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TITLE Remarks on the geodesic-Einstein metrics of a relative ample line bundle
KIAS AUTHORS Wan, Xueyuan
JOURNAL MATHEMATISCHE ZEITSCHRIFT, 2020
ARCHIVE  
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 (chi, 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 (chi, 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.
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