Lefschetz pencil structures for 2-calibrated manifolds

Proceedings of the American Mathematical Society, 2016

We find a family of five dimensional completely solvable compact manifolds that constitute the first examples of K-contact manifolds which satisfy the Hard Lefschetz Theorem and have a model of Tievsky type just as Sasakian manifolds but do not admit any Sasakian structure.

2000

The paper deals with relations between the Hard Lefschetz property, (non)vanishing of Massey products and the evenness of odd-degree Betti numbers of closed symplectic manifolds. It is known that closed symplectic manifolds can violate all these properties (in contrast with the case of Kaehler manifolds). However, the relations between such homotopy properties seem to be not analyzed. This analysis may shed a new light on topology of symplectic manifolds. In the paper, we summarize our knowledge in tables (different in the simply-connected and in symplectically aspherical cases). Also, we discuss the variation of symplectically harmonic Betti numbers on some 6-dimensional manifolds.

Geometry & Topology, 2009

We show that for each k > 3 there are infinitely many finite type Stein manifolds diffeomorphic to Euclidean space R 2k which are pairwise distinct as symplectic manifolds.

Journal of Symplectic Geometry, 2004

We prove that the vanishing spheres of the Lefschetz pencils constructed by Donaldson on symplectic manifolds of any dimension are conjugated under the action of the symplectomorphism group of the fiber. More precisely, a number of generalized Dehn twists may be used to conjugate those spheres. This implies the non-existence of homologically trivial vanishing spheres in these pencils. To develop the proof, we discuss some basic topological properties of the space of asymptotically holomorphic transverse sections.

Forum Mathematicum, 2013

In this paper, we study compact convex Lefschetz fibrations on compact convex symplectic manifolds (i.e., Liouville domains) of dimension 2n + 2 which are introduced by Seidel and later also studied by McLean. By a result of Akbulut-Arikan, the open book on ∂W , which we call convex open book, induced by a compact convex Lefschetz fibration on W carries the contact structure induced by the convex symplectic structure (i.e., Liouville structure) on W . Here we show that, up to a Liouville homotopy and a deformation of compact convex Lefschetz fibrations on W , any simply connected embedded Lagrangian submanifold of a page in a convex open book on ∂W can be assumed to be Legendrian in ∂W with the induced contact structure. This can be thought as the extension of Giroux's Legendrian realization (which holds for contact open books) for the case of convex open books. Moreover, a result of Akbulut-Arikan implies that there is a one-to-one correspondence between convex stabilizations of a convex open book and convex stabilizations of the corresponding compact convex Lefschetz fibration. We also show that the convex stabilization of a compact convex Lefschetz fibration on W yields a compact convex Lefschetz fibration on a Liouville domain W ′ which is exact symplectomorphic to a positive expansion of W . In particular, with the induced structures ∂W and ∂W ′ are contactomorphic.

2002

For any compact complex manifold M with a compatible symplectic form ω, we consider the homomorphisms L 1,0 : H 1,0 (M ) −→ H n,n−1 (M ) and L 0,1 : H 0,1 (M ) −→ H n−1,n (M ) given by the cup product with [ω] n−1 , n being the complex dimension of M and H * , * (M ) the Dolbeault cohomology of M .

2015

In his paper Cycles for the Dynamical Study of Foliated Manifolds and Complex Manifolds, Denis Sullivan proves that a closed manifold supports a symplectic structure if and only if it admits a distribution of cones of bivectors satisfying two conditions. We prove a similar result for contact structures. It relies on a suitable variant of the symplectization process that produces a S 1 -invariant nondegenerate 2-form on the closed manifold S 1 ×M that is closed for a twisted differential.

Comptes Rendus Mathematique, 2004

The notion of 2-calibrated structure, generalizing contact structures, smooth taut foliations, etc., is defined. Approximately holomorphic geometry as introduced by S. Donaldson for symplectic manifolds is extended to 2-calibrated manifolds. An estimated transversality result that enables to study the geometry of such manifolds is presented. To cite this article: A. Ibort, D.

Functional Analysis and Its Applications, 1995

We show that if a contact open book $(\Sigma,h)$ on a $(2n+1)$-manifold $M$ ($n\geq1$) is induced by a Lefschetz fibration $\pi:W \to D^2$, then there is a one-to-one correspondence between positive stabilizations of $(\Sigma,h)$ and \emph{positive stabilizations} of $\pi$. More precisely, any positive stabilization of $(\Sigma,h)$ is induced by the corresponding positive stabilization of $\pi$, and conversely any positive stabilization of $\pi$ induces the corresponding positive stabilization of $(\Sigma,h)$. We define \emph{exact open books} as boundary open books of compatible exact Lefschetz fibrations, and show that any exact open book carries a contact structure. Moreover, we prove that there is a one-to-one correspondence (similar to the one above) between \emph{convex stabilizations} of an exact open book and \emph{convex stabilizations} of the corresponding compatible exact Lefschetz fibration. We also show that convex stabilization of compatible exact Lefschetz fibrations ...