Related papers: Rational smoothness, cellular decompositions and G…
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 define `simplicial GKM orbifold complexes' and study some of their topological properties. We introduce the concept of filtration of regular graphs and `simplicial graph complexes', which have close relations with…
Let T be a compact torus and X a nice compact T-space (say a manifold or variety). We introduce a functor assigning to X a "GKM-sheaf" F_X over a "GKM-hypergraph" G_X. Under the condition that X is equivariantly formal, the ring of global…
We study the GKM theory for a equivariant stratified space having orbifold structures in tis successive quotients. Then, we introduce the notion of an \emph{almost simple polytope}, as well as a \emph{divisive toric variety} generalizing…
This paper contains two results concerning the equivariant K-theory of toric varieties. The first is a formula for the equivariant K-groups of an arbitrary affine toric variety, generalizing the known formula for smooth ones. In fact, this…
We construct differential equivariant K-theory of representable smooth orbifolds as a ring valued functor with the usual properties of a differential extension of a cohomology theory. For proper submersions (with smooth fibres) we construct…
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…
We present a description of the equivariant $K$-theory of a smooth projective spherical variety. This provides an integral $K$-theory version of Brion's calculation of equivariant Chow-cohomology of such varieties. We consider the…
In this note, we describe motivic cell structures arising from the Bialynicki-Birula decomposition. This provides a description of the stable A^1-homotopy types of smooth projective G_m-varieties where the G_m-action has isolated fixed…
In this article we describe the equivariant and ordinary topological $K$-ring of a toric bundle with fiber a $T$-{\it cellular} toric variety. This generalizes the results in \cite{su} on $K$-theory of smooth projective toric bundles. We…
This work is devoted to the study of the foundations of quantum K-theory, a K-theoretic version of quantum cohomology theory. In particular, it gives a deformation of the ordinary K-ring K(X) of a smooth projective variety X, analogous to…
Given an action of a reductive group on a normal variety, we construct all invariant open subsets admitting a good quotient with a quasiprojective or a divisorial quotient space. Our approach extends known constructions like Mumford's…
An action of a complex reductive group $\mathrm G$ on a smooth projective variety $X$ is regular when all regular unipotent elements in $\mathrm G$ act with finitely many fixed points. Then the complex $\mathrm G$-equivariant cohomology…
In this article we describe the $G_{comp}\times G_{comp}$-equivariant topological $K$-ring of a {\em cellular} toroidal embedding $\mathbb{X}$ of a complex connected reductive algebraic group $G$. In particular, our results extend the…
Toric orbifolds are a topological generalization of projective toric varieties associated to simplicial fans. We introduce some sufficient conditions on the combinatorial data associated to a toric orbifold to ensure the existence of an…
We introduce equivariant versions of uniform rationality: given an algebraic group G, a G-variety is called G-uniformly rational (resp. G-linearly uniformly rational) if every point has a G-invariant open neighborhood equivariantly…
In this paper, we study the all genus Gromov-Witten theory for any GKM orbifold $X$. We generalize the Givental formula which is studied in the smooth case in \cite{Giv2} \cite{Giv3} \cite{Giv4} to the orbifold case. Specifically, we…
We give a $K$-theoretic criterion for a quasi-projective variety to be smooth. If $\mathbb{L}$ is a line bundle corresponding to an ample invertible sheaf on $X$, it suffices that $K_q(X) = K_q(\mathbb{L})$ for all $q\le\dim(X)+1$.
We give a combinatorial algorithm for equivariant embedded resolution of singularities of a toric variety defined over a perfect field. The algorithm is realized by a finite succession of blowings-up with smooth invariant centres that…