On Thom spaces, Massey products and non-formal symplectic manifolds

A note on hamiltonian Lie group actions and Massey products

In this note we show that the property of having only vanishing triple Massey products in the equivariant cohomology is inherited by the set of fixed points of hamiltonian circle actions on closed symplectic manifolds. This result can be considered in a more general context of characterizing homotopic properties of Lie group actions. In particular, it can be viewed as a partial answer to the Allday-Puppe question about finding conditions ensuring the "formality" of G-actions.

The symplectic Thom conjecture

In this paper, we demonstrate a relation among Seiberg-Witten invariants which arises from embedded surfaces in four-manifolds whose self-intersection number is negative. These relations, together with Taubes' basic theorems on the Seiberg-Witten invariants of symplectic manifolds, are then used to prove the symplectic Thom conjecture: a symplectic surface in a symplectic four-manifold is genus-minimizing in its homology class. Another corollary of the relations is a general adjunction inequality for embedded surfaces of negative self-intersection in four-manifolds.

Non-formal co-symplectic manifolds

We study the formality of the mapping torus of an orientation-preserving diffeomorphism of a manifold. In particular, we give conditions under which a mapping torus has a non-zero Massey product. As an application we prove that there are non-formal compact co-symplectic manifolds of dimension $m$ and with first Betti number $b$ if and only if $m=3$ and $b \geq 2$, or $m \geq 5$ and $b \geq 1$. Explicit examples for each one of these cases are given.

Hamiltonian Thom 同痕引理的一个证明

A proof of the Hamiltonian Thom Isotopy Lemma

本文针对紧辛流形中单参数紧辛子流形族，给出其所有成员彼此 Hamiltonian 同痕这一结论的完整证明。

D-流形的余截面局部化与 $(-2)$-移位辛导出概形，再探

Cosection Localization for D-Manifolds and $(-2)$-Shifted Symplectic Derived Schemes, Revisited

本文延续作者先前工作，为具有满射余截面的导出流形构造约化虚基本类，为$(-2)$-移位辛导出概形构造余截面局部化虚基本类，并利用Oh-Thomas结果证明代数与微分几何构造同调等价，应用于Calabi-Yau四维流形Donaldson-Thomas理论中约化不变量的构造与整性。

Symplectic Harmonic theory and the Federer-Fleming deformation theorem

In this article, we initiate a geometric measure theoretic approach to symplectic Hodge theory. In particular, we apply one of the central results in geometric measure theory, the Federer-Fleming deformation theorem, together with the cohomology theory of normal cur- rents on a differential manifold, to establish a fundamental property on symplectic Harmonic forms. We show that on a closed symplectic manifold, every real primitive cohomology class of positive degrees admits a symplectic Harmonic representative not supported on the entire mani- fold. As an application, we use it to investigate …