Mathematical Research Letters

Volume 17 (2010)

Number 5

Form–type Calabi–Yau equations

Pages: 887 – 903



Jixiang Fu (Institute of Mathematics, Fudan University, Shanghai, 200433, China)

Zhizhang Wang (Institute of Mathematics, Fudan University, Shanghai 200433, China)

Damin Wu (Department of Mathematics, The Ohio State University, 1179 University Drive)


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 Kähler manifold.

Published 1 January 2010