On the limit of spectral measures associated to a test configuration

Abstract

We apply the integral formula of volumes to the family of graded linear series constructed from any test configuration. This solves the conjecture raised by Witt--Nyström so that the sequence of spectral measures for the induced ${\mathbb{C}}^{\ast }$-action on the central fiber converges to the canonical Duistermatt--Heckman measure defined by the associated weak geodesic ray. As a consequence, we show that the algebraic $p$-norm of the test configuration equals to the $L^p$-norm of tangent vectors. Using this result, We may give a natural energy theoretic explanation for the lower bound estimate on the Calabi functional by Donaldson and prove the analogous result for the Kähler--Einstein metric.

20 pages

References in corpus (10)

• CM Stability and the Generalized Futaki Invariant I
• A Brunn-Minkowski type inequality for Fano manifolds and the Bando-Mabuchi uniqueness theorem
• K-stability of constant scalar curvature polarization
• A stronger concept of K-stability
• K-stability of constant scalar curvature Kähler manifolds
• Probability measures related to geodesics in the space of Kähler metrics
• Bergman kernels and equilibrium measures for ample line bundles
• A variational approach to complex Monge-Ampere equations
• An obstruction to the existence of constant scalar curvature Kähler metrics
• The Calabi conjecture and K-stability

Cited by in corpus (1)

• Orthogonal projection of a test configuration to vector fields