K-stability of adjoint foliated structures

TL;DR

This paper develops a new framework for K-stability in adjoint foliated structures, establishing criteria and invariants to analyze their stability and boundedness.

Contribution

It introduces a notion of K-stability and Ding stability for adjoint foliated structures, along with valuative criteria and boundedness results.

Findings

K-stability is equivalent to Ding stability for these structures.

Reduction to special test configurations simplifies stability analysis.

K-semistable structures with bounded volume form a bounded family.

Abstract

We introduce a notion of K-stability for adjoint foliated structures via test configurations and the foliated Donaldson-Futaki invariant. We prove reduction to special test configurations for adjoint Fano foliated structures by showing that the mixed Donaldson-Futaki invariant is non-increasing along the birational procedure. We also introduce a notion of Ding stability for adjoint Fano foliated structures which we show is equivalent to our notion of K-stability. We then introduce mixed alpha, beta and delta-invariants and use the reduction theorem to establish valuative criteria for the K-stability of adjoint Fano foliated structures. To conclude, as an application, we show that K-semistable adjoint Fano foliated structures with bounded volume form a bounded family.