Klt varieties of general type with small volume

Abstract

By Hacon-McKernan-Xu, there is a positive lower bound in each dimension for the volume of all klt varieties with ample canonical class. We show that these bounds must go to zero extremely fast as the dimension increases, by constructing a klt $n$-fold with ample canonical class whose volume is less than $1/2^{2^n}$. These examples should be close to optimal. We also construct a klt Fano variety of each dimension $n$ such that $H^0(X,-mK_X)=0$ for all $1\leq m < b$ with $b$ roughly $2^{2^n}$. Here again there is some bound in each dimension, by Birkar's theorem on boundedness of complements, and we are showing that the bound must increase extremely fast with the dimension.