Stability and coercivity for toric polarizations

Abstract

We introduce uniform K-stability and its relationship with the coercivity property of the K-energy functional, for general polarized manifolds. Since the automorphism groups are not necessarily finite, size of the norm measuring uniformity should be reduced with respect to the group action. About this point we explain that it is enough to take the reduced norm for a single sub-torus, actually the center, in the cscK problem. Our main theorem then describes the slope of the reduced J-functional along any torus-equivariant test configuration. In the toric case it is shown that the uniform stability is indeed equivalent to the coercivity of the K-energy. In the Fano manifolds case existence of the KE metric implies the uniform stability.

20 pages. We place the slope formula as the main theorem, in view of the previous research. Accordingly, presentations are fixed and Related discussions are added. Proof of the theorems in the previous versions are not changed

References in corpus (4)

• Tian's properness conjectures and Finsler geometry of the space of Kahler metrics
• A variational approach to the Yau-Tian-Donaldson conjecture
• Kähler-Einstein metrics along the smooth continuity method
• Orthogonal projection of a test configuration to vector fields

Cited by in corpus (15)

• On the constant scalar curvature Kähler metrics, general automorphism group
• Geodesic rays and stability in the cscK problem
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• Mabuchi Solitons and Relative Ding Stability of Toric Fano Varieties
• The inverse Monge-Ampere flow and applications to Kahler-Einstein metrics
• Uniform K-stability of polarized spherical varieties
• Relative Ding Stability and an Obstruction to the Existence of Mabuchi Solitons
• On K-stability of some del Pezzo surfaces of Fano index 2
• Generalized Kähler Einstein metrics and uniform stability for toric Fano manifolds
• Relative Ding and $K$-stability of toric Fano manifolds in low dimensions
• On Calabi's extremal metric and properness
• Openness of uniform K-stability in the Kähler cone
• An effective weighted K-stability condition for polytopes and semisimple principal toric fibratons
• A uniform version of the Yau-Tian-Donaldson correspondence for extremal Kähler metrics on polarized toric manifolds
• Kähler-Einstein metrics and Ding functional on $\mathbb{Q}$-Fano group compactifications