Singularities of linear systems and boundedness of Fano varieties

By C. Birkar. Annals of Mathematics, 2016

291 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/37b0ec9dc369dcda843cf88755bf44522a11a7c9

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. H. Blum
2. Mattias Jonsson
3. Brian Lehmann
4. Sho Tanimoto
5. Chi Li
6. C. Birkar
7. Ziquan Zhuang
8. Yuchen Liu

Citations in the corpus

• Anti-pluricanonical systems on Fano varieties (2016, methodology): “ pairs and treat 1.3 and 1.11 inductively (eg, 7.2, 7.5, 7.9, 7.15), and give a criterion for a family of Fano varieties to be bounded (7.13) which is also a crucial ingredient of the proof of 1.5 in [5]. In Section 8, we discuss complements in the relative setting (8.2). In Section 9, we prove 1.6.”
• ACC for Minimal Log Discrepancies of Exceptional Singularities (2019, unknown): “By [4, Theorem 1.1], X is bounded.”
• Geometric Manin’s conjecture and rational curves (2017, background): “ional on a version of the BorisovAlexeev-Borisov Conjecture. [HJ16, Theorem 1.1] noticed that when L is big and semiample one can apply the weaker boundedness statements of [Bir16a] and [DC16]. Since [Bir16b] has settled the Borisov-Alexeev-Borisov Conjecture, the semiampleness restriction is no lon”
• Thresholds, valuations, and K-stability (2017, result): “me is established. It is also reminiscent of results in [JM12] on the existence of valuations computing log canonical thresholds of graded sequence of ideals, and related to a recent result by Birkar [Bir16] on the existence of Q-divisors achieving the inmum in the denition of lct(L) in the Q-Fano c”