Equivariant Hirzebruch class for singular varieties

Abstract

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.

Equivariant Hirzebruch Classes and Molien Series of Quotient Singularities

“…We can check that in our examples (and in many others) after the substitution T := 1 + S and y := −1 − δ the numerator of H(y, T ) is a polynomial with nonnegative coefficients. That is always the case for simplicial toric varieties by [Web16a,Thm. 13.1].…”

“…So far there is no proof. Moreover, in [Web16a], it was noticed that there is another, stronger positivity of local equivariant Hirzebruch classes. Positivity was proven for simplicial toric varieties, while for various Schubert cells in G/P it was only observed in the results of computations.…”

Hirzebruch Class and Bia Lynicki-Birula Decomposition

“…If the torus acts on X then naturally the Hirzebruch class (as any characteristic class) lifts to equivariant cohomology. The properties of the equivariant Hirzebruch class are studied in [Web15]. An example of computation is given in [MW15].…”

“…The non-equivariant case was studied in [BSY10]. The equivariant version for a torus action was developed in [Web15]. Let us list the formal properties which determine this class.…”

Elliptic classes, McKay correspondence and theta identities

“…Each fixed point component gives a local summand of the global invariant. The local equivariant Chern-Schwartz-MacPherson classes were studied in [26] and the local contributions to the Hirzebruch class were described in [27]. The role of the local contributions to the elliptic class in the Landau-Ginzburg model is demonstrated in [16].…”