[Paper Review] A gap theorem for minimal log discrepancies of non-canonical singularities in dimension three

Chen Jiang|arXiv (Cornell University)|Apr 21, 2019

Algebraic Geometry and Number Theory49 references23 citations

TL;DR

This paper establishes a 1-gap theorem for minimal log discrepancies (mld) in 3-dimensional non-canonical singularities, proving there exists a positive δ > 0 such that if a normal quasi-projective Q-Gorenstein 3-fold has mld < 1, then mld ≤ 1 − δ. The proof reduces the problem to isolated hyperquotient singularities and uses a modified version of Mori and Reid's classification method for terminal 3-folds, leading to boundedness of non-canonical klt Calabi–Yau 3-folds modulo flops and uniform boundedness of global indices.

ABSTRACT

We show that there exists a positive real number $\delta>0$ such that for any normal quasi-projective $\mathbb{Q}$-Gorenstein $3$-fold $X$, if $X$ has worse than canonical singularities, that is, the minimal log discrepancy of $X$ is less than $1$, then the minimal log discrepancy of $X$ is not greater than $1-\delta$. As applications, we show that the set of all non-canonical klt Calabi-Yau $3$-folds are bounded modulo flops, and the global indices of all klt Calabi-Yau $3$-folds are bounded from above.

Motivation & Objective

• To resolve the 1-gap conjecture for minimal log discrepancies in dimension 3, proving that mld cannot accumulate at 1 from below for non-canonical singularities.
• To establish boundedness of non-canonical klt Calabi–Yau 3-folds modulo flops, extending Alexeev's 2D result to dimension 3.
• To prove that the global indices of all klt Calabi–Yau 3-folds are bounded from above, generalizing results from canonical and surface cases.
• To remove the rational connectedness assumption in prior boundedness results for klt Calabi–Yau 3-folds.

Proposed method

• Reduces the problem to the case of extremely non-canonical singularities, where all but one exceptional divisor have log discrepancy > 1.
• Focuses on isolated hyperquotient singularities that are quotients of cDV singularities in A⁴, using a modified version of the classification method for 3-dimensional terminal singularities.
• Applies the terminal lemma and non-canonical lemma to analyze the structure of such singularities and derive contradictions if mld > 1 − δ.
• Uses the Global ACC theorem to ensure uniform klt thresholds and applies log boundedness techniques via MMP with scaling.
• Employs log resolution and relative MMP over a base scheme to contract the exceptional divisor and construct a bounded family modulo flops.
• Leverages effective bounds from prior work (e.g., δ₀ = 1/19 from Liu and Xiao) to derive an effective δ = 1/13 for the main gap theorem.

Experimental results

Research questions

• RQ1 Is there a positive gap δ > 0 such that mld(X) ≤ 1 − δ for any 3-fold X with non-canonical singularities (i.e., mld(X) < 1)?
• RQ2 Can the set of non-canonical klt Calabi–Yau 3-folds be bounded modulo flops, without assuming rational connectedness?
• RQ3 Are the global indices of all klt Calabi–Yau 3-folds uniformly bounded above, regardless of the singularity type?
• RQ4 Can the 1-gap conjecture in dimension 3 be established using a modified classification strategy for terminal singularities?

Key findings

• The paper proves the existence of δ > 0 such that for any normal quasi-projective Q-Gorenstein 3-fold X with mld(X) < 1, it holds that mld(X) ≤ 1 − δ, with an effective value δ = 1/13.
• The set of all non-canonical klt Calabi–Yau 3-folds is bounded modulo flops, even without the rational connectedness assumption.
• The global indices of all klt Calabi–Yau 3-folds are bounded from above by a uniform positive integer m, so that mKX ∼ 0.
• The proof establishes the 1-gap theorem for 3-dimensional isolated hyperquotient singularities by adapting the classification method of 3-fold terminal singularities.
• The boundedness result is achieved via log boundedness of pairs (Y, (1−a)E) and a relative MMP with scaling over a base scheme, leading to a family bounded modulo flops.
• The method yields an effective bound δ = 1/13, derived from the gap δ₀ = 1/19 in minimal log discrepancies of isolated cyclic quotient singularities in dimensions 3 and 5.