On projectivized vector bundles and positive holomorphic sectional curvature

Abstract

Abstract. We generalize a construction of Hitchin to prove that, given any compact Kähler manifold M with positive holomorphic sectional curvature and any holomorphic vector bundle E over M , the projectivized vector bundle P(E) admits a Kähler metric with positive holomorphic sectional curvature.

RC-positivity, rational connectedness and Yau’s conjecture

“…For some related topics on positive holomorphic sectional curvature, we refer to [27,28,2,49,3,48,1] and the references therein.…”

A note on the almost-one-half holomorphic pinching

“…It could be possible that any Kähler manifold with H > 0 is in fact projective, again it is an open question. We also remark that some generalization of Hitchin's construction of Kähler metrics of H > 0 in higher dimensions has been obtained in [2].…”

“…Where each of (CP ki , g ki ) has nonnegative bisectional curvature and each of (N li , h li ) is a compact irreducible Hermitian symmetric spaces of rank ≥ 2 with its canonical Kähler-Einstein metric. Now consider a time t 1 < t 0 close to t = 0, it follows from Step 2 that g ∞ (t 1 ) is close to 1 2 -holmorphic pinching and also have the same decomposition as (2). Indeed the decomposition (2) is reduced to exactly the list in the conclusion of Proposition 1.2.…”

Hermitian metrics of positive holomorphic sectional curvature on fibrations

“…. }, over P 1 , motivated the statement and proof of the main theorem in [AHZ16], establishing the existence of a Kähler metric of positive holomorphic sectional curvature on an arbitrary projectivized holomorphic vector bundle over a compact Kähler base manifold of positive holomorphic sectional curvature. The main result of this note is the following full-fledged positive curvature analog for Cheung's theorem.…”