Rigidity of invariant complex structures

Abstract

A Kähler solvmanifold is a connected Kähler manifold ( M , j , ⟨ , ⟩ ) (M,j,\left \langle , \right \rangle ) admitting a transitive solvable group of automorphisms. In this paper we study the isomorphism classes of Kähler structures ( j , ⟨ , ⟩ ) (j,\left \langle , \right \rangle ) turning M M into a Kähler solvmanifold. In the case when ( M , j , ⟨ , ⟩ ) (M,j,\left \langle , \right \rangle ) is irreducible and simply connected we show that any Kähler structure on M M , having the same group of automorphisms, is isomorphic to ( j , ⟨ , ⟩ ) (j,\left \langle , \right \rangle ) .