scholarly journals Singular hermitian metrics on holomorphic vector bundles

2015 ◽  
Vol 53 (2) ◽  
pp. 359-382 ◽  
Author(s):  
Hossein Raufi
Keyword(s):  
Vector Bundles ◽  
Hermitian Metrics ◽  
Holomorphic Vector Bundles

scholarly journals On a functional of Kobayashi for Higgs bundles

2020 ◽  
Vol 17 (13) ◽  
pp. 2050200
Author(s):  
Sergio A. H. Cardona ◽  
Claudio Meneses
Keyword(s):  
Mean Curvature ◽  
Vector Bundles ◽  
Kähler Manifold ◽  
Higgs Bundle ◽  
Inner Product ◽  
Higgs Bundles ◽  
Hermitian Metrics ◽  
Yang Mills ◽  
Holomorphic Vector Bundles ◽  
Kahler Manifold

We define a functional [Formula: see text] for the space of Hermitian metrics on an arbitrary Higgs bundle over a compact Kähler manifold, as a natural generalization of the mean curvature energy functional of Kobayashi for holomorphic vector bundles, and study some of its basic properties. We show that [Formula: see text] is bounded from below by a nonnegative constant depending on invariants of the Higgs bundle and the Kähler manifold, and that when achieved, its absolute minima are Hermite–Yang–Mills metrics. We derive a formula relating [Formula: see text] and another functional [Formula: see text], closely related to the Yang–Mills–Higgs functional, which can be thought of as an extension of a formula of Kobayashi for holomorphic vector bundles to the Higgs bundles setting. Finally, using 1-parameter families in the space of Hermitian metrics on a Higgs bundle, we compute the first variation of [Formula: see text], which is expressed as a certain [Formula: see text]-Hermitian inner product. It follows that a Hermitian metric on a Higgs bundle is a critical point of [Formula: see text] if and only if the corresponding Hitchin–Simpson mean curvature is parallel with respect to the Hitchin–Simpson connection.

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]).

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
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