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Let $(M,J)$ be a $n$-dimensional complex manifold: a $p$-K\"ahler structure (resp. $p$-symplectic structure) on $M$ is a real, closed $(p,p)$-transverse form $\Omega$ (resp. real, closed $2p$-form whose $(p,p)$-component is transverse). We…

We show that the almost complex structure underlying a non-Kahler, nearly Kahler 6-manifold (in particular, the standard almost complex structure of S^6) cannot be compatible with any symplectic form, even locally.

The symplectization of an overtwisted contact structure in Euclidean 3--space is shown to be an exotic symplectic structure on Euclidean 4--space. The technique can be extended to produce exotic symplectic structures in higher dimensional…

We study Farber's topological complexity for monotone symplectic manifolds. More precisely, we estimate the topological complexity of 4-dimensional spherically monotone manifolds whose Kodaira dimension is not $-\infty$.

We give two constructions of surfaces in simply-connected 4-manifolds with non simply-connected complements. One is an iteration of the twisted rim surgery introduced by the first author. We also construct, for any group G satisfying some…

We define and study branched shadows of 4-manifolds as a combination of branched spines of 3-manifolds and Turaev's shadows. We use these objects to combinatorially represent 4-manifolds equipped with $Spin^c$-structures and homotopy…

We prove that, for any n, there are simply-connected four-manifolds which admit n-tuples of symplectic forms whose first Chern classes have pairwise different divisibilities in integral cohomology. It follows that the moduli space of…

We present the various constructions of new symplectic $4$-manifolds with non-negative signatures using the complex surfaces on the BMY line $c_1^2 = 9\chi_h$, the Cartwright-Steger surfaces, the quotients of Hirzebruch's certain…

We present a handlebody construction of small symplectic caps, and hence of small closed symplectic 4-manifolds. We use this to construct handlebody descriptions of symplectic embeddings of rational homology balls in…

We analyze four-dimensional symplectic manifolds of type $X=S^1 \times M^3$ where $M^3$ is an open $3$-manifold admitting inequivalent fibrations leading to inequivalent symplectic structures on $X$. For the case where $M^3 \subset S^3$ is…

We study left invariant contact forms and left invariant symplectic forms on Lie groups. We give the classification of all symplectic structures on nilpotent Lie algebras up the dimension 6.

This article deals with 3-forms on 6-dimensional manifodls, the first dimension where the classification of 3-forms is not trivial. There are three classes of multisymplectic 3-forms there. We study the class which is closely related to…

In this paper we analyze in detail a collection of motivating examples to consider $b^m$-symplectic forms and folded-type symplectic structures. In particular, we provide models in Celestial Mechanics for every $b^m$-symplectic structure.…

Symplectic fillings of standard tight contact structures on lens spaces are understood and classified. The situation is different if one considers non-standard tight structures (i.e. those that are virtually overtwisted), for which a…

Homogeneous compatible almost complex structures on symplectic manifolds are studied, focusing on those which are special, meaning that their Chern-Ricci form is a multiple of the symplectic form. Non Chern-Ricci flat ones are proven to be…

We introduce a new technique that is used to show that the complex projective plane blown up at 6, 7, or 8 points has infinitely many distinct smooth structures. None of these smooth structures admit smoothly embedded spheres with…

We find geometric conditions on a four-dimensional almost Hermitian manifold under which the almost complex structure is a harmonic map or a minimal isometric imbedding of the manifold into its twistor space.

We classify four-dimensional compact solvmanifolds up to diffeomorphism, while determining which of them have complex analytic structures. In particular, we shall see that a four-dimensional compact solvmanifold S can be written, up to…

A myriad of irreducible symplectic 4-manifolds with abelian non-cyclic fundamental group is constructed. The botany of manifolds with finite non-cyclic fundamental groups is also studied.

We prove necessary and sufficient conditions for a smooth surface in a 4-manifold X to be pseudoholomorphic with respect to some almost complex structure on X. This provides a systematic approach to the construction of pseudoholomorphic…