On a borderline class of non-positively curved compact Kähler manifolds
Let \(M\) be a compact complex manifold, \({\mathcal F}(M)\) the space of all Kähler metrics on \(M\) with non-positive holomorphic bisectional curvature, and \({\mathcal C}(M)\) the linear span of the space of all Kähler metrics on \(M\). The authors introduce a semi-rigidly non-positively curved manifold \(M\), or simply semi-rigid manifold \(M\) if \({\mathcal F}(M)\) is not empty, and its linear span in \({\mathcal C}(M)\) is finite- dimensional. The main purpose of this article is to study a class of manifolds which are semi-rigid (but not rigid in general). One of the main results is the following. Theorem. Let \((M,g)\) be a \(n\)-dimensional compact Kähler manifold with: (1) \(n\geq 2\), and \((c^ 2_ 1- c_ 2)\cdot [\omega_ h]^{n-2}= 0\) for a Kähler metric \(h\); (2) \(g\in {\mathcal F}(M)\), (3) \(\{x\in M: \text{Ric}^ n_ g(x)\neq 0\}\) is dense in \(M\), and \(\{x\in M: \text{Ric}^{n-1}_ g(x)\neq 0\}\) is a Zariski open subset in \(M\). Then for any \(h\in {\mathcal F}(M)\), \((M,h)\) also satisfies the condition (3); and for any two isometric holomorphic immersions \(f: (\widetilde M,\widetilde g)\to (\mathbb{C}^{n+1},g_ 0)\) and \(f^{(h)}: (\widetilde M,\widetilde h)\to (\mathbb{C}^{n+1},g_ 0)\), there always exists an affine transformation \(\phi\) in \(\mathbb{C}^{n+1}\) such that \(f^{(h)}= \phi\circ f\). In particular, \(M\) is semi-rigid. The authors consider also a class of non-trivial examples of manifolds satisfying the conditions of the Theorem.
scientific article; zbMATH DE number 549753