Approximating delta invariants in the sense of complements of plt type

TL;DR

This paper demonstrates that the delta invariant of a log Fano pair can be approximated using lc places of complements of plt type, and establishes the existence of a divisorial valuation computing this invariant under certain conditions.

Contribution

It introduces a method to approximate delta invariants via lc places of plt type complements and proves the existence of a divisorial valuation computing the invariant.

Findings

Delta invariant can be approximated by lc places of plt type complements.

Existence of a divisorial valuation computing the delta invariant is established.

Applicable to log Fano pairs with delta invariant no greater than one.

Abstract

In this note, we will show that delta invariant of a log Fano pair can be approximated by lc places of complements of plt type if it is no greater than one. Give a log Fano pair with delta invariant no greater than one, under the assumption that delta invariant of the log Fano pair can be approximated by lc places of bounded complements of plt type, we show the existence of the divisorial valuation computing delta invariant of this log Fano pair.