非交换陈特征的一个注记

此注记利用Moscovici的一个想法和热核渐近展开技术,给出了Clifford模 上Dirac算子的整循环陈特征的一个计算公式.     

An Intrinsic Rigidity Theorem for Closed Minimal Hypersurfaces in S5 with Constant Nonnegative Scalar Curvature

Let M⁴ be a closed minimal hypersurface in \(\mathbb{S}^5\) with constant nonnegative scalar curvature. Denote by f₃ the sum of the cubes of all principal curvatures, by g the number of distinct principal curvatures. It is proved that if both f₃ and g are constant, then M⁴ is isoparametric. Moreover, the authors give all possible values for squared length of the second fundamental form of M⁴. This result provides another piece of supporting evidence to the Chern conjecture.     

Multiple Existence Results for the Self-Dual Chern–Simons–Higgs Vortex Equation

We study the asymptotic behavior for the condensate solutions of the self-dual Chern–Simons–Higgs equation as the Chern–Simons parameter tends to zero. By using these estimates, we establish existence results for solutions of non-topological type.     

Morse–Novikov cohomology of locally conformally Kähler manifolds

A locally conformally Kähler (LCK) manifold is a complex manifold admitting a Kähler covering, with the monodromy acting on this covering by holomorphic homotheties. We define three cohomology invariants, the Lee class, the Morse–Novikov class, and the Bott–Chern class, of an LCK-structure. These invariants play together the same role as the Kähler class in Kähler geometry. If these classes coincide for two LCK-structures, the difference between these structures can be expressed by a smooth potential, similar to the Kähler case. We show that the Morse–Novikov class and the Bott–Chern class of a Vaisman manifold vanish. Moreover, for any LCK-structure on a manifold, admitting a Vaisman structure, we prove that its Morse–Novikov class vanishes. We show that a compact LCK-manifold M$M$ with vanishing Bott–Chern class admits a holomorphic embedding into a Hopf manifold, if _dimCM?3${dim}_{\mathbb{C}}M?3$, a result which parallels the Kodaira embedding theorem.     

Hopf algebras of primitive Lie pseudogroups and Hopf cyclic cohomology

We associate to each infinite primitive Lie pseudogroup a Hopf algebra of ‘transverse symmetries,’ by refining a procedure due to Connes and the first author in the case of the general pseudogroup. The affiliated Hopf algebra can be viewed as a ‘quantum group’ counterpart of the infinite-dimensional primitive Lie algebra of the pseudogroup. It is first constructed via its action on the étale groupoid associated to the pseudogroup, and then realized as a bicrossed product of a universal enveloping algebra by a Hopf algebra of regular functions on a formal group. The bicrossed product structure allows to express its Hopf cyclic cohomology in terms of a bicocyclic bicomplex analogous to the Chevalley-Eilenberg complex. As an application, we compute the relative Hopf cyclic cohomology modulo the linear isotropy for the Hopf algebra of the general pseudogroup, and find explicit cocycle representatives for the universal Chern classes in Hopf cyclic cohomology. As another application, we determine all Hopf cyclic cohomology groups for the Hopf algebra associated to the pseudogroup of local diffeomorphisms of the line.     

Quillen superconnections and connections on supermanifolds

Given a supervector bundle $E=E_0\oplus E_1\to M$, we exhibit a parametrization of Quillen superconnections on $E$ by graded connections on the Cartan–Koszul supermanifold $(M,\Omega \left(M\right))$. The relation between the curvatures of both kind of connections, and their associated Chern classes, is discussed in detail. In particular, we find that Chern classes for graded vector bundles on split supermanifolds can be computed through the associated Quillen superconnections.     

ON THE CHERN CONNECTION OF FINSLER SUBMANIFOLDS

This paper studies the induced Chern connection of submanifolds in a Finsler manifold and gets the relations between the induced Chern connection and the Chern connection of the induced Finsler metric.Then the authors point out a difference between Finsler submanifolds and Riemann submanifolds.     

Perturbed Dolbeault operators and the homology Todd class

This paper discusses the role played by perturbed Dolbeault operators in relating the coherent sheaf and elliptic operator perspectives on the homology of projective varieties. Among the consequences are index formulas for perturbed Dolbeault operators.

     

Caractère de Chern bivariant

We construct a bivariant Chern character with values in Jones-Kassel's bivariant cyclic cohomology. This is done forK-theoretic objects such as idempotents, bimodules, quasi-homomorphisms à la Cuntz and extensions of algebras.