ACC for minimal log discrepancies of terminal threefolds

TL;DR

This paper proves the ACC conjecture for minimal log discrepancies of terminal threefolds within a specific range, establishes uniform boundedness of certain divisors, and confirms related conjectures on complements and thresholds.

Contribution

It establishes the ACC for minimal log discrepancies of terminal threefolds in a new range and proves the uniform boundedness of divisors computing these discrepancies.

Findings

Proved ACC for minimal log discrepancies in $[1- ext{delta},+ ext{infinity})$ for threefolds.

Showed the uniform boundedness of divisors computing minimal log discrepancies.

Established the set of accumulation points of threefold canonical thresholds.

Abstract

We prove that the ACC conjecture for minimal log discrepancies holds for threefolds in $[1-\delta,+\infty)$, where $\delta>0$ only depends on the coefficient set. We also study Reid's general elephant for pairs, and show Shokurov's conjecture on the existence of $(\epsilon,n)$-complements for threefolds for any $\epsilon\geq 1$. As a key important step, we prove the uniform boundedness of divisors computing minimal log discrepancies for terminal threefolds. We show the ACC for threefold canonical thresholds, and that the set of accumulation points of threefold canonical thresholds is equal to $\{0\}\cup\{\frac{1}{n}\}_{n\in\mathbb Z_{\ge 2}}$ as well.