Abstract
In this paper, we introduce the associated geodesic-Einstein flow for a relative ample line bundle L over the total space \(\mathcal {X}\) of a holomorphic fibration and obtain a few properties of that flow. In particular, we prove that the pair \((\mathcal {X}, L)\) is nonlinear semistable if the associated Donaldson type functional is bounded from below and the geodesic-Einstein flow has long-time existence property. We also define the associated S-classes and C-classes for \((\mathcal {X}, L)\) 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.
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An appendix by Xu Wang.
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Wan, X., Wang, X. Remarks on the geodesic-Einstein metrics of a relative ample line bundle. Math. Z. 296, 987–1010 (2020). https://doi.org/10.1007/s00209-020-02463-2
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DOI: https://doi.org/10.1007/s00209-020-02463-2