On a class of solvable Kähler algebras of nonpositive curvature
The authors study the class of solvable Kähler algebras associated to simply connected solvable Lie groups with left invariant Kähler metrics of non-positive, sectional or holomorphic curvature, and without flat de Rham factor. In Section 2 it is shown, using modifications, that this class is in one to one correspondence with normal \(j\)-algebras of non-positive, either sectional or holomorphic curvature. Sections 3 and 4 are devoted to the classification of the above class of Kähler algebras in dimension 6. In Theorem 3.3, a one parameter family of Kähler metrics \(\langle , \rangle_t\) of non-positive sectional or holomorphic curvature on a solvable Lie group \(S\) is given; the metric is symmetric for one fixed \(t\) and different values of \(t\) give rise to non-isometric Riemannian manifolds (see Proposition 3.5). In Theorem 4.1, the full classification of simply connected homogeneous Kähler manifolds of dimension 6 without flat de Rham factor, admitting a simply transitive solvable Lie group of holomorphic isometries and having non-positive, either sectional or holomorphic curvature, is given. The resulting Riemannian manifolds are symmetric with the exception of the one parameter group considered in Theorem 3.3.
scientific article; zbMATH DE number 1003884