A gap theorem for minimal log discrepancies of non-canonical singularities in dimension three

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.

39 pages, comments are welcome; v2: 40 pages, slightly modified, more discussion on computation of $\delta$ added; v3: final version, to appear in J. Algebraic Geom

References in corpus (5)

• ACC for minimal log discrepancies of exceptional singularities
• Birational boundedness of rationally connected Calabi-Yau 3-folds
• Toward the equivalence of the ACC for $a$-log canonical thresholds and the ACC for minimal log discrepancies
• On equivalent conjectures for minimal log discrepancies on smooth threefolds
• An optimal gap of minimal log discrepancies of threefold non-canonical singularities

Cited by in corpus (7)

• ACC for minimal log discrepancies of exceptional singularities
• Birational boundedness of rationally connected Calabi-Yau 3-folds
• Toward the equivalence of the ACC for $a$-log canonical thresholds and the ACC for minimal log discrepancies
• Boundedness of elliptic Calabi-Yau varieties with a rational section
• Shokurov's conjecture on conic bundles with canonical singularities
• On global ACC for foliated threefolds
• An optimal gap of minimal log discrepancies of threefold non-canonical singularities