Form-Type Calabi-Yau Equations

TL;DR

This paper introduces a new differential equation on balanced hermitian manifolds inspired by superstring theory, generalizing the Calabi-Yau equation, and proves existence and uniqueness results in specific geometric contexts.

Contribution

It formulates a novel Calabi-Yau type equation on balanced hermitian manifolds and establishes foundational existence and uniqueness results for solutions.

Findings

Existence and uniqueness on complex tori.

Uniqueness and openness results on Kähler manifolds.

Abstract

Motivated from mathematical aspects of the superstring theory, we introduce a new equation on a balanced, hermitian manifold, with zero first Chern class. Solving the equation, one will obtain, in each Bott--Chern cohomology class, a balanced metric which is hermitian Ricci--flat. This can be viewed as a differential form level generalization of the classical Calabi--Yau equation. We establish the existence and uniqueness of the equation on complex tori, and prove certain uniqueness and openness on a general K\"ahler manifold.