Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convexity of the Mabuchi functional in big cohomology classes

Eleonora Di Nezza, Stefano Trapani, Antonio Trusiani

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.

Convexity of the Mabuchi functional in big cohomology classes

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 in the big setting for finite entropy potentials.

Paper Structure

This paper contains 12 sections, 29 theorems, 215 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Envelopes and model potentials
  4. Plurisubharmonic geodesics
  5. Entropy
  6. Convexity of the Mabuchi functional in big classes
  7. From model to divisorial singularities
  8. Transcendental Fujita Approximation
  9. (Almost) Convexity of the Mabuchi functional
  10. On the Condition B and the Yau-Tian-Donaldson Conjecture
  11. Monge-Ampère measures on contact sets
  12. Geodesic Distance

Key Result

Theorem 1.1

Let $u_0,u_1\in{\rm PSH}(X,\theta)$ with minimal singularities and let $(u_t)_{t\in [0,1]}$ be the weak geodesic connecting $u_0$ and $u_1$. Let $\varphi\in \mathcal{E} (X,\theta)$ be such that $\theta_\varphi^n={\rm Vol}(\theta)\omega^n$, $\sup_X \varphi=0$. Then $u_t$ has minimal singularities and

Theorems & Definitions (64)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.4
  • ...and 54 more