K-moduli with real coefficients

TL;DR

This paper extends algebraic K-stability theory to log Fano real pairs, constructing a proper K-moduli space for K-polystable pairs with fixed invariants, by reducing problems to the rational coefficient case.

Contribution

It develops K-stability theory for log Fano real pairs and constructs a K-moduli space, generalizing known results from rational to real coefficients.

Findings

Established algebraic K-stability theory for log Fano $ ext{R}$-pairs.

Constructed a proper K-moduli space parametrizing K-polystable pairs.

Reduced problems to the rational coefficient case for broader applicability.

Abstract

In this paper, we develop an algebraic K-stability theory (e.g. special test configuration theory and optimal destabilization theory) for log Fano $\mathbb R$-pairs, and construct a proper K-moduli space to parametrize K-polystable log Fano $\mathbb R$-pairs with some fixed invariants (e.g. dimension, volume, coefficients). All of these are well-known for log Fano $\mathbb Q$-pairs, and the strategy in this paper is trying to reduce the problems (in many cases) to $\mathbb Q$-coefficients case rather than rebuilding the whole program as in $\mathbb Q$-coefficients case.