorphic sectional curvature. Some main theorems in Yang's proof hold under a weaker condition called weak RC-positivity. It is therefore natural to ask if (uniform) weak RC-positivity implies (uniform) RC-positivity. Another motivation for studying this problem is to understand the relation between rational connectedness of X and (uniform) RC-positivity of the holomorphic tangent bundle TX. In this paper, we obtain results in this direction. In particular, we show that if a vector bundle E is uniformly weakly RC-positive, then S^kE⊗ det E is uniformly RC-positive for any k≥ 0, and S^kE is uniformly RC-positive for k large. We also discuss an approach that might lead to a solution to the question of whether weak RC-positivity of E implies RC-positivity of E.

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Related

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