Related papers: Hirzebruch class and Bialynicki-Birula decompositi…

The paper is based on a talk. Complete exposition is given in "Equivariant Hirzebruch class for singular varieties". Starting from the classical theory we describe Hirzebruch class and the related Todd genus of a complex singular algebraic…

The Hirzebruch $td_y(X)$ class of a complex manifold X is a formal combination of Chern characters of the sheaves of differential forms multiplied by the Todd class. The related $\chi_y$-genus admits a generalization for singular complex…

The central object of investigation of this paper is the Hirzebruch class, a deformation of the Todd class, given by Hirzebruch (for smooth varieties) in his celebrated book "Topological Methods in Algebraic Geometry". The generalization…

Our goal in this work is to develop aspects of Bialynicki-Birula and Morse-Bott theory that can be extended from the classical setting of smooth manifolds to that of complex analytic spaces with a holomorphic $\mathbb{C}^*$ action. We…

We generalize the Bia{\l}ynicki-Birula decomposition from actions of $G_m$ on smooth varieties to actions of linearly reductive group ${\bf G}$ on finite type schemes and algebraic spaces. We also provide a relative version and briefly…

In this paper we study some new theories of characteristic homology classes for singular complex algebraic varieties. First we introduce a natural transformation T_{y}: K_{0}(var/X) -> H_{*}(X,Q)[y] commuting with proper pushdown, which…

Complex projective algebraic varieties with $\mathbb{C}^*$-actions can be thought of as geometric counterparts of birational transformations. In this paper we describe geometrically the birational transformations associated to rational…

In this paper we give a formula for the Hirzebruch $\chi_y$-genus $\chi_y(X)$ and similarly for the motivic Hirzebruch class $T_{y*}(X)$ for possibly singular varieties $X$, using the Vandermonde matrix. Motivated by the notion of secondary…

We study the bounded endomorphisms of $\ell_{N}^2(G)=\ell^2(G)\times \dots \times\ell^2(G)$ that commute with translations, where $G$ is a discrete abelian group. It is shown that they form a C*-algebra isomorphic to the C*-algebra of…

We provide a notion of algebraic rational cell with applications to intersection theory on singular varieties with torus action. Based on this notion, we study the algebraic analogue of $\mathbb{Q}$-filtrable varieties: algebraic varieties…

We investigate several aspects of the Bialynicki-Birula decomposition of a smooth complete $\mathbb{G}_m$-variety with finite fixed locus. Our results include novel characterizations of when the Bialynicki-Birula decomposition is filterable…

In this paper, we use the parametrised strict deformation quantization of C*-bundles obtained in a previous paper, and give more examples and applications of this theory. In particular, it is used here to classify H_3-twisted noncommutative…

We systematically produce algebraic varieties with torus action by constructing them as suitably embedded subvarieties of toric varieties. The resulting varieties admit an explicit treatment in terms of toric geometry and graded ring…

Suppose that an algebraic torus $G$ acts algebraically on a projective manifold $X$ with generically trivial stabilizers. Then the Zariski closure of the set of pairs $\{(x,y)\in X\times X\mid y=gx \text{for some}g\in G\}$ defines a nonzero…

We consider a Hamiltonian action of n-dimensional torus, T^n, on a compact symplectic manifold (M,\omega) with d isolated fixed points. For every fixed point p there exists (though not unique) a class a_p in H^*_{T}(M; Q) such that the…

We obtain general formulae expressing Hirzebruch genera of a manifold with Z/p-action in terms of invariants of this action (the sets of weights of fixed points). As an illustration, we consider numerous particular cases of well-known…

Given a torus action on a compact space X, a fundamental result of Borel and Atiyah-Segal asserts that the equivariant cohomology of X is concentrated in the fixed locus X^T, up to inverting enough Chern classes. We prove an analogue for…

In the first part of the paper, we build a foundation for further work on Hamiltonian actions on symplectic orbifolds. Most importantly we prove the orbifold versions of the abelian connectedness and convexity theorems. In the second half,…

The aim of this paper is to study the behavior of Hodge-theoretic (intersection homology) genera and their associated characteristic classes under proper morphisms of complex algebraic varieties. We obtain formulae that relate (parametrized…

We announce the following result and give several applications: A Hamiltonian $T$-space (for $T$ a torus) with isolated fixed points is cobordant to a disjoint union of weighted projective spaces which are constructed from its fixed point…