Chern-Mather Class Research Articles

Linear optimization on varieties and Chern-Mather classes

The linear optimization degree gives an algebraic measure of complexity of optimizing a linear objective function over an algebraic model. Geometrically, it can be interpreted as the degree of a projection map on the affine conormal variety. Fixing an affine variety, our first result shows that the geometry of this conormal variety, expressed in terms of bidegrees, completely determines the Chern-Mather classes of the given variety. We also show that these bidegrees coincide with the linear optimization degrees of generic affine sections.

Characteristic Classes of Homogeneous Essential Isolated Determinantal Varieties

The (homogeneous) Essentially Isolated Determinantal Variety is the natural generalization of generic determinantal variety, and is fundamental example to study non-isolated singularities. In this paper we study the characteristic classes on these varieties. We give explicit formulas of their Chern-Schwartz-MacPherson classes and Chern-Mather classes via standard Schubert calculus. As corollaries we obtain formulas for their (generic) sectional Euler characteristics, characteristic cycles and polar classes.

The Chern-Mather class of the multiview variety

ABSTRACTThe multiview varietyassociated to a collection of N cameras records which sequences of image points in ℙ2N can be obtained by taking pictures of a given world point x∈ℙ3 with the cameras. In order to reconstruct a scene from its picture under the different cameras, it is important to be able to find the critical points of the function which measures the distance between a general point u∈ℙ2N and the multiview variety. In this paper we calculate a specific degree 3 polynomial that computes the number of critical points as a function of N. In order to do this, we construct a resolution of the multiview variety and use it to compute its Chern-Mather class.

Projective duality and a Chern-Mather involution

We observe that linear relations among Chern-Mather classes of projective varieties are preserved by projective duality. We deduce the existence of an explicit involution on a part of the Chow group of projective space, encoding the effect of duality on Chern-Mather classes. Applications include Plücker formulae, constraints on self-dual varieties, generalizations to singular varieties of classical formulas for the degree of the dual and the dual defect, formulas for the Euclidean distance degree, and computations of Chern-Mather classes and local Euler obstructions for cones.

Singular Chern Classes of Schubert Varieties via Small Resolution

We discuss a method for calculating the Chern-Schwartz-MacPherson (CSM) class of a Schubert variety in the Grassmannian using small resolutions introduced by Zelevinsky. As a consequence, we show how to compute the Chern-Mather class and local Euler obstructions using small resolutions instead of the Nash blowup. The algorithm obtained for CSM classes also allows us to prove new cases of a positivity conjecture of Aluffi and Mihalcea.

Shadows of blow-up algebras

We study different notions of blow-up of a scheme X along a subscheme Y, depending on the datum of an embedding of X into an ambient scheme. The two extremes in this theory are the ordinary blow-up, corresponding to the identity, and the `quasi-symmetric blow-up', corresponding to an embedding into a nonsingular variety M. We prove that this latter blow-up is intrinsic of Y and X, and is universal with respect to the requirement of being embedded as a subscheme of the ordinary blow-up of some ambient space along Y. We consider these notions in the context of the theory of characteristic classes of singular varieties. We prove that if X is a hypersurface in a nonsingular variety and Y is its `singularity subscheme', these two extremes embody respectively the conormal and characteristic cycles of X. Consequently, the first carries the essential information computing Chern-Mather classes, and the second is likewise a carrier for Chern-Schwartz-MacPherson classes. In our approach, these classes are obtained from Segre class-like invariants, in precisely the same way as other intrinsic characteristic classes such as those proposed by William Fulton and by W. Fulton and Kent Johnson. We also identify a condition on the singularities of a hypersurface under which the quasi-symmetric blow-up is simply the linear fiber space associated with a coherent sheaf.

On a naturality of Chern-Mather classes

Introduction. The Chern-Schwartz-MacPher- son class (or more precisely natural transformation) is the unique natural transformation from the co- variant constructible function functor to the covari- ant integral homology functor satisfying the normal- ization that the value of the characteristic function of a nonsingular compact complex analytic variety is equal to the Poincare dual of the total Chern cohomology class of the tangent bundle. The ex- istence of such a transformation was conjectured by Deligne and Grothendieck and was proved by MacPherson (10). The novelity of MacPherson's proof is introducing the notion of local Euler obstruc- tion (which was independently introduced by Kashi- wara (7) also) and assigning the Chern-Mather class to this local Euler obstruction, not to the character- istic function. Although the Chern-Mather class is a very geometrically simple homology class, the assign- ment of the Chern-Mather class to the characteristic function does not give such a natural transformation. It is often said that few functorial proper- ties are know for the Chern-Mather class (e.g., see (6, Note, page 377, right after Example 19.1.7)), al- though the assignment of the Chern-Mather class to the local Euler obstruction is perfectly natural, which is the main part of MacPherson's proof. In this paper, using this fine naturality of this assingment, we interpret the Chern-Mather class in the same way as the Chern-Schwartz-MacPherson class. Further- more, by introducing the notion of a q-deformed local Euler obstruction which unifies the charac- teristic function and the local Euler obstruction, we give a q-deformed Chern-Schwartz-MacPherson class natural transformation, which specializes to the Chern-Mather class natural transformation for q = 0 and the Chern-Schwartz-MacPherson class natural transformation for q = 1.

Differential forms with logarithmic poles and Chern-Schwartz-MacPherson classes of singular varieties

We express the Chern-Schwartz-MacPherson class of a possibly singular variety in terms of the total Chern class of a bundle of differential forms with logarithmic poles. As an application, we obtain a formula for the Chern-Schwartz-MacPherson class of a hypersurface of a nonsingular variety, in terms of the Chern-Mather class of a suitable sheaf.

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.

A formula for Segre classes of singular projective varieties

The Segre class of a singular projective variety X is that of the normal cone of the diagonal in the product X x X. This class was introduced by K. W. Johnson and W. Fulton to study immersions and embeddings. In our previous work we related the Segre classes and the Chern-Mather classes for hypersurfaces with codimension one singularities and Xn c P2 with isolated singularities. In this paper we generalize these results to the case of Xn c FN with singularities of codimension N — n (N < In). The notion of Segre classes (of cones) has become of increasing importance as a key ingredient for constructing or analyzing various invariants, e.g., in intersection theory and group representation theory, etc. The Segre class treated in this note is the Segre class of a singular projective variety, which was introduced by K. W. Johnson (and W. Fulton) to study immersions and embeddings of singular projective varieties [4]. This is the Segre class of the normal cone C^{X x X) of the diagonal Δ in the product X x X. We call this class Johnson's Segre class, denoted by S*(X). Another well-studied characteristic class of a singular variety is MacPherson's Chern class, the existence of which was conjectured by Deligne and Grothendieck. R. MacPherson [7] constructed this Chern class, using Chern-Mather classes and introducing the notion of local Euler obstruction. A. Dubson [2] gave a more concrete description for MacPherson's Chern class C*(X) : Let S?χ be a (in fact, any) Whitney stratification of X with the smooth part of X as the top-dimensional stratum and let C^(X) denote the Chern-Mather class of X. Then ms •