Hirzebruch manifolds and positive holomorphic sectional curvature
Annales de l'Institut Fourier, Volume 69 (2019) no. 6, p. 2589-2634

This paper is the first step in a systematic project to study examples of Kähler manifolds with positive holomorphic sectional curvature (H>0). Hitchin proved that any compact Kähler surface with H>0 must be rational and he constructed such examples on Hirzebruch surfaces M 2,k =(H k 1 ℂℙ 1 ). We generalize Hitchin’s construction and prove that any Hirzebruch manifold M n,k =(H k 1 ℂℙ n-1 ) admits a Kähler metric of H>0 in each of its Kähler classes. We demonstrate that 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 previous 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.

Cet article est la première étape d’un projet d’étude systématique d’exemples de variétés kähleriennes à courbure sectionnelle holomorphe positive (H>0). Hitchin a prouvé que tout surface kählerienne compacte avec H>0 doit être rationnelle et il a construit de tels exemples sur les surfaces de Hirzebruch. Nous généralisons la construction de Hitchin et prouvons que toute variété de Hirzebruch admet une métrique kählerienne avec H>0 dans chacune de ses classes kähleriennes.

Il semble intéressant d’étudier l’espace de toutes les métriques kähleriennes avec H>0 sur une variété kählerienne donnée. Nous donnons des exemples de dimension supérieure tels que certaines classes kähleriennes admettent des métriques kähleriennes avec H>0 et d’autres non.

Received : 2016-12-14
Revised : 2018-06-30
Accepted : 2019-01-17
Published online : 2019-10-29
DOI : https://doi.org/10.5802/aif.3303
Classification:  53C55,  32Q15
Keywords: Kähler manifolds, holomorphic sectional curvature, Kähler–Ricci flow
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     author = {Yang, Bo and Zheng, Fangyang},
     title = {Hirzebruch manifolds and positive holomorphic sectional curvature},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {69},
     number = {6},
     year = {2019},
     pages = {2589-2634},
     doi = {10.5802/aif.3303},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2019__69_6_2589_0}
}
Yang, Bo; Zheng, Fangyang. Hirzebruch manifolds and positive holomorphic sectional curvature. Annales de l'Institut Fourier, Volume 69 (2019) no. 6, pp. 2589-2634. doi : 10.5802/aif.3303. https://aif.centre-mersenne.org/item/AIF_2019__69_6_2589_0/

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