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# A Donaldson type functional on a holomorphic Finsler vector bundle

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*in*Mathematische Annalen · July 2015

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DOI: 10.1007/s00208-016-1472-4 · Source: arXiv

Cite this publicationAbstract

In this paper, we solve a problem of Kobayashi posed in \cite{Ko4} by
introducing a Donaldson type functional on the space $F^+(E)$ of strongly
pseudo-convex complex Finsler metrics on $E$ -- a holomorphic vector bundle
over a closed K\"ahler manifold $M$. This Donaldson type functional is a
generalization in the complex Finsler geometry setting of the original
Donaldson functional and has Finsler-Einstein metrics on $E$ as its only
critical points, at which this functional attains the absolute minimum.

- ... Chen's geodesic approximation lemma (cf. [5], Lemma 7; also [13], Lemma 2.3), we get the following theorem: Theorem 0.1. The functional L(·, ψ) attains its absolute minimum at the geodesic-Einstein metrics on L. The famous Donaldson-Uhlenbeck-Yau theorem reveals the deep relationship between the stability of a holomorphic vector bundle and the existence of Hermitian-Einstein metrics (cf. ...... As an application, in Section 3, we study the special triple (P (E), M, O P (E) (1)) associated to a holomorphic vector bundle E → M. By the Kobayashi correspondence (cf. [16], [12], [13]), a Finsler metric G on E induces a natural admissible metric on O P (E) (1). In this case, we can prove that the induced metric on O P (E) (1) is geodesic-Einstein if and only if G is FinslerEinstein. ...... Also recall that a Finsler-Einstein vector bundle E → M is semistable (cf. [13]). Here a natural question is whether a Finsler-Einstein vector bundle admits a Hermitian-Einstein metric. ...Article
- Oct 2017
- T AM MATH SOC

In this paper, we introduce notions of nonlinear stabilities for a relative ample line bundle over a holomorphic fibration and define the notion of a geodesic-Einstein metric on this line bundle, which generalize the classical stabilities and Hermitian-Einstein metrics of holomorphic vector bundles. We introduce a Donaldson type functional to show that this functional attains its absolute minimum at geodesic-Einstein metrics, and we also discuss the relationships between the existence of geodesic-Einstein metrics and the nonlinear stabilities of the line bundle. As an application, we will prove that a holomorphic vector bundle admits a Finsler-Einstein metric if and only if it admits a Hermitian-Einstein metric, which answers a problem posed by S. Kobayashi.

- Article
- Jan 1987

These notes are based on the lectures I delivered at the German Mathematical Society Seminar in Schloss Michkeln in DUsseldorf in June. 1986 on Hermitian-Einstein metrics for stable bundles and Kahler-Einstein metrics. The purpose of these notes is to present to the reader the state-of-the-art results in the simplest and the most comprehensible form using (at least from my own subjective viewpoint) the most natural approach. The presentation in these notes is reasonably self-contained and prerequisi tes are kept to a minimum. Most steps in the estimates are reduced as much as possible to the most basic procedures such as integration by parts and the maximum principle. When less basic procedures are used such as the Sobolev and Calderon-Zygmund inequalities and the interior Schauder estimates. references are given for the reader to look them up. A considerable amount of heuristic and intuitive discussions are included to explain why certain steps are used or certain notions introduced. The inclusion of such discussions makes the style of the presentation at some places more conversational than what is usually expected of rigorous mathemtical prese"ntations. For the problems of Hermi tian-Einstein metrics for stable bundles and Kahler-Einstein metrics one can use either the continuity method or the heat equation method. These two methods are so very intimately related that in many cases the relationship betwen them borders on equivalence. What counts most is the a. priori estimates. The kind of scaffolding one hangs the a. - Article
- Sep 2003
- J MATH KYOTO U

The purpose of this article is to reformulate the algebraic geometric concept of ampleness of a vector bundle E in differential geometric terms. - In the present paper, we shall be concerned with Einstein-Finsler bundles, and study the semi-stability of them.
- Article
- Jul 2015
- INT J MATH

In this paper, we present two kinds of total Chern forms $c(E,G)$ and $\mathcal{C}(E,G)$ as well as a total Segre form $s(E,G)$ of a holomorphic Finsler vector bundle $\pi:(E,G)\to M$ expressed by the Finsler metric $G$, which answers a question of J. Faran (\cite{Faran}) to some extent. As some applications, we show that the signed Segre forms $(-1)^ks_k(E,G)$ are positive $(k,k)$-forms on $M$ when $G$ is of positive Kobayashi curvature; we prove, under an extra assumption, that a Finsler-Einstein vector bundle in the sense of Kobayashi is semi-stable; we introduce a new definition of a flat Finsler metric, which is weaker than Aikou's one (\cite{Aikou}) and prove that a holomorphic vector bundle is Finsler flat in our sense if and only if it is Hermitian flat. - Article
- Jun 1983
- MANUSCRIPTA MATH

Einstein-Hermitian vector bundles are defined by a certain curvature condition. We prove that over a compact Khler manifold a bundle satisfying this condition is semistable in the sense of Mumford-Takemoto and a direct sum of stable Einstein-Hermitian subbundles. - Article
- Dec 2010
- ASIAN J MATH

The purpose of this paper is to investigate canonical metrics on a semi-stable vector bundle E over a compact Kahler manifold X. It is shown that, if E is semi-stable, then Donaldson's functional is bounded from below. This implies that E admits an approximate Hermitian-Einstein structure, generalizing a classic result of Kobayashi for projective manifolds to the Kahler case. As an application some basic properties of semi-stable vector bundles over compact Kahler manifolds are established, such as the fact that semi-stability is preserved under tensor product and certain exterior and symmetric products. - Article
- Jun 1975
- NAGOYA MATH J

A complex Finsler structure F on a complex manifold M is a function on the tangent bundle T(M) with the following properties. (We denote a point of T(M) symbolically by (z, ζ), where z represents the base coordinate and ζ the fibre coordinate.)