Toru Ohmoto

Equivariant Chern classes of singular algebraic varieties with group actions

We define equivariant Chern–Schwartz–MacPherson classes of a possibly singular algebraic G-variety over the base field C, or more generally over a field of characteristic 0. In fact, we construct a natural transformation C G ∗ from the G-equivariant constructible function functor F G to the G-equivariant homology functor H G ∗ or A G ∗ (in the sense of Totaro–Edidin–Graham). This C G ∗ may be regarded as MacPherson’s transformation for (certain) quotient stacks. The Verdier–Riemann–Roch formula takes a key role throughout.

Generating functions of orbifold Chern classes I: symmetric products

In this paper, for a possibly singular complex variety X , generating functions of total orbifold Chern homology classes of the symmetric products S n X are given. These are very natural “class versions” of known generating function formulae of (generalized) orbifold Euler characteristics of S n X . Our Chern classes work covariantly for proper morphisms. We state the result more generally. Let G be a finite group and G n the wreath product G ∼ S n . For a G -variety X and a group A , we show a “Dey–Wohlfahrt type formula“ for equivariant Chern–Schwartz–MacPherson classes associated to G n -representations of A (Theorem 1ċ1 and 1ċ2). When X is a point, our formula is just the classical one in group theory generating numbers |Hom( A , G n )|.

Self-intersection class for singularities and its application to fold maps

Let f : M → N be a generic smooth map with corank one singularities between manifolds, and let S(f) be the singular point set of f. We define the self-intersection class I(S(f)) ∈ H*(M; Z) of S(f) using an incident class introduced by Rimanyi but with twisted coefficients, and give a formula for I(S(f)) in terms of characteristic classes of the manifolds. We then apply the formula to the existence problem of fold maps.

Local invariants of singular surfaces in an almost complex four-manifold

In this paper we define two local invariants, the local self-intersection index and the Maslov index, for singular surfaces in an almost complex four-manifold and prove formulae involving these invariants, which generalize formulae of Lai and Givental.

Singularities in Geometry and Topology

The purpose of this paper is applying minimality of hyperplane arrangements to local system cohomology groups. It is well known that twisted cohomology groups with coefficients in a generic rank one local system vanish except in the top degree, and bounded chambers form a basis of the remaining cohomology group. We determine precisely when this phenomenon happens for two-dimensional arrangements.Let f(z,¯) be a mixed polar homogeneous polynomial of n variables z = (z1, . . . , zn). It defines a projective real algebraic va- riety V := {(z) 2 CP n 1 | f(z,¯) = 0} in the projective space CP n 1 . The behavior is different from that of the projective hypersurface. The topology is not uniquely determined by the degree of the variety even if V is non-singular. We study a basic property of such a variety.In this paper we give an introduction to our recent work on characteristic classes of complex hypersurfaces based on some talks given at conferences in Strasbourg, Oberwolfach and Kagoshima. We explain the relation between nearby cycles for constructible functions or sheaves as well as for (relative) Grothendieck groups of algebraic varieties and mixed Hodge modules, and the specialization of characteristic classes of singular spaces like the Chern-, Todd-, Hirzebruch- and motivic Chern-classes. As an application we get a description of the differences between the corresponding virtual and functorial characteristic classes of complex hypersurfaces in terms of vanishing cycles related to the singularities of the hypersurface.

On topological Radon transformations

We construct a functor, which we call the topological Radon transform, from a category of complex algebraic varieties with morphisms given by divergent diagrams, to constructible functions. The topological Radon transform is thus the composition of a pull-back and a push-forward of constructible functions. We show that the Chern-Schwartz-MacPherson transformation makes the topological Radon transform of constructible functions compatible with a certain homological Verdier-Radon transform. We use this set-up to prove, given a projective variety X, a formula for the Chern-Mather class of the dual variety in terms of that of X.

C1‐triangulations of semialgebraic sets

We show that every semialgebraic set admits a semialgebraic triangulation such that each closed simplex is C1 differentiable. As an application, we give a straightforward definition of the integration ∫Xω over a compact semialgebraic subset X of a differential form ω on an ambient semialgebraic manifold. This provides a significant simplification of the theory of semialgebraic singular chains and integrations without using geometric measure theory. Our results hold over every (possibly non-archimedian) real closed field.

Thom polynomials for open Whitney umbrellas of isotropic mappings

A smooth mapping f : L → (M2n, ω) of a smooth n-dimensional manifold L into a smooth 2n-dimensional symplectic manifold (M,ω) is called isotropic if f∗ω vanishes. In the last ten years, the local theory of singularities of isotropic mappings has been rapidly developed by Arnol’d, Givental’ and several authors, while it seems that the global theory of their singularities has not been well studied except for the work of Givental’ [G1] in the case of dimension 2 (cf. [A], [Au], [I2], [I-O]). In the present paper, we are concerned with typical singularities with corank 1 of isotropic maps f :Ln→(M2n, ω) (arbitrary dimension n), so-called open Whitney umbrellas of higher order, investigated by Givental’ [G2], Ishikawa [I1] and Zakalyukin [Z], and our purpose is to give their topological invariants from the viewpoint of “Thom polynomial theory” (cf. [T], [P], [K], [AVGL]). These are obtained as a variant of Porteous’ formulae on Thom polynomials for Ak-singularities [P]. Throughout this paper, manifolds are assumed to be paracompact Hausdorff spaces and of class C∞, and maps are also of class C∞.