Singularities of linear systems and boundedness of Fano varieties

Caucher Birkar

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arXiv · 1609.05543

Singularities of linear systems and boundedness of Fano varieties

Caucher Birkar

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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.

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Caucher Birkar. 2016-09-18. Singularities of linear systems and boundedness of Fano varieties. https://arxiv.org/abs/1609.05543

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