Klt varieties of general type with small volume

TL;DR

This paper demonstrates that the minimal volume of klt varieties of general type with ample canonical class decreases exponentially fast as the dimension increases, providing near-optimal examples and exploring bounds in higher dimensions.

Contribution

It constructs explicit examples of klt varieties with extremely small volume and boundedness properties, showing how these bounds must grow rapidly with dimension.

Findings

Constructed klt n-folds with volume less than 1/2^{2^n}.

Built klt Fano varieties with vanishing sections for certain multiples of the canonical bundle.

Showed bounds on volume and sections grow exponentially with dimension.

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.