On the Iitaka volumes of log canonical surfaces and threefolds

TL;DR

This paper investigates the set of Iitaka volumes for log canonical pairs, establishing the descending chain condition in low dimensions and exploring topological properties and specific cases for surfaces.

Contribution

It proves the DCC for Iitaka volumes of log canonical pairs in dimensions up to three and describes detailed properties for certain classes of surfaces.

Findings

Iitaka volume sets satisfy DCC for dimensions ≤3.

Sets of Iitaka volumes are closed and have finitely many accumulation points in certain cases.

Explicit descriptions of Iitaka volumes for specific surface classes.

Abstract

Given positive integers $d\geq\kappa$, and a subset $\Gamma\subset [0,1]$, let $\mathrm{Ivol}_{\mathrm{lc}}^{\Gamma}(d,\kappa)$ denote the set of Iitaka volumes of $d$-dimensional projective log canonical pairs $(X, B)$ such that the Iitaka--Kodaira dimension $\kappa(K_X+B)=\kappa$ and the coefficients of $B$ come from $\Gamma$. In this paper, we show that, if $\Gamma$ satisfies the descending chain condition, then so does $\mathrm{Ivol}_\mathrm{lc}^\Gamma(d,\kappa)$ for $d\leq 3$. In case $d\leq 3$ and $\kappa=1$, $\Gamma$ and $\mathrm{Ivol}_\mathrm{lc}^\Gamma(d,\kappa)$ are shown to share more topological properties, such as closedness in $\mathbb{R}$ and local finiteness of accumulation complexity. In higher dimensions, we show that the set of Iitaka volumes for $d$-dimensional klt pairs with Iitaka dimension $\geq d-2$ satisfies the DCC, partially confirming a conjecture of Zhan Li. We give a more detailed description of the sets of Iitaka volumes for the following classes of projective log canonical surfaces: (1) smooth properly elliptic surfaces, (2) projective log canonical surfaces with coefficients from $\{0\}$ or $\{0,1\}$. In particular, the minima as well as the minimal accumulation points are found in these cases.