Bounding the volumes of singular Fano threefolds

Let $(X,Δ)$ be an $n$-dimensional $ε$-klt log $\QQ$-Fano pair. We give an upper bound for the volume ${\rm Vol}(-(K_X+Δ))=(-(K_X+Δ))^n$ when $n=2$ or $n=3$ and $X$ is {$\QQ$-factorial} of $ρ(X)=1$. 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 $ε$-klt log $\QQ$-Fano varieties of a given dimension $n$.

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