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 $\frak g$ is an endomorphism of $\frak g$ satisfying \begin{equation} J^2=-I, \hspace{1.5cm} [Jx,Jy]=[x,y], \; \;\; \forall x,y \in \frak g. \label{abel} \end{equation} If $G$ is a Lie group with Lie algebra $\frak g$ 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 $\Gamma \backslash G$ with the complex structure induced by $J$ is called an abelian complex nilmanifold. A splitting $\frak g = \frak g_+ \oplus \frak g _- $, where $\frak g _±$ are Lie subalgebras of $\frak g$, gives rise to a product structure $E$ by setting $E\mid \frak g_±=\pm $Id. A complex product structure on $\frak g$ 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. \begin{thebibliography}{BDM} \frenchspacing \bibitem[AD]{AD} A. Andrada and I. Dotti, {\it Double products and hypersymplectic structures on $\Bbb R^{4n}$}, Communications in Math. Physics, {\bf 262} (2006), 1--16. \bibitem[BDM]{bdm} M. L. Barberis, I. G. Dotti and R. J. Miatello, {\it On certain locally homogeneous Clifford manifolds}, Ann. Glob. Anal. Geom. {\bf 13} (1995), 289--301. \bibitem[BD]{ba-do} M. L. Barberis and I. Dotti, {\it Abelian complex structures on solvable Lie algebras}, J. Lie Theory {\bf 14} (2004), 25--34. \end{thebibliography}