Bounding Volumes of Singular Fano threefolds

Let (X,[STIX]x1D6E5) be an n -dimensional [STIX]x1D716 -klt log Q -Fano pair. We give an upper bound for the volume Vol(X,[STIX]x1D6E5)=(-(KX+[STIX]x1D6E5))ⁿ when $n=2$ , or $n=3$ and X is Q -factorial of [STIX]x1D70C(X)=1 . This bound is essentially sharp for $n=2$ . The main idea is to analyze the covering families of tigers constructed in J. M c Kernan ( 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 [STIX]x1D716 -klt log Q -Fano varieties of a given dimension n .