Equivariant ℝ-test configurations and semistable limits of ℚ-Fano group compactifications

Yan Li; ZhenYe Li

Paper · Mar 11, 2021

Equivariant $R$ -test configurations and semistable limits of $Q$ -Fano group compactifications

Yan Li, ZhenYe Li

math.DGarXiv:2103.06439v6

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Abstract

Let $G$ be a connected, complex reductive group. In this paper, we classify $G \times G$ -equivariant normal $R$ -test configurations of a polarized $G$ -compactification. Then for $Q$ -Fano $G$ -compactifications, we express the H-invariant of its equivariant normal $R$ -test configurations in terms of the combinatory data. Based on, we compute the semistable limit of a K-unstable Fano $G$ -compactification. As an application, we show that for the two K-unstable Fano $S O_{4} (C)$ -compactifications, the corresponding semistable limits are indeed the limit spaces of the normalized Kähler-Ricci flow.

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Submitted: Mar 11, 2021
Last revised: Jun 1, 2022 (v6)
Subjects: math.DG

Comments: We added a section to show the equivariance of the minimizer. Also, minor statements are refined

Ricci flow Kähler–Ricci flow Kähler geometry Stability & uniqueness Kähler–Einstein & K-stability

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Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Kähler–Ricci flow Ricci flow started from a Kähler metric stays Kähler and reduces to a single scalar equation, a parabolic complex Monge–Ampère equation. Start with: Deformation of Kähler matrics to Kähler-Einstein metrics on compact Kähler manifolds (Cao, 1985) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

In the guidebook

5A.6
Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry
The Kähler–Ricci flow, and when to come back to it.
5A.5
Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.
10A.12
Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization
How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

[1] 5A.6
Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry
The Kähler–Ricci flow, and when to come back to it.

[2] 5A.5
Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

[3] 10A.12
Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization
How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

Equivariant $R$ -test configurations and semistable limits of $Q$ -Fano group compactifications

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Equivariant R-test configurations and semistable limits of Q-Fano group compactifications

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Equivariant $R$ -test configurations and semistable limits of $Q$ -Fano group compactifications