Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Dual complexes of qdlt Fano type models and strong complete regularity

Jihao Liu, Konstantin Loginov

Abstract

We introduce birational strong complete regularity and strong complete regularity, two numerical invariants for pairs of (relative) Fano type. They are defined using variants of qdlt Fano type models and the dimension of the dual complex of the reduced boundary, and can be viewed as Fano type refinements of Shokurov's complete regularity. We establish basic properties of these invariants and clarify its relation to models of qdlt Fano type appearing in K-stability. In particular, we prove that any pair with maximal birational strong complete regularity is $1$-complementary, and the thresholds where birational strong complete regularity or strong complete regularity jumps satisfy the ascending chain condition.

Dual complexes of qdlt Fano type models and strong complete regularity

Abstract

We introduce birational strong complete regularity and strong complete regularity, two numerical invariants for pairs of (relative) Fano type. They are defined using variants of qdlt Fano type models and the dimension of the dual complex of the reduced boundary, and can be viewed as Fano type refinements of Shokurov's complete regularity. We establish basic properties of these invariants and clarify its relation to models of qdlt Fano type appearing in K-stability. In particular, we prove that any pair with maximal birational strong complete regularity is -complementary, and the thresholds where birational strong complete regularity or strong complete regularity jumps satisfy the ascending chain condition.
Paper Structure (12 sections, 23 theorems, 130 equations)

This paper contains 12 sections, 23 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic notation
  4. Pairs
  5. Relative Nakayama-Zariski decomposition
  6. Strong complete regularity
  7. Definitions and variants
  8. Comparing complete regularities
  9. Reduced (birational) qdlt Fano type models
  10. Qdlt anti-ample models and birational strong complete regularity
  11. Complements
  12. ACC for strong complete regularity threshold

Key Result

Theorem 1.5

Let $(X/Z\ni z,B)$ be a pair such that Then $\mathrm{SReg}^{{\operatorname{bir}}}(X/Z\ni z,B)=\dim X-1-\dim z$, and there exists a $1$-complement $(X/Z\ni z,B^+)$ of $(X/Z\ni z,B)$ such that In particular, $(X/Z\ni z,B)$ is $1$-complementary.

Theorems & Definitions (69)

  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Definition 1.4: Strong complete regularity
  • Theorem 1.5
  • Definition 1.6
  • Theorem 1.7
  • Definition 2.1
  • Definition 2.2
  • Definition 2.7
  • ...and 59 more