On the Iitaka volumes of log canonical surfaces and threefolds

Abstract

Given positive integers $d\ge \hat{I}º$, and a subset $\hat{I}“\subset [0,1]$, let ${\mathrm{Ivol}}_{\mathrm{lc}}^{\hat{I}}“(d,\hat{I}º)$ denote the set of Iitaka volumes of $d$-dimensional projective log canonical pairs $(X,B)$ such that the Iitaka--Kodaira dimension $\hat{I}º(K_X+B)=\hat{I}º$ and the coefficients of $B$ come from $\hat{I}“$. In this paper, we show that, if $\hat{I}“$ satisfies the descending chain condition, then so does ${\mathrm{Ivol}}_{\mathrm{lc}}^{\hat{I}}“(d,\hat{I}º)$ for $d\le 3$. In case $d\le 3$ and $\hat{I}º=1$, $\hat{I}“$ and ${\mathrm{Ivol}}_{\mathrm{lc}}^{\hat{I}}“(d,\hat{I}º)$ 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 $\ge d-2$ 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.

38 pages