Product theorem on delta invariants via adding a general boundary

TL;DR

This work addresses how delta invariants behave under products of log Fano pairs and how the addition of a general boundary induces K-stability. It leverages twisted K-stability concepts and the limit δ_m → δ to transfer stability properties to product spaces, showing that a product is K-semistable exactly when each factor is, and that δ for the product equals the minimum of the factors' δ (or is bounded below by 1). The results extend the product theorem for delta invariants, providing a sharp formula in the unstable case and a robust lower bound, thereby clarifying how boundary perturbations influence product stability. The findings have implications for understanding the existence of Kähler–Einstein metrics on product varieties and for stability criteria in birational geometry.

Abstract

It's well-known that adding a general boundary would create K-stability. As an application, we reprove product theorem for delta invariants of Fano varieties.

Theorems & Definitions (9)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Theorem 2.2
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• proof : Proof of Theorem \ref{['corollary']}