ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

5. Differential Geometry and Global Analysis

1991 MS Classification: 53 Hasegawa, Keizo, Niigata University, Japan A class of compact Kählerian solvmanifolds and a general conjecture It is now known that a non-toral nilmanifold has no Kähler structure. For solvmanifolds, we can see that a 4-dimensional compact solvmanifold has Kähler structure iff it is a hyperelliptic surface or a complex torus. Furthermore, they are all aspherical, that is, of the form G/Γ where G is a simply connected solvable Lie group and Γ is a uniform discrete subgroup of G. We state here a general conjecture on compact Kählerian solvmanifolds and show that it also holds for compact aspherical solvmanifolds: Conjecture: A compact solvmanifold has a Kähler structure iff it is a finite quotient of a complex torus which is a holomorphic fiber bundle over a complex torus with fiber a complex torus.