Dual complexes of qdlt Fano type models and strong complete regularity

TL;DR

This paper introduces new invariants called birational strong complete regularity and strong complete regularity for pairs of Fano type, refining existing concepts and relating to K-stability, with foundational properties and chain condition results.

Contribution

It defines and studies two new numerical invariants for Fano type pairs, connecting them to K-stability and existing regularity concepts, and establishes their fundamental properties.

Findings

Pairs with maximal birational strong complete regularity are 1-complementary.

Thresholds for jumps in regularity satisfy the ascending chain condition.

Basic properties and relations to qdlt Fano models are clarified.

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.