Geodesic rays in the space of K\"ahler metrics with $T$-symmetry

TL;DR

This paper constructs a geodesic ray in the space of K"ahler metrics on a torus-symmetric manifold, showing convergence of associated polarizations to a singular mixed polarization.

Contribution

It introduces a new construction of a geodesic ray in the K"ahler metric space with a specific polarization convergence result.

Findings

Constructed a singular mixed polarization on the manifold.

Developed a family of complex structures forming a geodesic ray.

Proved convergence of K"ahler polarizations to the mixed polarization.

Abstract

Let $(M, \omega, J)$ be a K\"ahler manifold, equipped with an effective Hamiltonian torus action $\rho: T \rightarrow \mathrm{Diff}(M, \omega, J)$ by isometries with moment map $\mu: M \rightarrow \mathfrak{t}^{*}$. We first construct a singular mixed polarization $\mathcal{P}_{\mathrm{mix}}$ on $M$. Second, we construct a one-parameter family of complex structures $J_{t}$ on $M$ which are compatible with $\omega$. Furthermore, the path of corresponding K\"ahler metrics $g_{t}$ is a complete geodesic ray in the space of K\"ahler metrics of $M$, when $M$ is compact. Finally, we show that the corresponding family of K\"ahler polarizations $\mathcal{P}_{t}$ associated to $J_{t}$ converges to $\mathcal{P}_{\mathrm{mix}}$ as $t \rightarrow \infty$.