arXiv subjects

Ramón Vera

Publications and source records attributed to Ramón Vera.

7 recordsLinked to original sources

arXiv2021

Poisson structures of near-symplectic manifolds and their cohomology

We connect Poisson and near-symplectic geometry by showing that there is a singular Poisson structure on a near-symplectic 4-manifold. The Poisson structure $π$ is defined on the tubular neighbourhood of the singular locus $Z_ω$ of the 2-form $ω$, it is of maximal rank 4 and it vanishes on a degeneracy set containing $Z_ω$. We compute its smooth Poisson cohomology, which depends on the modular vector field and it is finite dimensional. We conclude with a discussion on the relation between the Poisson structure $π$ and the overtwisted contact structure associated to a near-symplectic 4-manifold.

math.SG↗

arXiv2020

Poisson Cohomology of Broken Lefschetz Fibrations

We compute the formal Poisson cohomology of a broken Lefschetz fibration by calculating it at fold and Lefschetz singularities. Near a fold singularity the computation reduces to that for a point singularity in 3 dimensions. For the Poisson cohomology around singular points we adapt techniques developed for the Sklyanin algebra. As a side result, we give compact formulas for the Poisson coboundary operator of an arbitrary Jacobian Poisson structure in 4 dimensions.

math.DG↗

arXiv2018

On Bott-Morse Foliations and their Poisson Structures in Dimension 3

We show that a Bott-Morse foliation in dimension 3 admits a linear, singular, Poisson structure of rank 2 with Bott-Morse singularities. We provide the Poisson bivectors for each type of singular component, and compute the symplectic forms of the characteristic distribution.

math.SG↗

arXiv2016

The Leray Dimension of a Convex Code

Convex codes were recently introduced as models for neural codes in the brain. Any convex code $\C$ has an associated minimal embedding dimension $d(\C)$, which is the minimal Euclidean space dimension such that the code can be realized by a collection of convex open sets. In this work we import tools from combinatorial commutative algebra in order to obtain better bounds on $d(\C)$ from an associated simplicial complex $Δ(\C)$. In particular, we make a connection to minimal free resolutions of Stanley-Reisner ideals, and observe that they contain topological information that provides stronger bounds on $d(\C)$. This motivates us to define the Leray dimension $d_L(\C),$ and show that it can be obtained from the Betti numbers of such a minimal free resolution. We compare $d_L(\C)$ to two previously studied dimension bounds, obtained from Helly's theorem and the simplicial homology of $Δ(\C)$. Finally, we show explicitly how $d_L(\C)$ can be computed algebraically, and illustrate this with examples.

math.CO↗

arXiv2016

Poisson and near-symplectic structures on generalized wrinkled fibrations in dimension 6

We show that generalized broken fibrations in arbitrary dimensions admit rank-2 Poisson structures compatible with the fibration structure. After extending the notion of wrinkled fibration to dimension 6 we prove that these wrinkled fibrations also admit compatible rank-2 Poisson structures. In the cases with indefinite singularities we can provide these wrinkled fibrations in dimension 6 with near-symplectic structures.

math.SG↗

arXiv2015

Near-symplectic 2n-manifolds

We give a generalization of the concept of near-symplectic structures to 2n dimensions. According to our definition, a closed 2-form ωon a 2n-manifold M is near-symplectic, if it is symplectic outside a submanifold Z of codimension 3, where ω^{n-1} vanishes. This extends the concept known in dimension 4. We depict how this notion relates to near-symplectic 4-manifolds and broken Lefschetz fibrations via some examples. We define a generalized broken Lefschetz fibration, or BLF, as a singular map with indefinite folds and Lefschetz-type singularities. We show that given such a map on a 2n-manifold over a symplectic base of codimension 2, then the total space carries such a near-symplectic structure, whose singular locus corresponds precisely to the singularity set of the fibration. A second part studies the geometry around the codimension--3 singular locus Z. We describe a splitting property of the normal bundle N_Z that is also present in dimension four. A tubular neighbourhood for Z is provided, which has as a corollary a Darboux-type theorem for near-symplectic forms.

math.SG↗

arXiv2015

Poisson structures on smooth 4-manifolds

We show that every closed oriented smooth 4-manifold admits a complete singular Poisson structure in each homotopy class of maps to the 2-sphere. The rank of this structure is 2 outside a small singularity set, which consists of finitely many circles and isolated points. The Poisson bivector has rank 0 on the singularities, where we give its local form explicitly.

math.DG↗