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 \(b_ 1(M_ 2) = 0\).
Reviewer: N.Bokan (Beograd)


53C55 Global differential geometry of Hermitian and Kählerian manifolds
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