ACC for Minimal Log Discrepancies of Exceptional Singularities

Abstract

In this paper, we study the ascending chain condition (ACC) conjecture for minimal log discrepancies (mlds), proposed by the third author. We show the ACC conjecture holds for singularities admitting $\epsilon $-plt blow-ups. In particular, this gives the ACC for mlds for exceptional singularities. The key ingredients in the proofs of our main results are the Birkar–Borisov–Alexeev–Borisov theorem, proved by Birkar, the boundedness of complements conjecture for arbitrary DCC coefficients, proposed by the third author and proved in this paper, and the existence of uniform $\mathbb {R}$-complementary rational polytopes.

Keywords

Minimal log discrepancy / Minimal model program / Ascending chain condition / Complements

Funding

National Science Fund for the Excellent Young Scientists(12322102)