On Ginzburg's bivariant Chern classes

The convolution product is an important tool in geometric representation theory. Ginzburg constructed the ``bivariant" Chern class operation from a certain convolution algebra of Lagrangian cycles to the convolution algebra of Borel-Moore homology. In this paper we prove a ``constructible function version" of one of Ginzburg's results; motivated by its proof, we introduce another bivariant algebraic homology theory on smooth morphisms of nonsingular varieties and show that the Ginzburg bivariant Chern class is the unique Grothendieck transformation from the Fulton-MacPherson bivariant theory of constructible functions to this new bivariant algebraic homology theory, modulo a reasonable conjecture. Furthermore, taking a hint from this conjecture, we introduce another bivariant theory of constructible functions, and we show that the Ginzburg bivariant Chern class is the unique Grothendieck transformation from to satisfying the ``normalization condition" and that it becomes the Chern-Schwartz-MacPherson class when restricted to the morphisms to a point.

     

Classes de Chern bivariantes et classes polaires relatives

W. Fulton and R. MacPherson conjectured the existence and the uniqueness of bivariant Chern classes. J.-P. Brasselet and C. Sabbah have given each one a construction and these constructions coincide. In this paper we give a relation between these bivariant Chern classes and the relative polar classes.     

Bivariant Chern classes for morphisms with nonsingular target varieties

W. Fulton and R. MacPherson posed the problem of unique existence of a bivariant Chern class—a Grothendieck transformation from the bivariant theory F of constructible functions to the bivariant homology theory H. J.-P. Brasselet proved the existence of a bivariant Chern class in the category of embeddable analytic varieties with cellular morphisms. In general however, the problem of uniqueness is still unresolved. In this paper we show that for morphisms having nonsingular target varieties there exists another bivariant theory of constructible functions and a unique bivariant Chern class γ: . Partially supported by Grant-in-Aid for Scientific Research (C) (No. 15540086+No. 17540088), the Japanese Ministry of Education, Science, Sports and Culture.     

Chern character for totally disconnected groups

In this paper we construct a bivariant Chern character for the equivariant KK-theory of a totally disconnected group with values in bivariant equivariant cohomology in the sense of Baum and Schneider. We prove in particular that the complexified left hand side of the Baum–Connes conjecture for a totally disconnected group is isomorphic to cosheaf homology. Moreover, it is shown that our transformation extends the Chern character defined by Baum and Schneider for profinite groups.     

On Ginzburg's Bivariant Chern Classes, II

For a morphism whose target variety is nonsingular, the Chern–Schwartz–MacPherson class homomorphism followed by capping with the pullback of the Segre class of the target variety is called the Ginzburg–Chern class. In this paper, using the Verdier–Riemann–Roch for Chern Class, we show that the correspondence assigning to a bivariant constructible function on any morphism with nonsingular target variety the Ginzburg–Chern class of it is the unique Grothendieck transformation satisfying the 'normalization condition' that for morphisms to a point it becomes the Chern–Schwartz–MacPherson class homomorphism, except for that the bivariant homology pullback is considered only for a smooth morphism.     

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.

     

On the uniqueness of bivariant Chern class and bivariant Riemann–Roch transformations

The existence of bivariant Chern classes was conjectured by W. Fulton and R. MacPherson and proved by J.-P. Brasselet for cellular morphisms of analytic varieties. However, its uniqueness has been unsolved since then. In this paper we show that restricted to morphisms whose target varieties are possibly singular but (rational) homology manifolds (such as orbifolds), the bivariant Chern classes (with rational coefficients) are uniquely determined. We also discuss some related results and problems.     

Bivariant Chern-Schwartz-MacPherson classes with values in Chow groups

Fulton and MacPherson asked if there exists a bivariant version of the Chern-Schwartz-MacPherson class. Brasselet solved this problem affirmatively in the category of analytic varieties and cellular morphisms. However, it has not been solved in the general case and the uniqueness of such a bivariant Chern class is still open. In this paper we show the unique existence of the bivariant Chern-Schwartz-MacPherson class with values in Chow groups. To be more precise, we show that there exists a unique Grothendieck transformation from the bivariant theory of constructible functions to Fulton-MacPherson's operational bivariant theory of Chow groups, provided that the compatibility with flat pullback is not required on the operational bivariant theory.     

Base independent algebraic cobordism

The purpose of this article is to show that the bivariant algebraic A-cobordism groups considered previously by the author are independent of the chosen base ring A. This result is proven by analyzing the bivariant ideal generated by the so called snc relations, and, while the alternative characterization we obtain for this ideal is interesting by itself because of its simplicity, perhaps more importantly it allows us to easily extend the definition of bivariant algebraic cobordism to divisorial Noetherian derived schemes of finite Krull dimension. As an interesting corollary, we define the corresponding homology theory called algebraic bordism. We also generalize projective bundle formula, the theory of Chern classes, the Conner–Floyd theorem and the Grothendieck–Riemann–Roch theorem to this setting. The general definitions of bivariant cobordism are based on the careful study of ample line bundles and quasi-projective morphisms of Noetherian derived schemes, also undertaken in this work.     

On the commutativity of localized Chern characters with refined Gysin homomorphisms

We give a simple proof of the fact that the localized Chern characters of Baum, Fulton and MacPherson commute with the refined Gysin homomorphisms of [3]. This has been proved in [3] in the context of bivariant intersection theory using the technique of deformation to the normal cone. Our proof is more elementary in the sense that it avoids such a deformation and relies on a commutativity formula of these Chern characters with effective Cartier divisors. From this formula we also derive easily that the localized Chern characters pass to rational equivalence.     

Bivariant Chern character and longitudinal index

In this paper we consider a family of Dirac-type operators on fibration P→B equivariant with respect to an action of an étale groupoid. Such a family defines an element in the bivariant K theory. We compute the action of the bivariant Chern character of this element on the image of Connes' map Φ in the cyclic cohomology. A particular case of this result is Connes' index theorem for étale groupoids [A. Connes, Noncommutative Geometry, Academic Press, 1994] in the case of fibrations.     

On Grothendieck transformations in Fulton-MacPherson’s bivariant theory

W. Fulton and R. MacPherson have introduced a notion unifying both covariant and contravariant theories, which they called a Bivariant Theory. A transformation between two bivariant theories is called a Grothendieck transformation. The Grothendieck transformation induces natural transformations for covariant theories and contravariant theories. In this paper we show some general uniqueness and existence theorems on Grothendieck transformations associated to given natural transformations of covariant theories. Our guiding or typical model is MacPherson’s Chern class transformation c_∗:F→H_∗. The existence of a corresponding bivariant Chern class γ:F→H was conjectured by W. Fulton and R. MacPherson, and was proved by J.-P. Brasselet under certain conditions.     

Waldhausen范畴的陈映射

本文讨论了关于 Ｗａｌｄｈａｕｓｅｎ范畴的Ｋ理论与循环同调理论的联系．主要是推广了 Ｄｅｎｎｉｓ迹映射和Ｊｏｎｅｓ－Ｇｏｏｄｗｉｌｌｉｅ Ｃｈｅｒｎ映射．证明了迹映射保持各自的乘法结构．最后讨论了双变量的情形．     

Chern class of Schubert cells in the flag manifold and related algebras

We discuss a relationship between Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds, the Fomin–Kirillov algebra, and the generalized nil-Hecke algebra. We show that the nonnegativity conjecture in the Fomin–Kirillov algebra implies the nonnegativity of the Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds for type A. Motivated by this connection, we also prove that the (equivariant) Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds are certain summations of the structure constants of the equivariant cohomology of Bott–Samelson varieties. We also discuss refined positivity conjectures of the Chern–Schwartz–MacPherson classes for Schubert cells motivated by the nonnegativity conjecture in the Fomin–Kirillov algebra.     

Equivariant–Bivariant Chern Character for Profinite Groups

For the action of a locally compact and totally disconnected group G on a pair of locally compact spaces X and Y we construct, by sheaf theoretic means, a new equivariant and bivariant cohomology theory. If we take for the first space Y an universal proper G-action then we obtain for the second space its delocalized equivariant homology. This is in exact formal analogy to the definition of equivariant K-homology by Baum, Connes, Higson starting from the bivariant equivariant Kasparov KK-theory. Under certain basic finiteness conditions on the first space Y we conjecture the existence of a Chern character from the equivariant Kasparov KK-theory of Y and X into our cohomology theory made two-periodic which becomes an isomorphism upon tensoring the KK-theory with the complex numbers. This conjecture is proved for profinite groups G. An essential role in our construction is played by a bivariant version of Segal localization which we establish for KK-theory.     

Well‐posedness of time‐space fractional stochastic evolution equations driven by α‐stable noise

This paper is devoted to the well‐posedness for time‐space fractional Ginzburg‐Landau equation and time‐space fractional Navier‐Stokes equations by α‐stable noise. The spatial regularity and the temporal regularity of the nonlocal stochastic convolution are firstly established, and then the existence and uniqueness of the global mild solution are obtained by the Banach fixed point theorem and Mittag‐Leffler functions, respectively. Numerical simulations for time‐space fractional Ginzburg‐Landau equation are provided to verify the analysis results.     

Thom elements and bivariant cyclic cohomology

We consider smooth dynamical systems with unital algebra. We show that the corresponding crossed product isH-unital, and use differentiable Thom elements for proving a Thom isomorphism in bivariant periodic cohomology.     

Representations of Gan–Ginzburg algebras

Given a quiver, a fixed dimension vector, and a positive integer n, we construct a functor from the category of D-modules on the space of representations of the quiver to the category of modules over a corresponding Gan–Ginzburg algebra of rank n. When the quiver is affine Dynkin, we obtain an explicit construction of representations of the corresponding wreath product symplectic reflection algebra of rank n. When the quiver is star-shaped, but not finite Dynkin, we use this functor to obtain a Lie-theoretic construction of representations of a “spherical” subalgebra of the Gan–Ginzburg algebra isomorphic to a rational generalized double affine Hecke algebra of rank n. Our functors are a generalization of the type A and type BC functors from [1] and [4], respectively.