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A foliation $(M,\mathcal{F})$ is said to be $2$--calibrated if it admits a closed 2-form $\omega$ making each leaf symplectic. By using approximately holomorphic techniques, a sequence $W_k$ of $2$--calibrated submanifolds of codimension--$2$ can be found for $(M, \mathcal{F}, \omega)$. Our main result says that the Lefschetz hyperplane theorem holds for the pairs $(F, F \cap W_k)$, with $F$ any leaf of $\mathcal{F}$. This is applied to draw important consequences on the transverse geometry of such foliations., Comment: Title and abstract modified. Section 2 on Lie groupoids and essential equivalence greatly reduced. bibliography updated. DOI added (to appear in Nagoya Math. J.)

We construct positive loops of Legendrian submanifolds in several instances. In particular, we partially recover G. Liu's result stating that any loose Legendrian admits a positive loop, under some mild topological assumptions on the Legendrian. Moreover, we show contractibility of the constructed loops under an extra topological assumption., Comment: 2 figures

Recently there has been renewed interest among differential geometers in the theory of Engel structures. We introduce holomorphic analogues of these structures, and pose the problem of classifying projective manifolds admitting them. Besides providing their basic properties and presenting two series of examples, we classify those satisfying certain positivity conditions.

In this article parametric versions of Wilson's plug and Kuperberg's plug are discussed. We show that there is a weak homotopy equivalence induced by the inclusion between the space of non-singular vector fields tangent to a foliation and the subspace of those without closed orbits, as long as the leaves of the foliation have dimension at least 3. We contrast this with the case of foliations by surfaces in 3-manifolds., Comment: 9 pages, 2 figures

The loop space associated to a Riemannian manifold admits a quasisymplectic structure (that is, a closed 2-form which is non-degenerate up to a finite-dimensional kernel). We show how to construct a compatible almost-complex structure. Finally conditions to have contact structures on loop spaces are studied.

A foliation is said to admit a foliated contact structure if there is a codimension 1 distribution in the tangent space of the foliation such that the restriction to any leaf is contact. We prove a version of the Weinstein conjecture in the presence of an overtwisted leaf. The result is shown to be sharp., 17 pages

In this article we prove that the inclusion of the space of Engel structures of a smooth $4$-fold into the space of full flags of its tangent bundle induces surjections in all homotopy groups. In particular, we construct Engel structures representing any given full flag., 21 pages, 7 figures

Open access version at https://arxiv.org/abs/1503.06221, In this article we establish efficient geometric criteria to decide whether a contact manifold is overtwisted. Starting with the original definition, we first relate overtwisted disks in different dimensions and show that a manifold is overtwisted if and only if the Legendrian unknot admits a loose chart. Then we characterize overtwistedness in terms of the monodromy of open book decompositions and contact surgeries. Finally, we provide several applications of these geometric criteria., The first author was supported by NSF grant DMS-1841913 and a BBVA Research Fellowship. The second author was supported by NSF grant DMS-1510305 and a Sloan Research Fellowship. The third author was supported by Spanish Research Projects SEV–2015–0554, MTM2016– 79400–P, and MTM2015–72876–EXP.

In this article, we introduce the topological study of codimension-1 foliations which admit contact structures on the leaves. A parametric existence h-principle for foliated contact structures is provided for any cooriented foliation in a closed oriented four-fold.

We define the concept of Lefschetz contact pencil and we show the existence of such structures on any contact manifold. The main idea of the proof is a generalization of the Donaldson arguments used in the symplectic case. We will analyze some of the applications of such existence theorem for the topology of approximately holomorphic contact submanifolds., Comment: 26 pages, 1 figure

We provide various definitions for the contact blow--up. Such different approaches to the contact blow--up are related. Some uniqueness and non--uniqueness results are also provided., 23 pages, 2 figures. Minor changes made in the last version

We define the concept of symplectic foliation on a symplectic manifold and provide a method of constructing many examples, by using asymptotically holomorphic techniques.

We prove a Lefschetz hyperplane theorem for the determinantal loci of a morphism between two holomorphic vector bundles $E$ and $F$ over a complex manifold under the condition that $E^*\ox F$ is Griffiths $k$-positive. We apply this result to find some homotopy groups of the Brill-Noether loci for a generic curve., Comment: (one small mistake corrected)

We use Donaldson's approximately holomorphic techniques to build embeddings of a closed symplectic manifold with symplectic form of integer class in the grassmannians Gr(r,N). We assure that these embeddings are asymptotically holomorphic in a precise sense. We study first the particular case of embeddings in the projective space $CP^N$ obtaining control on N. The main reason of our study is the construction of singular determinantal submanifolds as the intersection of the embedding with certain ``generalized Schur cycles'' defined on a product of grassmannians. It is shown that the symplectic type of these submanifolds is quite more general that the ones obtained by Auroux as zero sets of approximately holomorphic sections of ``very ample'' vector bundles., 40 pages, no figures, Latex2e