Hirzebruch manifolds and positive holomorphic sectional curvature

Abstract

This paper is the first step in a systematic project to study examples of Kähler manifolds with positive holomorphic sectional curvature (H > 0). Previously Hitchin proved that any compact Kähler surface with H > 0 must be rational and he constructed such examples on Hirzebruch surfaces M 2,k = P(H k ⊕ 1 CP 1 ). We generalize Hitchin's construction and prove that any Hirzebruch manifold M n,k = P(H k ⊕ 1 CP n−1 ) admits a Kähler metric of H > 0 in each of its Kähler classes. We demonstrate that the pinching behaviors of holomorphic sectional curvatures of new examples differ from those of Hitchin's which were studied in the recent work of Alvarez-Chaturvedi-Heier. Some connections to recent works on the Kähler-Ricci flow on Hirzebruch manifolds are also discussed.It seems interesting to study the space of all Kähler metrics of H > 0 on a given Kähler manifold. We give higher dimensional examples such that some Kähler classes admit Kähler metrics with H > 0 and some do not.It is the Riemannian sectional curvature restricted on any J-invariant real 2-plane (p165 [29]). Compact Kähler manifolds with positive holomorphic sectional (H > 0) form an interesting class of Kähler manifolds. For example, these manifolds are simply-connected, by the work of Tsukamoto [38]. An averaging argument due to Berger [6] showed that H > 0 implies positive scalar curvature, which further leads to the vanishing of its pluri-canonical ring by a Bochner-Kodaira type identity, see [26]. In 1975 Hitchin [24] proved that any Hirzebruch surface M 2,k = P(H k ⊕ 1 CP 1 ) admits a Kähler metrics with H > 0. Moreover, he proved that any rational surface admits a Kähler metric with positive scalar curvature.

A note on the almost-one-half holomorphic pinching

“…In a previous work of Zheng and the second named author ( [24]), we observe the following result, which follows from some pinching equality on H > 0 due to Berger [3] and recent works on nonnegative orthogonal bisectional curvature ( [7], [10], and [23]).…”

“…Here the local holomorphic pinching constant of a Kähler manifold (M, J, g) of H > 0 is the maximum of all λ ∈ (0, 1] such that 0 < λH(π , ) ≤ H(π) for any J−invariant real 2−planes π, π , ⊂ T p (M ) at any p ∈ M , while the global holomorphic pinching constant is the maximum of all λ ∈ (0, 1] such that there exists a positive constant C so that λC ≤ H(p, π) ≤ C holds for any p ∈ M and any J-invariant real 2-plane π ⊂ T p (M ). Obviously the global holomorphic pinching constant is no larger than the local one, and there are examples of Kähler metrics with different global and local holomorphic pinching constants on Hirzebruch manifolds ( [24]).…”

“…The Kähler-Einstein metrics on many simply-connected compact homogeneous Kähler manifolds (Kähler C-spaces) also have H > 0, and it seems very tedious to work with corresponding Lie algebras carefully to determine these holomorphic pinching constants except in lower dimensions. It was observed in [24] that the flag 3-manifold, the only Kähler C-space in dimesion 3 which is not Hermitian symmetric, has 1 4 -holomorphic pinching for its canonical Kähler-Einstein metric. Note that Alvarez-Chaturvedi-Heier [1] studied pinching constants of Hitchin's examples of Kähler metrics with H > 0 on a Hirzebruch surface.…”

“…However, it remains unknown what is the best pinching constant among all Kähler metrics with H > 0 on such a surface. We refer the interested reader to [1] and [24] for more discussions.…”

“…Proof of Claim 2.2. Note that Berger's inequality [3] (see also Lemma 2.5 in [24] for an exposition) implies the orthogonal bisectional curvature of (M k , g k (0)) is bounded from below by − 1 4k . Now the proof is based on a maximum principle developed in [12].…”