Form–type Calabi–yau Equations

As important examples in the superstring theory and non-Kähler complex geometry, the complex manifolds #k(S 3×S3) for any k ≥ 2 [4, 11] also admit a non-vanishing holomorphic three-form [4] and a balanced metric [5]. Moreover, we know that #k(S 3 × S3) satisfies the ∂∂̄–lemma [4]. A natural question to ask is, whether #k(S 3 ×S3) admit a balanced metric ω0 such that (1.2) holds. Such a metric ω0, if exists, will play an important role in the superstring theory and hermitian geometry. More generally, let X (n ≥ 3) be a complex n-dimensional manifold with a nonvanishing holomorphic n-form Ω and with a balanced metric ω0. We want to look for a balanced metric ω such that