On a Spectral Sequence for Twisted Cohomologies

Let(Ω*(M), d) be the de Rham cochain complex for a smooth compact closed manifolds M of dimension n. For an odd-degree closed form H, there is a twisted de Rham cochain complex(Ω*(M), d + H∧) and its associated twisted de Rham cohomology H*(M, H). The authors show that there exists a spectral sequence {Ep,qr, dr} derived from the filtration Fp(Ω*(M)) = i≥pΩi(M) of Ω*(M), which converges to the twisted de Rham cohomology H*(M, H). It is also shown that the differentials in the spectral sequence can be given in terms of cup products and specific elements of Massey products as well,which generalizes a result of Atiyah and Segal. Some results about the indeterminacy of differentials are also given in this paper.     

Auslander－Gorenstein滤环上的全律模

本文给出了Ａｕｓｌａｎｄｅｒ－Ｇｏｒｅｎｓｔｅｉｎ滤环上全律模（ｈｏｌｏｎｏｍｉｃｍｏｄｕｌｅｓ）的一个特征簇刻划，由此得到了通过特征理想的极小素因子（有限个）计算余维数的公式。     

Auslander－Gorenstein滤环上的全律模

本文给出了Ａｕｓｌａｎｄｅｒ－Ｇｏｒｅｎｓｔｅｉｎ滤环上全律模（ｈｏｌｏｎｏｍｉｃｍｏｄｕｌｅｓ）的一个特征簇刻划，由此得到了通过特征理想的极小素因子（有限个）计算余维数的公式。     

Filtered deformations of the Frank algebras

In this paper we prove the rigidity of simple Lie algebras of the Frank series of characteristic 3 with the standard grading of depth 2 with respect to filtered deformations.     

Symplectic resolutions, Lefschetz property and formality

We introduce a method to resolve a symplectic orbifold(M,ω) into a smooth symplectic manifold . Then we study how the formality and the Lefschetz property of are compared with that of (M,ω). We also study the formality of the symplectic blow-up of (M,ω) 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 of Babenko and Taimanov.     

On a Filtered Multiplicative Bases of Group Algebras II

We give an explicit list of all p-groups G with a cyclic subgroup of index p ², such that the group algebra KG over the field K of characteristic p has a filtered multiplicative K-basis. We also prove that such a K-basis does not exist for the group algebra KG, in the case when G is either a non-Abelian powerful p-group or a two generated p-group (p2) with a central cyclic commutator subgroup. This paper is a continuation of the paper which appeared in Arch. Math. (Basel) 74 (2000), 217–285.     

Symplectic Structures and Cohomologies on Some Solvmanifolds

We consider the question of existence of symplectic and Kahler structures on compact homogeneous spaces of solvable triangular Lie groups. The aim of the article is to clarify the situation with examples in this area. We prove that it is impossible to complete the construction of examples in the well-known article by Benson and Gordon on the structure of compact solvmanifolds with Kahler structure. We do this by proving the absence of lattices (and thereby a compact form) in the Lie groups of the above-mentioned article. We construct a new (similar) example for which, unlike the above examples, a compact form exists. We consider one class of solvable Lie groups, namely the class of almost abelian groups, and obtain for this class a characterization of those Lie groups for which the cohomologies of their compact solvmanifolds are isomorphic to the cohomologies of the corresponding Lie algebras. Until recently, such isomorphism has been known only for one specific class of Lie groups, namely the class of triangular groups. We give examples of new (almost abelian) Lie groups with such isomorphism.     

An Example Using Improved Lefschetz Duality

A theorem of Lambrechts and Stanley is used to find the rational cohomology of the complement of an embedding S~(4n-1)→ S~(2n)× S~m as a module and demonstrate that it is not necessarily determined by the map induced on cohomology by the embedding, nor is it a trivial extension. This demonstrates that the theorem is an improvement on the classical Lefschetz duality.     

Normal Form in Filtered Lie Algebra Representations

This paper gives a setup for normal form theory and the computation of normal forms with emphasis on the dual character of the transformation generators and the objects to be transformed into normal form. Spectral sequence techniques will be used to define unique normal forms. Theoretical estimates are given for the rate of convergence in terms of the spectral sequence.Mathematics Subject Classifications (2000) primary: 34E, secondary: 18G40.     

On the bilinear and cubic forms of some symplectic connected sums

Let ( M1 , M2 , N ) be three symplectic manifolds and suppose that we can do the symplectic connected sum of M1 and M2 along their submanifold N to obtain M1#NM2 . In this paper, we consider the bilinear and cubic forms of H* (M1#NM2 , Z) when dim M1#NM2 = 4, 6. Under some conditions, we get some relations of the bilinear and the cubic forms between M1#NM2 and M1■M2 .     

Examples of non-Kähler Hamiltonian circle manifolds with the strong Lefschetz property

In this paper we construct six-dimensional compact non-Kähler Hamiltonian circle manifolds which satisfy the strong Lefschetz property themselves but nevertheless have a non-Lefschetz symplectic quotient. This provides the first known counterexamples to the question whether the strong Lefschetz property descends to the symplectic quotient. We also give examples of Hamiltonian strong Lefschetz circle manifolds which have a non-Lefschetz fixed point submanifold. In addition, we establish a sufficient and necessary condition for a finitely presentable group to be the fundamental group of a strong Lefschetz manifold. We then use it to show the existence of Lefschetz four-manifolds with non-Lefschetz finite covering spaces.     

An obstruction to quantizing compact symplectic manifolds

We prove that there are no nontrivial finite-dimensional Lie representations of certain Poisson algebras of polynomials on a compact symplectic manifold. This result is used to establish the existence of a universal obstruction to quantizing a compact symplectic manifold, regardless of the dimensionality of the representation.

     

Flag Duality

We introduce a duality on complex flag manifolds that extendsthe usual point-hyperplane duality of complex projective spaces. Thishas consequences for the structure of the linear cycle spaces of flagdomains, especially when those flag domains are not measurable.     

On the Existence of n-Harmonic Spheres

Using the bubbling argument of Sacks and Uhlenbeck, we prove the existence of n-harmonic maps from the n-sphere to Riemannian manifolds. An application is made to a problem concerning manifolds with strongly pth moment stable stochastic dynamical systems.     

Hodge structures for orbifold cohomology

We construct a polarized Hodge structure on the primitive part of Chen and Ruan's orbifold cohomology for projective -orbifolds satisfying a ``Hard Lefschetz Condition'. Furthermore, the total cohomology forms a mixed Hodge structure that is polarized by every element of the Kähler cone of . Using results of Cattani-Kaplan-Schmid this implies the existence of an abstract polarized variation of Hodge structure on the complexified Kähler cone of .

This construction should be considered as the analogue of the abstract polarized variation of Hodge structure that can be attached to the singular cohomology of a crepant resolution of , in light of the conjectural correspondence between the (quantum) orbifold cohomology and the (quantum) cohomology of a crepant resolution.

     

Vertical Cohomologies and Their Application to Completely Integrable Hamiltonian Systems

Some functorial and topological properties of vertical cohomologies and their application to completely integrable Hamiltonian systems are studied.     

Symplectically hyperbolic manifolds

A symplectic form is called hyperbolic if its pull-back to the universal cover is a differential of a bounded one-form. The present paper is concerned with the properties and constructions of manifolds admitting hyperbolic symplectic forms. The main results are:

• If a symplectic form represents a bounded cohomology class then it is hyperbolic.

• The symplectic hyperbolicity is equivalent to a certain isoperimetric inequality.

• The fundamental group of symplectically hyperbolic manifold is non-amenable.

We also construct hyperbolic symplectic forms on certain bundles and Lefschetz fibrations, discuss the dependence of the symplectic hyperbolicity on the fundamental group and discuss some properties of the group of symplectic diffeomorphisms of a symplectically hyperbolic manifold.

Symmetries of Lagrangian fibrations

We construct fiber-preserving anti-symplectic involutions for a large class of symplectic manifolds with Lagrangian torus fibrations. In particular, we treat the K3 surface and the six-dimensional examples constructed by Castaño-Bernard and Matessi (2009) [8], which include a six-dimensional symplectic manifold homeomorphic to the quintic threefold. We interpret our results as corroboration of the view that in homological mirror symmetry, an anti-symplectic involution is the mirror of duality. In the same setting, we construct fiber-preserving symplectomorphisms that can be interpreted as the mirror to twisting by a holomorphic line bundle.