Regularity of weak minimizers of the K-energy and applications to properness and K-stability

By Robert J. Berman, Tam'as Darvas, Chinh H. Lu. Annales Scientifiques de l'Ecole Normale Supérieure, 2016

116 citations recorded by the source corpus.

Let $(X,\omega)$ be a compact K\"ahler manifold and $\mathcal H$ the space of K\"ahler metrics cohomologous to $\omega$. If a cscK metric exists in $\mathcal H$, we show that all finite energy minimizers of the extended K-energy are smooth cscK metrics, partially confirming a conjecture of Y.A. Rubinstein and the second author. As an immediate application, we obtain that existence of a cscK metric in $\mathcal H$ implies J-properness of the K-energy, thus confirming one direction of a conjecture of Tian. Exploiting this properness result we prove that an ample line bundle $(X,L)$ admitting a cscK metric in $c_1(L)$ is $K$-polystable.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ffa6aa01d09003c7acda2aae0534b3fc7ba5f288

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In Applied Mathematics

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Citations in the corpus

Geodesic rays and stability in the cscK problem (2020, background): “] in the smooth case and shows that, when Xis smooth Fano, one could avoid the use of valuative criterion by a direct estimate of slope of J (see (145)).2 Proof of Theorem 5.12. By the previous works [5, 6, 19, 20, 28] on analytic criterions for the existence of cscK, we just need to show that M is ”
Uniform K-stability and asymptotics of energy functionals in Kähler geometry (2016, background): “nstein manifold without vector elds. This was rst established in the Fano case by [PSSW08]; an elegant proof can be found in [DR15]. As a special case of a very recent result of Berman, Darvas and Lu [BDL16], coercivity of the Mabuchi functional also holds for general polarized varieties admitting a”