Test configurations for K-stability and geodesic rays

By Duong H. Phong, Jacob Sturm. 2006

114 citations recorded by the source corpus.

Let $X$ be a compact complex manifold, $L\to X$ an ample line bundle over $X$, and ${\cal H}$ the space of all positively curved metrics on $L$. We show that a pair $(h_0,T)$ consisting of a point $h_0\in {\cal H}$ and a test configuration $T=({\cal L}\to {\cal X}\to {\bf C})$, canonically determines a weak geodesic ray $R(h_0,T)$ in ${\cal H}$ which emanates from $h_0$. Thus a test configuration behaves like a vector field on the space of K\"ahler potentials ${\cal H}$. We prove that $R$ is non-trivial if the ${\bf C}^\times$ action on $X_0$, the central fiber of $\cal X$, is non-trivial. The ray $R$ is obtained as limit of smooth geodesic rays $R_k\subset{\cal H}_k$, where ${\cal H}_k\subset{\cal H}$ is the subspace of Bergman metrics.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7362fd8695fdfafc4d46ed6c56322d60117b1064

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Geometric pluripotential theory on Kähler manifolds (2019, methodology): “For results about the quantization of the geometry of the space of Kähler metrics, we refer to the original papers [18, 42, 66, 89, 90, 54], as well as the survey [91] along with references therein.”
Extremal metrics and K-stability (PhD thesis) (2006, background): “In fact as was shown in [28] such a test-configuration can be used to define a weak geodesic ray of metrics in the Kähler class although we will not use this.”
Filtrations and test-configurations (2011, background): “For more in this direction see for example Phong–Sturm [22].”
Geodesic rays and stability in the cscK problem (2020, background): “Let Φm be the geodesic ray associated to (Xm,Lm) (as constructed in [59] or in [5, Theorem 6.”
C1,1 regularity for degenerate complex Monge–Ampère equations and geodesic rays (2017, background): “2) below, which is related to the asymptotic behavior of theMabuchi energy (see also [15, 33] for earlier results along these lines).”