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arXiv · 2004.02832

Existence of cscK metrics on smooth minimal models

Zakarias Sjöström Dyrefelt
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Abstract

Given a compact K\"ahler manifold $X$ it is interesting to ask whether it admits a constant scalar curvature K\"ahler (cscK) metric. In this short note we show that there always exist cscK metrics on compact K\"ahler manifolds with nef canonical bundle, thus on all smooth minimal models, and also on the blowup of any such manifold. This confirms an expectation of Jian-Shi-Song \cite{JianShiSong} and extends their main result from $K_X$ semi-ample to $K_X$ nef, with a direct proof that does not appeal to the Abundance conjecture. As a byproduct we obtain that the connected component $\mathrm{Aut}_0(X)$ of a compact K\"ahler manifold with $K_X$ nef is either trivial or a complex torus.

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Zakarias Sjöström Dyrefelt. 2020-04-06. Existence of cscK metrics on smooth minimal models. https://arxiv.org/abs/2004.02832

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