Let (X, Δ) be an n-dimensional ϵ-klt log ℚ-Fano pair. We give an upper bound for the volume Vol(X, Δ) = (-(KX + Δ))n when n = 2 , or n = 3 and X is ℚ-factorial of ρ(X)=1 . This bound is essentially sharp for . The main idea is to analyze the covering families of tigers constructed in J.Â McKernan (Boundedness of log terminal fano pairs of bounded index, preprint, 2002, arXiv:0205214). Existence of an upper bound for volumes is related to the Borisov-Alexeev-Borisov Conjecture, which asserts boundedness of the set of ϵ-klt log ℚ-Fano varieties of a given dimension.
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