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Abstract. Combining the approach to Thom polynomials via classifying spaces of singularities with the Fulton-Lazarsfeld theory of cone classes and positive polynomials for ample vector bundles, we show that the coefficients of the Schur function expansions of the Thom polynomials of stable singularities are nonnegative with positive sum. In the present paper, we study the Schur function expansions of Thom polynomials from a "qualitative" point of view. In contrast to [16], [13], [14], where the Szűcs-Rimányi approach via symmetries of singularities was used, we follow here the Kazarian approach [9] to Thom polynomials. In fact, both approaches rely on suitable "classifying spaces of singularities". We substitute the jet automorphism group by the group of the linear transformations GL m × GL n . This allows one to extend the definition of Thom polynomials for maps f : M → N of complex manifolds to pairs of vector bundles. It is convenient to pass to homotopy theory, where each pair of bundles can be pulled back from the universal pair of bundles on BGL m × BGL n .

Abstract. We develop an equivariant version of the Hirzebruch class for singular varieties. When the group acting is a torus we apply Localization Theorem of Atiyah-Bott and Berline-Vergne. The localized Hirzebruch class is an invariant of a singularity germ. The singularities of toric varieties and Schubert varieties are of special interest. We prove certain positivity results for simplicial toric varieties. The positivity for Schubert varieties is illustrated by many examples, but it remains mysterious.The main goal of the paper is to show how a theory of global invariants can be applied to study local objects equipped with an action of a large group of symmetries. The theory of global invariants we are going to discuss is the theory of characteristic classes, more precisely the Hirzebruch class and χ y -genus. By [Yok94, BSY10] the Hirzebruch class admits a generalization for singular varieties. Let X be an algebraic variety in a compact complex algebraic manifold M . Suppose that X is preserved by a torus T acting on M . For simplicity assume that the fixed point set M T is discrete. Then by Localization Theorem of Atiyah-Bott and Berline-Vergne the χ y -genus of X can be written as a sum of contributions coming from fixed points. The contribution of a fixed point p ∈ M T is equal to the equivariant Hirzebruch class restricted to that point and divided by the Euler class of the tangent space at pThe local contributions to χ y -genus are fairly computable and they are expressed by polynomials in characters of the torus. We will describe all the necessary components of the described construction, give various examples and we will discuss positivity property of the localized Hirzebruch class. A relation with the Bia lynicki-Birula decomposition [BB73] will be given in a subsequent paper [Web14]. The reader can find useful to look at the article [Web13], where an elementary and self-contained introduction to equivariant characteristic classes is given.

We generalize the notion of Thom polynomials from singularities of maps between two complex manifolds to invariant cones in representations, and collections of vector bundles. We prove that the generalized Thom polynomials, expanded in the products of Schur functions of the bundles, have nonnegative coefficients. For classical Thom polynomials associated with maps of complex manifolds, this gives an extension of our former result for stable singularities to nonnecessary stable ones. We also discuss some related aspects of Thom polynomials, which makes the article expository to some extent.