Interpolation of characteristic classes of singular hypersurfaces

We show that the Chern–Schwartz–MacPherson class of a hypersurface X in a nonsingular variety M ‘interpolates’ between two other notions of characteristic classes for singular varieties, provided that the singular locus of X is smooth and that certain numerical invariants of X are constant along this locus. This allows us to define a lift of the Chern–Schwartz–MacPherson class of such ‘nice’ hypersurfaces to intersection homology. As another application, the interpolation result leads to an explicit formula for the Chern–Schwartz–MacPherson class of X in terms of its polar classes.     

Modification systems and integration in their Chow groups

We introduce a notion of integration on the category of proper birational maps to a given variety X, with value in an associated Chow group. Applications include new birational invariants; comparison results for Chern classes and numbers of nonsingular birational varieties; ‘stringy’ Chern classes of singular varieties; and a zeta function specializing to the topological zeta function. In its simplest manifestation, the integral gives a new expression for Chern–Schwartz–MacPherson classes of possibly singular varieties, placing them into a context in which a ‘change-of-variable’ formula holds.     

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.     

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 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.     

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.     

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.     

On the existence of a Gysin morphism for the Blow-up of an ordinary singularity

In this paper we characterize the Blowing-up maps of ordinary singularities for which there exists a natural Gysin morphism, i.e. a bivariant class \(\theta \in Hom_{D(Y)}(R\pi _*\mathbb {Q}_X, \mathbb {Q}_Y)\), compatible with pullback and with restriction to the complement of the singularity.     

A Remark on an Algebraic Riemann—Roch Formula for Flat Bundles

Let f:X S be a smooth projective morphism over an algebraically closed field, with X and S regular. When E, ) is a flat bundle over X, then its Gauss–Manin bundles on S have a flat connection and one may ask for a Riemann–Roch formula relating the algebraic Chern–Simons and Cheeger–Simons invariants. We give an answer for X = Y × S, f = projection. The method of proof is inspired by the work of Hitchin and Simpson.     

Extensions and renormalized traces

It has been shown by Nistor (Doc Math J DMV 2:263–295, 1997) that given any extension of associative algebras over \mathbb C{\mathbb C}, the connecting morphism in periodic cyclic homology is compatible, under the Chern–Connes character, with the index morphism in lower algebraic K-theory. The proof relies on the abstract properties of cyclic theory, essentially excision, which does not provide explicit formulas a priori. Avoiding the use of excision, we explain in this article how to get explicit formulas in a wide range of situations. The method is connected to the renormalization procedure introduced in our previous work on the bivariant Chern character for quasihomomorphisms Perrot (J Geom Phys 60:1441–1473, 2010), leading to “local” index formulas in the sense of non-commutative geometry. We illustrate these principles with the example of the classical family index theorem: we find that the characteristic numbers of the index bundle associated to a family of elliptic pseudodifferential operators are expressed in terms of the (fiberwise) Wodzicki residue.     

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.

     

Euler characteristics of general linear sections and polynomial Chern classes

We obtain a precise relation between the Chern–Schwartz–MacPherson class of a subvariety of projective space and the Euler characteristics of its general linear sections. In the case of a hypersurface, this leads to simple proofs of formulas of Dimca–Papadima and Huh for the degrees of the polar map of a homogeneous polynomial, extending these formula to any algebraically closed field of characteristic \(0\), and proving a conjecture of Dolgachev on ‘homaloidal’ polynomials in the same context. We generalize these formulas to subschemes of higher codimension in projective space. We also describe a simple approach to a theory of ‘polynomial Chern classes’ for varieties endowed with a morphism to projective space, recovering properties analogous to the Deligne–Grothendieck axioms from basic properties of the Euler characteristic. We prove that the polynomial Chern class defines homomorphisms from suitable relative Grothendieck rings of varieties to \(\mathbb{Z }[t]\).     

Remarks on Ginzburg's bivariant Chern classes

The convolution product is an important tool in the 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 give some remarks on the Ginzburg bivariant Chern classes.

     

Structure of the set of stationary solutions of phase-lock equations in superconductivity

In previous article [M. Zhan, Phase-lock equations and its connections to Ginzburg–Landau equations of superconductivity, J. Nonlinear Anal. 42 (2000) 1063–1075], we introduced a system of equations (phase-lock equations) to model the superconductivity phenomena. We investigated its connection to Ginzburg–Landau equations and proved the existence and uniqueness of both weak and strong solutions. In this article, we study the steady-state problem associated with the phase-lock equations. We prove that the steady-state problem has multiple solutions and show that the solution set enjoys some structural properties as proved by Foias and Teman for the Navier–Stokes equations in [C. Foias, R. Teman, Structure of the set of stationary solutions of the Navier–Stokes equations, Commun. Pure Appl. Math. XXX (1977) 149–164].     

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.     

Riemannian Geometry of Conical Singular Sets

In this paper, we study a class of singular Riemannian manifolds. The singular set itself is a smooth manifold with a cone-like neighborhood. By imposing a reasonable convergence condition on the metric, we can determine the local geometrical structure near the singular set. In general, the curvature near the singular set is unbounded. We prove that a bounded curvature assumption would have a strong implication on the geometrical and topological structures near the singular set. We also establish the Gauss–Bonnet–Chern formula, which can be applied to the study of singular Eistein 4-manifolds.     

Modules over rings with a power basis

A homological category is constructed and also a functor into it from the category of finitely generated modules over a ring with a power basis such that from an object that corresponds to a module the latter is uniquely determined to within free direct summands, and from a morphism corresponding to a module homomorphism this homomorphism can be uniquely reconstructed to within norms.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 71, pp. 259–281, 1977.     

Index Theory and Non-Commutative Geometry I. Higher Families Index Theory

We prove an index theorem for foliated manifolds. We do so by constructing a push forward map in cohomology for a k-oriented map from an arbitrary manifold to the space of leaves of an oriented foliation, and by constructing a Chern–Connes character from the k-theory of the compactly supported smooth functions on the holonomy groupoid of the foliation to the Haefliger cohomology of the foliation. Combining these with the Connes–Skandalis topological index map and the classical Chern character gives a commutative diagram from which the index theorem follows immediately.     

A generalization of Fulton–MacPherson configuration spaces

We construct a wonderful compactification of the variety parameterizing n distinct labeled points in X away from D, where X is a nonsingular variety and D is a nonsingular proper subvariety. When D is empty, it coincides with the Fulton–MacPherson configuration space.