Degeneration of Kahler polarizations to mixed polarizations on toric varieties

TL;DR

This paper investigates how Kahler polarizations on toric varieties degenerate into mixed and real polarizations through Hamiltonian actions, and studies the convergence of associated holomorphic sections.

Contribution

It constructs a family of polarizations on toric varieties that degenerate from Kahler to mixed and real polarizations, analyzing their properties and convergence behavior.

Findings

Polarizations $\\shP_{k}$ are singular mixed for $1\le k < n$

The polarization $\shP_{n}$ coincides with a previously studied real polarization

Holomorphic sections $\shH_{k,t}^{T}$ converge to $\shH_{k}^{0}$

Abstract

Let $(X, \omega, J)$ be a toric variety of dimension $2n$ determined by a Delzant polytope. In this paper, we first construct the polarizations $\shP_{k}$ by the Hamiltonian $T^{k}$-action on $X$ (see Theorem 3.11). We will show that $\shP_{k}$ is a singular mixed polarization for $1\le k < n$, and $\shP_{n}$ is a singular real polarization which coincides with the real polarization studied in \cite{BFMN} on the open dense subset of $X$. Then for each $1\le k \le n$, we will find a one-parameter family of K\"ahler polarizations $\shP_{k,t}$ on $X$ that converges to $\shP_{k}$ (see Theorem 3.12). Finally, we will show that $\shH_{k,t}^{T}$ the space of $T^{k}$-invariant $J_{k,t}$-holomorphic sections converges to $\shH_{k}^{0}$ (see Theorem 3.18).