Paper
Uniform RC-positivity of direct image bundles
Authors
Kuang-Ru Wu
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 and (uniform) RC-positivity of the holomorphic tangent bundle .
In this paper, we obtain results in this direction. In particular, we show that if a vector bundle is uniformly weakly RC-positive, then is uniformly RC-positive for any , and is uniformly RC-positive for large. We also discuss an approach that might lead to a solution to the question of whether weak RC-positivity of implies RC-positivity of .