scholarly journals Approximations and examples of singular Hermitian metrics on vector bundles

2017 ◽  
Vol 55 (1) ◽  
pp. 131-153 ◽  
Author(s):  
Genki Hosono
Keyword(s):  
Vector Bundles ◽  
Hermitian Metrics

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.

scholarly journals Singular hermitian metrics on vector bundles

1998 ◽  
Vol 1998 (502) ◽  
pp. 93-122 ◽  
Author(s):  
MARC ANDREA A. DE CATALDO
Keyword(s):  
Vector Bundles ◽  
Hermitian Metrics

THE OPTIMAL JET EXTENSION OF OHSAWA–TAKEGOSHI TYPE

2018 ◽  
Vol 239 ◽  
pp. 153-172 ◽  
Author(s):  
GENKI HOSONO
Keyword(s):  
Vector Bundles ◽  
Extension Theorem ◽  
Optimal Estimate ◽  
Hermitian Metrics

We prove the $L^{2}$ extension theorem for jets with optimal estimate following the method of Berndtsson–Lempert. For this purpose, following Demailly’s construction, we consider Hermitian metrics on jet vector bundles.

scholarly journals ON PROJECTIVE MANIFOLDS WITH PSEUDO-EFFECTIVE TANGENT BUNDLE

2021 ◽  
pp. 1-30
Author(s):  
Genki Hosono ◽  
Masataka Iwai ◽  
Shin-ichi Matsumura
Keyword(s):  
Minimal Surfaces ◽  
Tangent Bundle ◽  
Vector Bundles ◽  
Structure Theorem ◽  
Projective Manifold ◽  
Hermitian Metrics ◽  
Rationally Connected ◽  
Projective Manifolds ◽  
Positively Curved ◽  
Tangent Bundles

Abstract In this paper, we develop the theory of singular Hermitian metrics on vector bundles. As an application, we give a structure theorem of a projective manifold X with pseudo-effective tangent bundle; X admits a smooth fibration $X \to Y$ to a flat projective manifold Y such that its general fibre is rationally connected. Moreover, by applying this structure theorem, we classify all the minimal surfaces with pseudo-effective tangent bundle and study general nonminimal surfaces, which provide examples of (possibly singular) positively curved tangent bundles.

Banach Lie Algebroids

2011 ◽  
Vol 57 (2) ◽  
pp. 409-416
Author(s):  
Mihai Anastasiei
Keyword(s):  
Differential Equations ◽  
Smooth Manifold ◽  
Vector Bundles ◽  
Second Order ◽  
Lie Algebroids ◽  
Second Order Differential Equations

Banach Lie AlgebroidsFirst, we extend the notion of second order differential equations (SODE) on a smooth manifold to anchored Banach vector bundles. Then we define the Banach Lie algebroids as Lie algebroids structures modeled on anchored Banach vector bundles and prove that they form a category.

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