Test Configurations for K-Stability and Geodesic Rays

Abstract

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

27 pages, no figure; references added; typos corrected

References in corpus (1)

• CM Stability and the Generalized Futaki Invariant I