## Nonpositively curved Kähler metrics on product manifolds.(English)Zbl 0779.53045

The main purpose of this note is to study a metric with certain curvature conditions on the product of two compact complex manifolds $$M = M_ 1 \times M_ 2$$. Let $${\mathcal F}(M)$$ denote the space of all those Kähler metrics on $$M$$ with nonpositive bisectional curvature. Let $$\pi_ i$$ be the projection map from $$M$$ to $$M_ i$$ with $$q_ i = h^{1,0}(M_ i)$$, $$i = 1,2$$. Let $$\{\varphi_ 1,\dots,\varphi_{q_ 1}\}$$ and $$\{\psi_ 1,\dots,\psi_{q_ 2}\}$$ be the basis of global holomorphic 1 forms on $$M_ 1$$ and $$M_ 2$$ respectively, and $$\omega_ g$$ the Kähler form of a metric $$g$$. If $${\mathcal F}(M) \neq \emptyset$$ and $$g \in {\mathcal F}(M)$$ the author proves that there exist $$g_ i \in {\mathcal F}(M_ i)$$ and a constant $$q_ 1 \times q_ 2$$-matrix $$(a_{ij})$$ such that $\omega_ g = \pi^*_ 1\omega_{g_ 1} + \pi^*_ 2\omega_{g_ 2} + \rho + \bar\rho,$ where $$\rho = \sum^{q_ 1}_{i=1}\sum^{q_ 2}_{j=1}a_{ij}\varphi_ i \wedge \bar\psi_ j$$. Furthermore, for any $$g_ i \in {\mathcal F}(M_ i)$$ and any global holomorphic 1-forms $$\eta_ 1,\dots,\eta_ r$$ on $$M$$, $$\omega = \pi_ 1^*\omega_ 1 + \pi^*_ 2\omega_ 2 + \sqrt{-1}\sum^ r_{i=1}\eta_ i\wedge\bar\eta_ i$$ is the Kähler form of some $$g \in {\mathcal F}(M)$$. Therefore $\text{codim}_{\mathbb{R}}({\mathcal F}(M_ 1) \times {\mathcal F}(M_ 2);\;{\mathcal F}(M)) = {1\over 2}\cdot b_ 1(M_ 1) \cdot b_ 1(M_ 2).$ In particular $${\mathcal F}(M) = {\mathcal F}(M_ 1) \times {\mathcal F}(M_ 2)$$ if and only if $$b_ 1(M_ 1) = 0$$ or $$b_ 1(M_ 2) = 0$$.