Special representatives of complexified K\"ahler classes

TL;DR

This paper explores special representatives of complexified Kähler classes inspired by mirror symmetry, providing a moment map perspective, potential links to Landau-Ginzburg models, and existence results in toric cases.

Contribution

It introduces a new class of special representatives for complexified Kähler classes, extending constant scalar curvature and extremal notions, with a moment map interpretation and existence proofs.

Findings

Established a moment map interpretation for these representatives.

Proved existence results in certain toric cases.

Discussed potential links to Landau-Ginzburg models.

Abstract

Motivated by constructions appearing in mirror symmetry, we study special representatives of complexified K\"ahler classes, which extend the notions of constant scalar curvature and extremal representatives for usual K\"ahler classes. In particular, we provide a moment map interpretation, discuss a possible correspondence with compactified Landau-Ginzburg models, and prove existence results for such special complexified K\"ahler forms and their large volume limits in certain toric cases.