Hermitian–Einstein metrics on holomorphic vector bundles over Hermitian manifolds

2005 ◽  
Vol 53 (3) ◽  
pp. 315-335 ◽  
Author(s):  
Zhang Xi
Keyword(s):  
Vector Bundles ◽  
Einstein Metrics ◽  
Holomorphic Vector Bundles ◽  
Hermitian Manifolds

Hermitian–Einstein metrics and jumping lines forSp(n+1)-homogeneous bundles over ℙ2n+1

1993 ◽  
Vol 114 (3) ◽  
pp. 443-451
Author(s):  
Al Vitter
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Vector Bundles ◽  
Einstein Metrics ◽  
Einstein Metric ◽  
Holomorphic Vector Bundles ◽  
Kähler Form ◽  
Points Of View ◽  
The Stability ◽  
Complex Projective

Stable holomorphic vector bundles over complex projective space ℙnhave been studied from both the differential-geometric and the algebraic-geometric points of view.On the differential-geometric side, the stability ofE-→ ℙncan be characterized by the existence of a unique hermitian–Einstein metric onE, i.e. a metric whose curvature matrix has trace-free part orthogonal to the Fubini–Study Kähler form of ℙn(see [6], [7], and [13]). Very little is known about this metric in general and the only explicit examples are the metrics on the tangent bundle of ℙnand the nullcorrelation bundle (see [9] and [10]).

scholarly journals STABILITY AND HERMITIAN–EINSTEIN METRICS FOR VECTOR BUNDLES ON FRAMED MANIFOLDS

2012 ◽  
Vol 23 (09) ◽  
pp. 1250091
Author(s):  
MATTHIAS STEMMLER
Keyword(s):  
Vector Bundles ◽  
Einstein Metrics ◽  
Coherent Sheaf ◽  
Einstein Metric ◽  
Complex Manifolds ◽  
Holomorphic Vector Bundles ◽  
Compact Complex Manifolds ◽  
Torsion Free

We adapt the notions of stability of holomorphic vector bundles in the sense of Mumford–Takemoto and Hermitian–Einstein metrics in holomorphic vector bundles for canonically polarized framed manifolds, i.e. compact complex manifolds X together with a smooth divisor D such that KX ⊗ [D] is ample. It turns out that the degree of a torsion-free coherent sheaf on X with respect to the polarization KX ⊗ [D] coincides with the degree with respect to the complete Kähler–Einstein metric gX\D on X\D. For stable holomorphic vector bundles, we prove the existence of a Hermitian–Einstein metric with respect to gX\D and also the uniqueness in an adapted sense.

EXISTENCE OF EXTREMAL METRICS ON ALMOST HOMOGENEOUS MANIFOLDS OF COHOMOGENEITY ONE — III

2003 ◽  
Vol 14 (03) ◽  
pp. 259-287 ◽  
Author(s):  
DANIEL GUAN
Keyword(s):  
Vector Bundles ◽  
Einstein Metrics ◽  
Extremal Metrics ◽  
Cohomogeneity One ◽  
Holomorphic Vector Bundles ◽  
Extremal Metric ◽  
First Chern Class ◽  
The Stability ◽  
Kähler Einstein Metrics ◽  
Calabi Extremal Metrics

In this paper we prove that on certain manifolds Nn with nonnegative first Chern class the existence of extremal metric in a Kähler class is the same as the stability of the Kähler class. We also obtain many new Kähler classes with extremal metrics, in particular, the Kähler-Einstein metrics for Nn with n > 2. We also compare the problem of finding Calabi extremal metrics with the similar problem of finding Hermitian–Einstein metrics on the holomorphic vector bundles. We explain the geodesic stability and found that the stability for the manifold is much more complicated

Holomorphic Vector Bundles and the Geometry of ℙn

1980 ◽  
pp. 1-138
Author(s):  
Christian Okonek ◽  
Michael Schneider ◽  
Heinz Spindler
Keyword(s):  
Vector Bundles ◽  
Holomorphic Vector Bundles

Complex Finsler structures on tensor products

2016 ◽  
Vol 16 (1) ◽  
Author(s):  
Adnène Ben Abdesselem ◽  
Ines Adouan
Keyword(s):  
Vector Bundles ◽  
Tensor Products ◽  
Holomorphic Vector Bundles

AbstractGiven two holomorphic vector bundles E

scholarly journals Holomorphic Vector Bundles on Holomorphically Convex Complex Manifolds

2006 ◽  
Vol 13 (1) ◽  
pp. 7-10
Author(s):  
Edoardo Ballico
Keyword(s):  
Vector Bundle ◽  
Complex Manifold ◽  
Vector Bundles ◽  
Open Neighborhood ◽  
Holomorphic Vector Bundle ◽  
Stein Manifold ◽  
Complex Manifolds ◽  
Holomorphic Vector Bundles

Abstract Let 𝑋 be a holomorphically convex complex manifold and Exc(𝑋) ⊆ 𝑋 the union of all positive dimensional compact analytic subsets of 𝑋. We assume that Exc(𝑋) ≠ 𝑋 and 𝑋 is not a Stein manifold. Here we prove the existence of a holomorphic vector bundle 𝐸 on 𝑋 such that is not holomorphically trivial for every open neighborhood 𝑈 of Exc(𝑋) and every integer 𝑚 ≥ 0. Furthermore, we study the existence of holomorphic vector bundles on such a neighborhood 𝑈, which are not extendable across a 2-concave point of ∂(𝑈).

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