A structure theorem for abelian complex nilmanifolds

A. ANDRADA, M.L. BARBERIS, I. DOTTI

Rio de Janeiro

Congreso; First Joint Meeting AMS-SBM; 2008

Sociedad Brasilera de Matemática y American Mathematical Society

An {it abelian} complex structure on a real Lie algebra $rak g$ is an endomorphism of $rak g$ satisfying egin{equation} J^2=-I, hspace{1.5cm} [Jx,Jy]=[x,y], ; ;; orall x,y in rak g. label{abel} end{equation} If $G$ is a Lie group with Lie algebra $rak g$ these conditions imply the vanishing of the Nijenhuis tensor on the invariant almost complex manifold $(G,J)$, that is, $J$ is integrable on $G$. If $Gamma subset G$ is any discrete co-compact subgroup of $G$ then the nilmanifold $Gamma ackslash G$ with the complex structure induced by $J$ is called an abelian complex nilmanifold. A splitting $rak g = rak g_+ oplus rak g _- $, where $rak g _±$ are Lie subalgebras of $rak g$, gives rise to a product structure $E$ by setting $Emid rak g_±=pm $Id. A complex product structure on $rak g$ is a pair of a product structure $E$ and a complex structure $J$ such that $JE=-EJ$. Abelian complex structures on Lie algebras were first considered in cite{bdm} and further studied in cite{ba-do}. Our interest arises from properties of the complex manifolds obtained by considering this class of structures. For instance, a procedure is given in cite{AD} to construct hypersymplectic structures on $Bbb R^{4n}$ by using abelian complex product structures. It is the aim of this work to improve the results in cite{ba-do}. We show that any nilmanifold with an abelian complex structure fibers holomorphically over a torus with fiber a nilmanifold with a complex product structure. We exhibit some new examples as an application of our result. egin{thebibliography}{BDM} renchspacing ibitem[AD]{AD} A. Andrada and I. Dotti, {it Double products and hypersymplectic structures on $Bbb R^{4n}$}, Communications in Math. Physics, {f 262} (2006), 1--16. ibitem[BDM]{bdm} M. L. Barberis, I. G. Dotti and R. J. Miatello, {it On certain locally homogeneous Clifford manifolds}, Ann. Glob. Anal. Geom. {f 13} (1995), 289--301. ibitem[BD]{ba-do} M. L. Barberis and I. Dotti, {it Abelian complex structures on solvable Lie algebras}, J. Lie Theory {f 14} (2004), 25--34. end{thebibliography}