Kähler metrics of negative holomorphic bisectional curvature on Kodaira surfaces

To, Wing-Keung; Yeung, Sai-Kee

doi:10.1112/blms/bdq117

Cited by 6 publications

(2 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The case of negative holomorphic bisectional curvature is not so rigid. For example, recently To and Yeung [92] have constructed such Kähler metrics on any Kodaira surface.…”

Section: Holomorphic Curvatures Of Twistor Spacesmentioning

confidence: 99%

Curvature Properties of the Chern Connection of Twistor Spaces

Davidov¹,

Grantcharov²,

Mus˘karov³

2009

Rocky Mountain J. Math.

View full text Add to dashboard Cite

The twistor space Z of an oriented Riemannian 4-manifold M admits a natural 1-parameter family of Riemannian metrics h t compatible with the almost complex structures J 1 and J 2 introduced, respectively, by Atiyah, Hitchin and Singer, and Eells and Salamon. In this paper we compute the first Chern form of the almost Hermitian manifold (Z, h t , J n ), n = 1, 2 and find the geometric conditions on M under which the curvature of its Chern connection D n is of type (1, 1). We also describe the twistor spaces of constant holomorphic sectional curvature with respect to D n and show that the Nijenhuis tensor of J 2 is D 2parallel provided the base manifold M is Einstein and self-dual.

show abstract

“…The case of negative holomorphic bisectional curvature is not so rigid. For example, recently To and Yeung [92] have constructed such Kähler metrics on any Kodaira surface.…”

Section: Holomorphic Curvatures Of Twistor Spacesmentioning

confidence: 99%

Curvature Properties of the Chern Connection of Twistor Spaces

Davidov¹,

Grantcharov²,

Mus˘karov³

2009

Rocky Mountain J. Math.

View full text Add to dashboard Cite

show abstract

“…The case of negative holomorphic bisectional curvature is not so rigid. For example, recently To and Yeung [31] have constructed such Kähler metrics on any Kodaira surface.…”

Section: Introductionmentioning

confidence: 99%

Holomorphic curvatures of twistor spaces

Ali

Davidov

Мушкаров

2014

Int. J. Geom. Methods Mod. Phys.

View full text Add to dashboard Cite

We study the twistor spaces of oriented Riemannian 4-manifolds as a source of almost Hermitian 6-manifolds of constant or strictly positive holomorphic, Hermitian and orthogonal bisectional curvatures. In particular, we obtain explicit formulas for these curvatures in the case when the base manifold is Einstein and self-dual, and observe that the "squashed" metric on CP 3 is a non-Kähler Hermitian-Einstein metric of positive holomorphic bisectional curvature. This shows that a recent result of Kalafat and Koca [M. Kalafat and C. Koca, Einstein-Hermitian 4-manifolds of positive bisectional curvature, preprint (2012), arXiv: 1206.3941v1 [math.DG]] in dimension four cannot be extended to higher dimensions. We prove that the Hermitian bisectional curvature of a non-Kähler Hermitian manifold is never a nonzero constant which gives a partial negative answer to a question of Balas and Gauduchon [A. Balas and P. Gauduchon, Any Hermitian metric of constant non-positive (Hermitian) holomorphic sectional curvature on a compact complex surface is Kähler, Math. Z. 190 (1985) 39-43]. Finally, motivated by an integrability result of Vezzoni [L. Vezzoni, On the Hermitian curvature of symplectic manifolds, Adv. Geom. 7 (2007) 207-214] for almost Kähler manifolds, we study the problem when the holomorphic and the Hermitian bisectional curvatures of an almost Hermitian manifold coincide. We extend the result of Vezzoni to a more general class of almost Hermitian manifolds and describe the twistor spaces having this curvature property.

show abstract

Hyperbolicity of Moduli Spaces of Log-Canonically Polarized Manifolds

Yeung

2020

J Geom Anal

View full text Add to dashboard Cite

Kähler metrics of negative holomorphic bisectional curvature on Kodaira surfaces

Abstract: We construct a Kähler metric of negative holomorphic bisectional curvature on any compact complex surface that admits a Kodaira fibration. This is achieved by considering the curvature of the Weil-Petersson metric on an associated Teichmüller space of punctured Riemann surfaces.

Cited by 6 publications

References 8 publications

Curvature Properties of the Chern Connection of Twistor Spaces

Curvature Properties of the Chern Connection of Twistor Spaces

Holomorphic curvatures of twistor spaces

Hyperbolicity of Moduli Spaces of Log-Canonically Polarized Manifolds

Contact Info

Product

Resources

About