Pluri-subharmonic functions on complex tori, Ricci curvature and convexity

TL;DR

This paper explores the relationship between Ricci curvature, convexity, and pluri-subharmonic functions on complex tori and toric manifolds, revealing how curvature signs relate to volume convexity and submanifold properties.

Contribution

It provides a new characterization of Ricci curvature sign via volume convexity in toric manifolds and discusses broader links between curvature, submanifolds, and pluri-subharmonic functions.

Findings

Ricci curvature sign characterized by volume convexity in toric manifolds

Relationships established between curvature, volume, and convexity

Connections made between submanifold types and pluri-subharmonic functions

Abstract

We show that, in toric manifolds, one can characterize the sign of the Ricci curvature in terms of the convexity of the volume functional. More generally we discuss relationships between (i) Ricci curvature and volume, (ii) totally real and Lagrangian submanifolds, (iii) pluri-subharmonic functions and convexity.