On minimal log discrepancies and kollár components

Abstract

In this article, we prove a local implication of boundedness of Fano varieties. More precisely, we prove that $d$ -dimensional $a$ -log canonical singularities with standard coefficients, which admit an $\epsilon$ -plt blow-up, have minimal log discrepancies belonging to a finite set which only depends on $d,\,a$ and $\epsilon$ . This result gives a natural geometric stratification of the possible mld's in a fixed dimension by finite sets. As an application, we prove the ascending chain condition for minimal log discrepancies of exceptional singularities. We also introduce an invariant for klt singularities related to the total discrepancy of Kollár components.

Running a Minimal Model Program

“…Among the results on conjectures surrounding the MMP in the last few years there have been important results on effectiveness of Iitaka fibrations [BZ16], log discrepancies of singularities [Mor21], and termination of flips [HM20].…”

The Jordan property for local fundamental groups

“…By [Bir19], we know that for every n-dimensional klt singularity x ∈ X, we may find a boundary B ≥ 0 on X so that (X, B) is log canonical, it is not klt, and m(K X + B) ∼ 0, where m is a constant only depending on n. Indeed, while the statement of [Bir19, Theorem 1.8] does not mention it explicitly, the complements constructed there are strictly log canonical, see Step 1 in the proof of [Bir19, Proposition 8.1]. Hence, a n-dimensional exceptional klt singularity admits a 1 m -plt blow-up in the sense of [Mor18]. From the proof of Theorem 4.11, it follows that there is an exact sequence 1 → Z k → π loc 1 (X, x) → N (n) → 1, where N (n) is a group whose order is bounded by a constant that only depends on n. Moreover, if we assume that (X, B) is ǫ-log canonical for some ǫ > 0, and argue as in the proof of [Mor18, Theorem 1], it follows that k is bounded by a constant only depending on n and ǫ.…”

Bounding toric singularities with normalized volume

“…The combinatorial nature of toric singularities allows us to use methods from convex geometry to study them [6]. There are two important invariants for log terminal singularities, the normalized volume [22] and the minimal log discrepancy [28, 29]. The former is related to K‐stability [20], while the latter is related to termination of flips [31].…”