Analytic test configurations and geodesic rays

By J. Ross, David Witt Nyström. 2011

120 citations recorded by the source corpus.

Starting with the data of a curve of singularity types, we use the Legendre transform to construct weak geodesic rays in the space of locally bounded metrics on an ample line bundle L over a compact manifold. Using this we associate weak geodesics to suitable filtrations of the algebra of sections of L. In particular this works for the natural filtration coming from an algebraic test configuration, and we show how in this case we recover the weak geodesic ray of Phong-Sturm.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0e0736497d46e8be398b8627b55e7a31423ef909

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Log-concavity of volume and complex Monge–Ampère equations with prescribed singularity (2018, unknown): “Thesewere introduced byRoss andWitt Nyström [35] in their construction of geodesic rays, using slightly different notation.”
Smooth and singular Kahler-Einstein metrics (2014, background): “Recently, Ross–Witt-Nyström introduced the notion of an analytic test configuration [212].”
Filtrations and test-configurations (2011, background): “A key new ingredient in the proof is the Okounkov body [21], and the concave (in our case convex) transform of a filtration introduced by Boucksom–Chen [4], which was also used in the context of test-configurations by Witt Nyström [31].”
Geodesic rays and stability in the cscK problem (2020, background): “For the construction of such geodesic rays, we refer to [33, 59, 62] and references therein.”