Remarks on the geodesic-Einstein metrics of a relative ample line bundle
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Mathematische Zeitschrift
Remarks on the geodesic-Einstein metrics of a relative ample line bundle Xueyuan Wan1 · Xu Wang2 Received: 27 November 2018 / Accepted: 23 November 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we introduce the associated geodesic-Einstein flow for a relative ample line bundle L over the total space X of a holomorphic fibration and obtain a few properties of that flow. In particular, we prove that the pair (X , L) is nonlinear semistable if the associated Donaldson type functional is bounded from below and the geodesic-Einstein flow has longtime existence property. We also define the associated S-classes and C-classes for (X , L) and obtain two inequalities between them when L admits a geodesic-Einstein metric. Finally, in the appendix of this paper, we prove that a relative ample line bundle is geodesic-Einstein if and only if an associated infinite rank bundle is Hermitian–Einstein.
Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Geodesic-Einstein flow and nonlinear semistable . . . . . . . . . . . . . . . . 2.1 Some properties of the geodesic-Einstein flow . . . . . . . . . . . . . . . 2.2 Nonlinear semistable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The case of trω c(φ) ≥ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 S-class and C-class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Some inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Positivity of classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Appendix: Hermitian–Einstein metrics on quasi-vector bundles (By Xu Wang) References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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An appendix by Xu Wang.
B
Xueyuan Wan [email protected] Xu Wang [email protected]
1
School of Mathematics, KIAS, Heogiro 85, Dongdaemun-gu, Seoul 02455, Republic of Korea
2
Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway
123
X. Wan, X. Wang
Introduction Let E be a holomorphic vector bundle over a compact Kähler manifold (M, ω). A Hermitian metric h on E is called a Hermitian–Einstein metric if ω Fh = λId for some constant λ, ¯ · h −1 ) ∈ A1,1 (M, End(E)) is the Chern curvature of the Hermitian metric where Fh = ∂(∂h h. 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
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