Any Hermitian metric of constant non-positive (Hermitian) holomorphic sectional curvature on a compact complex surface is K�hler

1985 ◽  
Vol 190 (1) ◽  
pp. 39-43 ◽  
Author(s):  
Andrew Balas ◽  
Paul Gauduchon
Keyword(s):  
Sectional Curvature ◽  
Complex Surface ◽  
Holomorphic Sectional Curvature ◽  
Hermitian Metric ◽  
Compact Complex Surface

Holomorphic curvatures of twistor spaces

2014 ◽  
Vol 11 (03) ◽  
pp. 1450022
Author(s):  
Danish Ali ◽  
Johann Davidov ◽  
Oleg Mushkarov
Keyword(s):  
Complex Surface ◽  
Hermitian Manifold ◽  
Negative Answer ◽  
Einstein Metric ◽  
Holomorphic Sectional Curvature ◽  
Almost Hermitian Manifold ◽  
Twistor Spaces ◽  
Bisectional Curvature ◽  
Holomorphic Bisectional Curvature ◽  
Compact Complex Surface

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 ℂℙ3 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.

On Hyperbolicity of Domains with Strictly Pseudoconvex Ends

2014 ◽  
Vol 66 (1) ◽  
pp. 197-204
Author(s):  
Adam Harris ◽  
Martin Kolář
Keyword(s):  
Sectional Curvature ◽  
Level Set ◽  
Unbounded Domains ◽  
Holomorphic Sectional Curvature ◽  
Sufficient Condition ◽  
Bounded Subset ◽  
Hermitian Metric ◽  
Bounded Curvature ◽  
Kobayashi Hyperbolicity ◽  
Negative Holomorphic Sectional Curvature

AbstractThis article establishes a sufficient condition for Kobayashi hyperbolicity of unbounded domains in terms of curvature. Specifically, when Ω ⊂ ℂn corresponds to a sub-level set of a smooth, real-valued function Ψ such that the form ω = is Kähler and has bounded curvature outside a bounded subset, then this domain admits a hermitian metric of strictly negative holomorphic sectional curvature.

HERMITIAN METRIC WITH CONSTANT HOLOMORPHIC SECTIONAL CURVATURE ON CONVEX DOMAINS

2000 ◽  
Vol 11 (06) ◽  
pp. 849-855 ◽  
Author(s):  
WING SUM CHEUNG ◽  
BUN WONG
Keyword(s):  
Sectional Curvature ◽  
Convex Domain ◽  
Blow Up ◽  
Euclidean Ball ◽  
Holomorphic Sectional Curvature ◽  
Convex Domains ◽  
Hermitian Metric ◽  
Bounded Convex Domain ◽  
Bounded Convex ◽  
Negative Holomorphic Sectional Curvature

Let D be a bounded convex domain in [Formula: see text] with a Hermitian metric [Formula: see text] of constant negative holomorphic sectional curvature such that all components [Formula: see text] blow up to infinity on the boundary of D. Then D is biholomorphic to the Euclidean ball.

scholarly journals Non Kählerian surfaces with a cycle of rational curves

2021 ◽  
Vol 8 (1) ◽  
pp. 208-222
Author(s):  
Georges Dloussky
Keyword(s):  
Deformation Theory ◽  
Complex Surface ◽  
Rational Curves ◽  
Intersection Form ◽  
Connected Component ◽  
Hirzebruch Surface ◽  
Compact Complex Surface ◽  
A Chain

Abstract Let S be a compact complex surface in class VII0 + containing a cycle of rational curves C = ∑Dj . Let D = C + A be the maximal connected divisor containing C. If there is another connected component of curves C ′ then C ′ is a cycle of rational curves, A = 0 and S is a Inoue-Hirzebruch surface. If there is only one connected component D then each connected component Ai of A is a chain of rational curves which intersects a curve Dj of the cycle and for each curve Dj of the cycle there at most one chain which meets Dj . In other words, we do not prove the existence of curves other those of the cycle C, but if some other curves exist the maximal divisor looks like the maximal divisor of a Kato surface with perhaps missing curves. The proof of this topological result is an application of Donaldson theorem on trivialization of the intersection form and of deformation theory. We apply this result to show that a twisted logarithmic 1-form has a trivial vanishing divisor.

scholarly journals On a $K$-space of constant holomorphic sectional curvature

1973 ◽  
Vol 25 (3) ◽  
pp. 297-306 ◽  
Author(s):  
Yoshiyuki Watanabe ◽  
Kichiro Takamatsu
Keyword(s):  
Sectional Curvature ◽  
Holomorphic Sectional Curvature

Jacobi fields and geodesic spheres

1979 ◽  
Vol 82 (3-4) ◽  
pp. 233-240 ◽  
Author(s):  
L. Vanhecke ◽  
T. J. Willmore
Keyword(s):  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Holomorphic Sectional Curvature ◽  
Principal Curvatures ◽  
Jacobi Fields ◽  
Geodesic Spheres ◽  
Jacobi Vector ◽  
Nearly Kähler ◽  
Nearly Kähler Manifolds

SynopsisThis is a contribution to the general problem of determining the extent to which the geometry of a riemannian manifold is determined by properties of its geodesic spheres. In particular we show that total umbilicity of geodesic spheres determines riemannian manifolds of constant sectional curvature; quasi-umbilicity of geodesic spheres determines Kähler and nearly-Kähler manifolds of constant holomorphic sectional curvature; and the condition that geodesic spheres have only two different principal curvatures, one having multiplicity 3, determines manifolds locally isometric to the quaternionic projective spaces. The use of Jacobi vector fields leads to a unified treatment of these different cases.

scholarly journals On totally umbilicalCR-submanifolds of a Kaehler manifold

1993 ◽  
Vol 16 (2) ◽  
pp. 405-408
Author(s):  
M. A. Bashir
Keyword(s):  
Sectional Curvature ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Lens Space ◽  
Holomorphic Sectional Curvature ◽  
Mean Curvature Vector ◽  
Kaehler Manifold ◽  
Totally Umbilical ◽  
The Mean

LetMbe a compact3-dimensional totally umbilicalCR-submanifold of a Kaehler manifold of positive holomorphic sectional curvature. We prove that if the length of the mean curvature vector ofMdoes not vanish, thenMis either diffeomorphic toS3orRP3or a lens spaceLp,q3.

scholarly journals On projectivized vector bundles and positive holomorphic sectional curvature

2018 ◽  
Vol 146 (7) ◽  
pp. 2877-2882 ◽  
Author(s):  
Angelynn Alvarez ◽  
Gordon Heier ◽  
Fangyang Zheng
Keyword(s):  
Sectional Curvature ◽  
Vector Bundles ◽  
Holomorphic Sectional Curvature
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