On the Iitaka volumes of log canonical surfaces and threefolds

Abstract

Given positive integers $d⩾\kappa$ and a subset $\Gamma \subset [0,1]$, let ${Ivol}_{\mathrm{lc}}^{\Gamma }(d,\kappa )$ 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 ${Ivol}_{\mathrm{lc}}^{\Gamma }(d,\kappa )$ for $d⩽3$. In case $d⩽3$ and $\kappa =1$, $\Gamma$ and ${Ivol}_{\mathrm{lc}}^{\Gamma }(d,\kappa )$ are shown to share more topological properties, such as closedness in $R$ 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 $⩾d-2$ 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 $\left\{0\right\}$ or $\{0,1\}$. In particular, the minima as well as the minimal accumulation points are found in these cases.