Convexity of the Mabuchi functional in big cohomology classes

TL;DR

This paper investigates the convexity properties of the Mabuchi functional in big cohomology classes, introducing an invariant linked to Fujita approximations and exploring its implications for the Yau-Tian Donaldson conjecture.

Contribution

It defines a new invariant related to transcendental Fujita approximations and establishes convexity of the Mabuchi functional under certain conditions, advancing understanding in complex geometry.

Findings

Vanishing of the invariant relates to the Yau-Tian Donaldson conjecture.

Established (almost) convexity of the Mabuchi functional along weak geodesics.

Provided an explicit expression for the distance $d_p$ in the big setting for finite entropy potentials.

Abstract

We study the Mabuchi functional associated to a big cohomology class. We define an invariant associated to transcendental Fujita approximations, whose vanishing is related to the Yau-Tian Donaldson conjecture. Assuming vanishing (finiteness) of this invariant we establish (almost) convexity along weak geodesics. As an application, we give an explicit expression of the distance $d_p$ in the big setting for finite entropy potentials.