Bounding the volumes of singular Fano threefolds

Abstract

Let $(X,\Delta )$ be an $n$-dimensional $\epsilon$-klt log $\QQ$-Fano pair. We give an upper bound for the volume $\mathrm{Vol}(-(K_X+\Delta ))=(-(K_X+\Delta ){)}^n$ when $n=2$ or $n=3$ and $X$ is {$\QQ$-factorial} of $\rho (X)=1$. This bound is essentially sharp for $n=2$. Existence of an upper bound for anticanonical volumes is related the Borisov-Alexeev-Borisov Conjecture which asserts boundedness of the set of $\epsilon$-klt log $\QQ$-Fano varieties of a given dimension $n$.