Symplectic analog of Calabi's conjecture for Calabi--Yau threefolds

TL;DR

This paper proposes a symplectic analog of Calabi's conjecture for Calabi--Yau threefolds, focusing on deformations of complex structures via symplectic methods instead of the classical complex Monge--Ampère equation.

Contribution

It introduces a new symplectic deformation approach as an analog to Calabi's conjecture, extending the classical complex geometric framework.

Findings

Formulation of a symplectic analog of Calabi's conjecture

Deformation of complex structures via symplectic methods

Potential new pathways for Calabi--Yau geometry

Abstract

In this paper we state an analog of Calabi's conjecture proved by Yau. The difference with the classical case is that we propose deformation of the complex structure, whereas the complex Monge--Amp\`{e}re equation describes deformation of the K\"{a}hler (symplectic) structure.