Inequalities \`a la Fr\"olicher and cohomological decompositions

We study Bott-Chern and Aeppli cohomologies of a vector space endowed with two anti-commuting endomorphisms whose square is zero. In particular, we prove an inequality \`a la Fr\"olicher relating the dimensions of the Bott-Chern and Aeppli cohomologies to the dimensions of the Dolbeault cohomologies. We prove that the equality in such an inequality \`a la Fr\"olicher characterizes the validity of the so-called cohomological property of satisfying the $\partial\overline{\partial}$-Lemma. As an application, we study cohomological properties of compact either complex, or symplectic, or, more in general, generalized-complex manifolds.Comment: to appear in J. Noncommut. Geo

Adapted complex tubes on the symplectization of pseudo-Hermitian manifolds

Let $(M,\omega)$ be a pseudo-Hermitian space of real dimension $2n+1$, that is \RManBase is a \CR-manifold of dimension $2n+1$ and $\omega$ is a contact form on $M$ giving the Levi distribution $HT(M)\subset TM$. Let $M^\omega\subset T^*M$ be the canonical symplectization of $(M,\omega)$ and $M$ be identified with the zero section of $M^\omega$. Then $M^\omega$ is a manifold of real dimension $2(n+1)$ which admit a canonical foliation by surfaces parametrized by $\mathbb{C}\ni t+i\sigma\mapsto \phi_p(t+i\sigma)=\sigma\omega_{g_t(p)}$, where p\inM is arbitrary and $g_t$ is the flow generated by the Reeb vector field associated to the contact form $\omega$. Let $J$ be an (integrable) complex structure defined in a neighbourhood $U$ of $M$ in $M^\omega$. We say that the pair $(U,J)$ is an {adapted complex tube} on $M^\omega$ if all the parametrizations $\phi_p(t+i\sigma)$ defined above are holomorphic on $\phi_p^{-1}(U)$. In this paper we prove that if $(U,J)$ is an adapted complex tube on $M^\omega$, then the real function $E$ on $M^\omega\subset T^*M$ defined by the condition $\alpha=E(\alpha)\omega_{\pi(\alpha)}$, for each $\alpha\in M^\omega$, is a canonical equation for $M$ which satisfies the homogeneous Monge-Amp\`ere equation $(dd^c E)^{n+1}=0$. We also prove that if $M$ and $\omega$ are real analytic then the symplectization $M^\omega$ admits an unique maximal adapted complex tube.Comment: 6 page

Contact Calabi-Yau manifolds and Special Legendrian submanifolds

We consider a generalization of Calabi-Yau structures in the context of $\alpha$-Sasakian manifolds. We study deformations of a special class of Legendrian submanifolds and classify invariant contact Calabi-Yau structures on 5-dimensional nilmanifolds. Finally we generalize to codimension $r$.Comment: 16 pages, no figures. Final version to appear in "Osaka J. Math.

Oka principle for Levi flat manifolds

The name of Oka principle, or Oka-Grauert principle, is traditionally used to refer to the holomorphic incarnation of the homotopy principle: on a Stein space, every problem that can be solved in the continuous category, can be solved in the holomorphic category as well. In this note, we begin the study of the same kind of questions on a Levi-flat manifold; more precisely, we try to obtain a classification of CR-bundles on a semiholomorphic foliation of type (n, 1). Our investigation should only be considered a preliminary exploration, as it deals only with some particular cases, either in terms of regularity or bidegree of the bundle, and partial results