Circular spherical divisors and their contact topology

2021

In this note we study the contact geometry of symplectic divisors. We show the contact structure induced on the boundary of a divisor neighborhood is invariant under toric and interior blow-ups and blow-downs. We also construct an open book decomposition on the boundary of a concave divisor neighborhood and apply it to the study of universally tight contact structures of contact torus bundles.

arXiv: Symplectic Geometry, 2020

This paper investigates the contact topology associated to symplectic log Calabi-Yau pairs $ (X,D,\omega ) $. We classify, up to toric equivalence, all circular spherical divisors with $ b^+\ge 1 $ that can be embedded symplectically into a symplectic rational surface and show they are all realized as symplectic log Calabi-Yau pairs if their complements are minimal. We apply such embeddability and rigidity results to determine the Stein fillability of all contact torus bundles induced as the boundaries of circular spherical divisors with $ b^+\ge 1 $. When $ D $ is negative definite, we give explicit betti number bounds for Stein fillings of its boundary contact torus bundle. Also we show that a large family of contact torus bundles are universally tight, generalizing a conjecture by Golla and Lisca.

2014

We investigate the notion of symplectic divisorial compactification for symplectic 4-manifolds with either convex or concave type boundary. This is motivated by the notion of compactifying divisors for open algebraic surfaces. We give a sufficient and necessary criterion, which is simple and also works in higher dimensions, to determine whether an arbitrarily small concave/convex neighborhood exist for an ω-orthogonal symplectic divisor (a symplectic plumbing). If deformation of symplectic form is allowed, we show that a symplectic divisor has either a concave or convex neighborhood whenever the symplectic form is exact on the boundary of its plumbing. As an application, we classify symplectic compactifying divisors having finite boundary fundamental group. We also obtain a finiteness result of fillings when the boundary can be capped by a symplectic divisor with finite boundary fundamental group.

Journal of Symplectic Geometry

We investigate the notion of symplectic divisorial compactification for symplectic 4-manifold with either convex or concave type boundary. This is motivated by the notion of compactifying divisors for open algebraic surfaces. We classify symplectic compactifying divisor having finite boundary fundamental group.

arXiv (Cornell University), 2022

In this paper we are interested in the isotopy classes of symplectic log Calabi-Yau divisors in a fixed symplectic rational surface. We give several equivalent definitions and prove the stability, finiteness and rigidity results. Motivated by the problem of counting toric actions, we obtain a general counting formula of symplectic log Calabi-Yau divisors in a restrictive region of c 1 -nef cone. A detailed count in the case of 2-and 3-point blow-ups of complex projective space for all symplectic forms is also given. In our framework the complexity of the combinatorics of analyzing Delzant polygons is reduced to the arrangement of homology classes. Then we study its relation with almost toric fibrations. We raise the problem of realizing all symplectic log Calabi-Yau divisors by some almost toric fibrations and verify it together with another conjecture of Symington in a special region.

Journal of Fixed Point Theory and Applications

In this paper, we compute the embedded contact homology (ECH) capacities of the disk cotangent bundles D * S 2 and D * RP 2. We also find sharp symplectic embeddings into these domains. In particular, we compute their Gromov widths. In order to do that, we explicitly calculate the ECH chain complexes of S * S 2 and S * RP 2 using a direct limit argument on the action inspired by Bourgeois's Morse-Bott approach and ideas from Nelson-Weiler's work on the ECH of prequantization bundles. Moreover, we use integrable systems techniques to find explicit symplectic embeddings. In particular, we prove that the disk cotangent bundles of a hemisphere and of a punctured sphere are symplectomorphic to an open ball and a symplectic bidisk, respectively.

Journal of the Korean Mathematical Society

For a closed symplectic 4-manifold X, let Diff 0 (X) be the group of diffeomorphisms of X smoothly isotopic to the identity, and let Symp(X) be the subgroup of Diff 0 (X) consisting of symplectic automorphisms. In this paper we show that for any finitely given collection of positive integers {n 1 , n 2 ,. .. , n k } and any non-negative integer m, there exists a closed symplectic (or Kähler) 4-manifold X with b + 2 (X) > m such that the homologies H i of the quotient space Diff 0 (X)/Symp(X) over the rational coefficients are non-trivial for all odd degrees i = 2n 1 − 1,. .. , 2n k − 1. The basic idea of this paper is to use the local invariants for symplectic 4-manifolds with contact boundary, which are extended from the invariants of Kronheimer for closed symplectic 4-manifolds, as well as the symplectic compactifications of Stein surfaces of Lisca and Matić.

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.

International Mathematics Research Notices

We introduce the Kodaira dimension of contact 3-manifolds and establish some basic properties. Contact 3-manifolds with distinct Kodaira dimensions behave differently when it comes to the geography of various kinds of symplectic fillings. On the other hand, we also prove that, given any contact 3-manifold, there is a lower bound of $2\chi +3\sigma $ for all of its minimal symplectic fillings. This is motivated by Stipsicz’s result in [38] for Stein fillings. Finally, we discuss various aspects of exact self-cobordisms of fillable contact 3-manifolds.

Topology and its Applications, 1998

We discuss several applications of Seiberg-Witten theory in conjunction with an embedding theorem (proved elsewhere) for complex 2-dimensional Stein manifolds with boundary. We show that a closed, real 2-dimensional surface smoothly embedded in the interior of such a manifold satisfies an adjunction inequality, regardless of the sign of its self-intersection. This inequality gives constraints on the minimum genus of a smooth surface representing a given 2-homology class. We also discuss consequences for the contact structures existing on the boundaries of these Stein manifolds. We prove a slice version of the Bennequin-Eliashberg inequality for holomorphically tillable contact structures, and we show that there exist families of homology 3-spheres with arbitrarily large numbers of homotopic, nonisomorphic tight contact structures. Another result we mention is that the canonical class of a complex 2-dimensional Stein manifold with boundary is invariant under self-diffeomorphisms fixing the boundary.