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A boundedness conjecture for minimal log discrepancies on a fixed germ

Mircea Mustata, Yusuke Nakamura

Abstract

We consider the following conjecture: on a klt germ (X,x), for every finite set I there is a positive integer N with the property that for every R-ideal J on X with exponents in I, there is a divisor E over X that computes the minimal log discrepancy of (X,J) at x and such that its discrepancy k_E is bounded above by N. We show that this implies Shokurov's ACC conjecture for minimal log discrepancies on a fixed klt germ and give some partial results towards the conjecture.

A boundedness conjecture for minimal log discrepancies on a fixed germ

Abstract

We consider the following conjecture: on a klt germ (X,x), for every finite set I there is a positive integer N with the property that for every R-ideal J on X with exponents in I, there is a divisor E over X that computes the minimal log discrepancy of (X,J) at x and such that its discrepancy k_E is bounded above by N. We show that this implies Shokurov's ACC conjecture for minimal log discrepancies on a fixed klt germ and give some partial results towards the conjecture.

Paper Structure

This paper contains 7 sections, 15 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Minimal log discrepancies: definition and basic facts
  3. Generic limits: bounding the order of the ideal of the point
  4. A proof of the conjecture in dimension 2
  5. A proof of the conjecture in the monomial case
  6. Connection with ACC
  7. Three equivalent conjectures

Key Result

Theorem 1.2

Let $X$ be a klt variety and $x\in X$ a point defined by the ideal $\mathfrak{m}_x$. For every finite subset $I\subset {\mathbf R}_{\geq 0}$, there is a positive integer $\ell$${\rm (}$depending on $(X,x)$ and $I$${\rm )}$ such that the following conditions hold:

Theorems & Definitions (34)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Proposition 3.1
  • ...and 24 more