Uniform RC-positivity of direct image bundles

TL;DR

This paper investigates the relationship between weak RC-positivity and RC-positivity of vector bundles, providing new results on how weak RC-positivity influences the positivity of symmetric powers and determinants, with implications for complex geometry.

Contribution

It establishes that uniformly weakly RC-positive bundles lead to uniformly RC-positive symmetric powers and determinants, advancing understanding of positivity properties in complex geometry.

Findings

Weak RC-positivity implies positivity of symmetric powers for large k

S^kE⊗det E is uniformly RC-positive for any k≥0

Discussion of potential approaches to equate weak RC-positivity with RC-positivity

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 K\"ahler 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 $S^kE\otimes \det E$ is uniformly RC-positive for any $k\geq 0$, 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 $E$.