Form–type Calabi–Yau equations

Abstract

Motivated from mathematical aspects of the superstring theory, we introduce a new equation on a balanced, hermitian manifold, with zero first Chern class. By 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 Khler manifold.