Symplectic Lefschetz fibrations on S^1 x M^3

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.

The $dδ$--lemma for weakly Lefschetz symplectic manifolds

For a symplectic manifold $(M,ω)$, not necessarily hard Lefschetz, we prove a version of the Merkulov $dδ$--lemma. We also study the $dδ$--lemma and related cohomologies for compact symplectic solvmanifolds.

Constraints on Lefschetz fibrations with four-dimensional fibers from Seiberg-Witten theory

We establish constraints on the topology of smooth Lefschetz fibrations with $4$-dimensional fibers, by studying the family Bauer-Furuta invariant. To compute this invariant, we analyze the framed bordism class of 1-dimensional Seiberg-Witten moduli spaces using the local index theorem by Bismut-Freed. Using this, we deduce new obstructions to the smooth isotopy to the identity for compositions of Dehn twists on $(-2)$-spheres in closed $4$-manifolds. We obtain several applications: (1) We exhibit the first examples of closed simply-connected symplectic $4$-manifolds admitting Torelli symplect…

Resolution of the canonical fiber metrics for a Lefschetz fibration

We consider the family of constant curvature fiber metrics for a Lefschetz fibration with regular fibers of genus greater than one. A result of Obitsu and Wolpert is refined by showing that on an appropriate resolution of the total space, constructed by iterated blow-up, this family is log-smooth, i.e. polyhomogeneous with integral powers but possible multiplicities, at the preimage of the singular fibers in terms of parameters of size comparable to the logarithm of the length of the shrinking geodesic.

共形辛结构与 Lefschetz 条件

Conformally symplectic structures and the Lefschetz condition

本文证明紧辛流形在次数 1 满足硬 Lefschetz 条件且具有相容近复结构时，局部共形辛结构必为全局共形辛结构，给出 Vaisman 定理的辛类比。

The Lefschetz property, formality and blowing up in symplectic geometry

In this paper we study the behaviour of the Lefschetz property under the blow-up construction. We show that it is possible to reduce the dimension of the kernel of the Lefschetz map if we blow up along a suitable submanifold satisfying the Lefschetz property. We use that, together with results about Massey products, to construct nonformal (simply connected) symplectic manifolds satisfying the Lefschetz property.

Symplectic resolutions, Lefschetz property and formality

We introduce a method to resolve a symplectic orbifold into a smooth symplectic manifold. Then we study how the formality and the Lefschetz property of the symplectic resolution are compared with that of the symplectic orbifold. We also study the formality of the symplectic blow-up of a symplectic orbifold along symplectic submanifolds disjoint from the orbifold singularities. This allows us to construct the first example of a simply connected compact symplectic manifold of dimension 8 which satisfies the Lefschetz property but is not formal, therefore giving a counter-example to a conjecture …

Lagrangian submanifolds and Lefschetz pencils

Given a Lagrangian submanifold in a symplectic manifold and a Morse function on the submanifold, we show that there is an isotopic Morse function and a symplectic Lefschetz pencil on the manifold extending the Morse function to the whole manifold. From this construction we define a sequence of symplectic invariants classifying the isotopy classes of Lagrangian spheres in a symplectic 4-manifold.

Formality and the Lefschetz property in symplectic and cosymplectic geometry

We review topological properties of Kähler and symplectic manifolds, and of their odd-dimensional counterparts, coKähler and cosymplectic manifolds. We focus on formality, Lefschetz property and parity of Betti numbers, also distinguishing the simply-connected case (in the Kähler/symplectic situation) and the $b_1=1$ case (in the coKähler/cosymplectic situation).

Transversal hard Lefschetz theorem on transversely symplectic foliations

We study the transversal hard Lefschetz theorem on a transversely symplectic foliation. This article extends the results of transversally symplectic flows (H.K.~Pak, "Transversal harmonic theory for transversally symplectic flows", J. Aust. Math. Soc. 84 (2008), 233--245) to the general transversely symplectic foliation.

具有和没有Hard Lefschetz条件的辛非Kähler流形

Symplectic non-Kähler manifolds with and without the Hard Lefschetz Condition

构造无Kähler结构的紧流形，使其同时存在满足和不满足Hard Lefschetz条件的辛形式，并证明这些结构位于同一连通分支中。

$L^{2}$-hard Lefschetz complete symplectic manifolds

For a complete symplectic manifold $M^{2n}$, we define the $L^{2}$-hard Lefschetz property on $M^{2n}$. We also prove that the complete symplectic manifold $M^{2n}$ satisfies $L^{2}$-hard Lefschetz property if and only if every class of $L^{2}$-harmonic forms contains a $L^{2}$ symplectic harmonic form. As an application, we get if $M^{2n}$ is a closed symplectic parabolic manifold which satisfies the hard Lefschetz property, then its Euler characteristic satisfies the inequality $(-1)^{n}χ(M^{2n})\geq0$.

圆盘上边界 Lefschetz 纤维化的分类

Classification of boundary Lefschetz fibrations over the disc

本文完全分类了圆盘上具有边界 Lefschetz 纤维化的四维流形，并由此确定了哪些流形能承载类型变化轨迹为单分量的稳定广义复结构。

辛非Kähler可解流形上的Hard Lefschetz条件

Hard Lefschetz Condition on symplectic non-Kähler solvmanifolds

本文构造了无Kähler结构的紧致复流形族，通过可解李群的紧致商得到，并证明其承载满足Hard Lefschetz条件的辛结构，同时计算了Kodaira维数及上同调生成元。