Yi Lin, Reyer Sjamaar
Consider a Hamiltonian action of a compact Lie group on a symplectic manifold which has the strong Lefschetz property. We establish an equivariant version of the Merkulov-Guillemin $dδ$-lemma and an improved version of the Kirwan-Ginzburg equivariant formality theorem, which says that every cohomology class has a \emph{canonical} equivariant extension.
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Lin and Sjamaar have used symplectic Hodge theory to obtain canonical equivariant extensions for Hamiltonian actions on closed symplectic manifolds that have the strong Lefschetz property. Here we obtain canonical equivariant extensions much more generally by means of classical Hodge theory.
Using the mirror theorem [CCIT15], we give a Landau-Ginzburg mirror description for the big equivariant quantum cohomology of toric Deligne-Mumford stacks. More precisely, we prove that the big equivariant quantum D-module of a toric Deligne-Mumford stack is isomorphic to the Saito structure associated to the mirror Landau-Ginzburg potential. We give a GKZ-style presentation of the quantum D-module, and a combinatorial description of quantum cohomology as a quantum Stanley-Reisner ring. We establish the convergence of the mirror isomorphism and of quantum cohomology in the big and equivariant …
In this paper we first consider the Hamiltonian action of a compact connected Lie group on an $H$-twisted generalized complex manifold $M$. Given such an action, we define generalized equivariant cohomology and generalized equivariant Dolbeault cohomology. If the generalized complex manifold $M$ satisfies the $\bar{\partial}\partial$-lemma, we prove that they are both canonically isomorphic to $(S\g^*)^G\otimes H_H(M)$, where $(S\g^*)^G$ is the space of invariant polynomials over the Lie algebra $\g$ of $G$, and $H_H(M)$ is the $H$-twisted cohomology of $M$. Furthermore, we establish an equiva…
Consider the Hamiltonian action of a compact connected Lie group on a transversely symplectic foliation which satisfies the transverse hard Lefschetz property. We establish an equivariant formality theorem and an equivariant symplectic $dδ$-lemma in this setting. As an application, we show that if the foliation is also Riemannian, then there exists a natural formal Frobenius manifold structure on the equivariant basic cohomology of the foliation.
Symplectic Hodge Theory on Lie Algebroids
本文研究了辛几何中李代数簇上的辛Hodge理论，探讨了Brylinksi条件、强Lefschetz条件和dδ引理的类比，并通过代数方法证明其等价性及几何应用。
In this paper, we develop symplectic Hodge theory on transversely symplectic foliations. In particular, we establish the symplectic $dδ$-lemma for any such foliations with the (transverse) $s$-Lefschetz property. As transversely symplectic foliations include many geometric structures, such as contact manifolds, co-symplectic manifolds, symplectic orbifolds, and symplectic quasi-folds as special examples, our work provides a unifying treatment of symplectic Hodge theory in these geometries. As an application, we show that on compact $K$-contact manifolds, the $s$-Lefschetz property implies a ge…
Suppose that a compact and connected Lie group $G$ acts on a complex Hodge manifold $M$ in a holomorphic and Hamiltonian manner, and that the action linearizes to a positive holomorphic line bundle $A$ on $M$. Then there is an induced unitary representation on the associated Hardy space and, if the moment map of the action is nowhere vanishing, the corresponding isotypical components are all finite dimensional. We study the asymptotic concentration behavior of the corresponding equivariant Szegö kernels near certain loci defined by the moment map.
Suppose given a complex projective manifold $M$ with a fixed Hodge form $Ω$. The Bohr-Sommerfeld Lagrangian submanifolds of $(M,Ω)$ are the geometric counterpart to semi-classical physical states, and their geometric quantization has been extensively studied. Here we revisit this theory in the equivariant context, in the presence of a compatible (Hamiltonian) action of a connected compact Lie group.
Let G be a compact connected Lie group, and (M,ω) a Hamiltonian G-space with proper moment map μ. We give a surjectivity result which expresses the K-theory of the symplectic quotient M//G in terms of the equivariant K-theory of the original manifold M, under certain technical conditions on μ. This result is a natural K-theoretic analogue of the Kirwan surjectivity theorem in symplectic geometry. The main technical tool is the K-theoretic Atiyah-Bott lemma, which plays a fundamental role in the symplectic geometry of Hamiltonian G-spaces. We discuss this lemma in detail and highlight the diffe…
Equivariant Morse theory for Lie algebra actions on Riemannian foliations
该研究探讨黎曼叶状结构上李代数作用的等变莫尔斯理论，建立相关引理与定理，推广基尔万定理并给出黎曼叶状结构莫尔斯不等式的新证明。
Cyclotomic Structures in Symplectic Topology
本文推广Cohen-Jones-Segal构造至等变Morse函数，发展等变Floer理论，并在辛流形上建立辛上同调谱提升的分圆结构，连接Floer同调与p-adic Hodge理论。
We set up an algebraic framework for the study of pseudoholomorphic discs bounding nonorientable Lagrangians, as well as equivariant extensions of such structures arising from a torus action. First, we define unital cyclic twisted $A_\infty$ algebras and prove some basic results about them, including a homological perturbation lemma which allows one to construct minimal models of such algebras. We then construct an equivariant extension of $A_\infty$ algebras which are invariant under a torus action on the underlying complex. Finally, we construct a homotopy retraction of the Cartan-Weil compl…
Cyclic homology, $S^1$-equivariant Floer cohomology, and Calabi-Yau structures
本文构造Fukaya范畴循环同调到$S^1$-等变上同调的循环开-闭映射，用以建立Calabi-Yau结构并证明非交换Hodge-de Rham退化。
It has been shown recently by Kapustin and Tomasiello that the mathematical notion of Hamiltonian actions on twisted generalized Kähler manifolds is in perfect agreement with the physical notion of general $(2,2)$ gauged sigma models with three-form fluxes. In this article, we study the twisted equivariant cohomology theory of Hamiltonian actions on $H$-twisted generalized complex manifolds. If the manifold satisfies the $\bar{\partial}\partial$-lemma, we establish the equivariant formality theorem. If in addition, the manifold satisfies the generalized Kähler condition, we prove the Kirwan in…
A polynomial assignment for a continuous action of a compact torus $T$ on a topological space $X$ assigns to each $p\in X$ a polynomial function on the Lie algebra of the isotropy group at $p$ in such a way that a certain compatibility condition is satisfied. The space ${\mathcal{A}}_T(X)$ of all polynomial assignments has a natural structure of an algebra over the polynomial ring of ${\rm Lie}(T)$. It is an equivariant homotopy invariant, canonically related to the equivariant cohomology algebra. In this paper we prove various properties of ${\mathcal{A}}_T(X)$ such as Borel localization, a C…
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