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Singularities of linear systems and boundedness of Fano varieties

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arxiv 1609.05543 v2 pith:ED2OWWXX submitted 2016-09-18 math.AG

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classification math.AG
keywords canonicalfanovarietiesconjectureepsilonlinearnumberparticular
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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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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. On anticanonical volumes of weak $\mathbb{Q}$-Fano terminal threefolds of Picard rank two

    math.AG 2025-01 conditional novelty 7.0 of 10

    A weak Q-Fano threefold of Picard rank two has anticanonical volume at most 72, except in one subcase, with equality only for the projective bundle P(O⊕O(3)) over P^2.

  2. The K\"ahler-Ricci flow and quantitative bounds for Donaldson-Futaki invariants of optimal degenerations

    math.DG 2019-09 conditional novelty 7.0 of 10

    For any Fano manifold, the Donaldson-Futaki invariant of its optimal degeneration is bounded below by -(1-R(X))/R(X) nV, where R(X) is the greatest Ricci lower bound.

  3. A Supplement to the anticanonical Volumes of weak $\mathbb{Q}$-Fano threefolds of Picard rank two

    math.AG 2025-01 reject novelty 6.0 of 10

    For weak Q-Fano threefolds of Picard rank two, the anticanonical volume is at most 64 except for one explicit projective bundle, which has volume 72.

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