On the Iitaka volumes of log canonical surfaces and threefolds

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.