Singularities of linear systems and boundedness of Fano varieties

TL;DR

This paper proves lower bounds for log canonical thresholds, confirms the Borisov-Alexeev-Borisov conjecture on boundedness of Fano varieties with certain singularities, and addresses related questions in algebraic geometry.

Contribution

It establishes the existence of positive lower bounds for log canonical thresholds, proves the boundedness of Fano varieties with specified singularities, and answers key questions regarding their automorphism groups and thresholds.

Findings

Confirmed the Borisov-Alexeev-Borisov conjecture.

Proved that automorphism groups of rationally connected varieties are Jordan.

Showed that the log canonical threshold of the anti-canonical system is computed by a divisor.

Abstract

We study log canonical thresholds (also called global log canonical threshold or $\alpha$-invariant) of $\mathbb{R}$-linear systems. We prove existence of positive lower bounds in different settings, in particular, proving a conjecture of Ambro. We then show that the Borisov-Alexeev-Borisov conjecture holds, that is, given a natural number $d$ and a positive real number $\epsilon$, the set of Fano varieties of dimension $d$ with $\epsilon$-log canonical singularities forms a bounded family. This implies that birational automorphism groups of rationally connected varieties are Jordan which in particular answers a question of Serre. Next we show that if the log canonical threshold of the anti-canonical system of a Fano variety is at most one, then it is computed by some divisor, answering a question of Tian in this case.