Let $G$ be a complex, linear algebraic group acting on an algebraic space $X$. The purpose of this paper is to prove a Riemann-Roch theorem (Theorem 5.3) which gives a description of the completion of the equivariant Grothendieck group $G_0(G,X)$ at any maximal ideal of the representation ring $R(G) \otimes \C$ in terms of equivariant cycles. The main new technique for proving this theorem is our non-abelian completion theorem (Theorem 4.3) for equivariant $K$-theory. Theorem 4.3 generalizes the classical localization theorems for diagonalizable group actions to arbitrary groups.
In this paper, we study the algebraic analogue of the topological Atiyah-Segal completion theorem. We verify this completion theorem for the algebraic equivariant $K$-theory of smooth projective schemes. We also show that the completion theorem fails in general for smooth non-projective schemes.
We show how derived completion can be used to prove an analogue of Atiyah-Segal completion for the $T$-equivariant Hermitian K-theory of a scheme $X$ with a trivial $T$-action, containing $\tfrac{1}{2}$ and satisfying the resolution property, where $T$ is a split torus of rank $t$. This result is an important first step towards a more general Atiyah-Segal completion theorem for Hermitian K-theory.
The purpose of this paper is to prove an equivariant Riemann-Roch theorem for schemes or algebraic spaces with an action of a linear algebraic group $G$. For a $G$-space $X$, this theorem gives an isomorphism between a completion of the equivariant Grothendieck group and a completion of equivariant equivariant Chow groups. The key to proving this isomorphism is a geometric description of completions of the equivariant Grothendieck group. Besides Riemann-Roch, this result has some purely $K$-theoretic applications. In particular, we prove a conjecture of Köck (in the case of regular schemes) an…
The goal of this paper is to prove a version of the non-abelian localization theorem for the rational equivariant K-theory of a smooth variety $X$ with the action of a linear algebraic group $G$. We then use this to prove a Riemann-Roch theorem which represents the completion of the higher equivariant K-theory of $X$ at various maximal ideals of the representation ring, in terms the equivariant higher Chow groups. This generalizes a result of Edidin and Graham to higher $K$-theory with rational coefficients.
In this paper we construct an equivariant embedding of the affine space $\mathbb{A}^n$ with the translation group action into a complete non-projective algebraic variety $X$ for all $n \geq 3$. The theory of toric varieties is used as the main tool for this construction. In the case of $n = 3$ we describe the orbit structure on the variety $X$.
We treat equivariant completions of toric contraction morphisms as an application of the toric Mori theory. For this purpose, we generalize the toric Mori theory for non-$\mathbb Q$-factorial toric varieties. So, our theory seems to be quite different from Reid's original combinatorial toric Mori theory. We also explain various examples of non-$\mathbb Q$-factorial contractions, which imply that the $\mathbb Q$-factoriality plays an important role in the Minimal Model Program. Thus, this paper completes the foundations of the toric Mori theory and show us a new aspect of the Minimal Model Prog…
In the mid 1980s, while working on establishing completion theorems for equivariant Algebraic K- Theory similar to the well-known Atiyah-Segal completion theorem for equivariant topological K-theory, the late Robert Thomason found the strong finiteness conditions that are required in such theorems to be too restrictive. Then he made a conjecture on the existence of a completion theorem in the sense of Atiyah and Segal for equivariant Algebraic G-theory, for actions of linear algebraic groups on schemes that holds without any of the strong finiteness conditions that are required in such theorem…
In this paper, we describe the category of bi-equivariant vector bundles on a bi-equivariant smooth (partial) compactification of a reductive algebraic group with normal crossing boundary divisors. Our result is a generalization of the description of the category of equivariant vector bundles on toric varieties established by A.A. Klyachko [Math. USSR. Izvestiya, {\bf 35}, No.2 (1990)]. As an application, we prove splitting of equivariant vector bundles of low rank on the wonderful compactification of an adjoint semisimple group in the sense of C. De Concini and C. Procesi [Lecture Note in Mat…
We survey recent results on open embeddings of the affine space $\mathbb{C}^n$ into a complete algebraic variety $X$ such that the action of the vector group $\mathbb{G}_a^n$ on $\mathbb{C}^n$ by translations extends to an action of $\mathbb{G}_a^n$ on $X$. We begin with Hassett-Tschinkel correspondence describing equivariant embeddings of $\mathbb{C}^n$ into projective spaces and give its generalization for embeddings into projective hypersurfaces. Further sections deal with embeddings into flag varieties and their degenerations, complete toric varieties, and Fano varieties of certain types.
In this paper, we define an action of the group of equivariant Cartier divisors on a toric variety X on the equivariant cycle groups of X, arising naturally from a choice of complement map on the underlying lattice. If X is nonsingular, this gives a lifting of the multiplication in equivariant cohomology to the level of equivariant cycles. As a consequence, one naturally obtains an equivariant cycle representative of the equivariant Todd class of any toric variety. These results extend to equivariant cohomology the results of Thomas and Pommersheim. In the case of a complement map arising from…
We prove an equivariant localization theorem over an algebraically closed field of characteristic zero for smooth quotient stacks by reductive groups $X/G$ in the setting of derived loop spaces as well as Hochschild homology and its cyclic variants. We show that the derived loop spaces of the stack $X/G$ and its classical $z$-fixed point stack $π_0(X^z)/G^z$ become equivalent after completion along a semisimple parameter $[z] \in G//G$, implying the analogous statement for Hochschild and cyclic homology of the dg category of perfect complexes $\text{Perf}(X/G)$. We then prove an analogue of th…
In this paper we develop an equivariant intersection theory for actions of algebraic groups on algebraic schemes. The theory is based on our construction of equivariant Chow groups. They are algebraic analogues of equivariant cohomology groups which satsify the formal properties of ordinary Chow groups. In addition, they enjoy many of the properties of equivariant cohomology. The principal results are: (1) We prove the existence of canonical intersection products on the Chow groups of geometric quotients of smooth varieties- even when the stabilizers of geometric points are non-reduced. (2) We…
We establish the analogue of the Friedlander-Mazur conjecture for Teh's reduced Lawson homology groups of real varieties, which says that the reduced Lawson homology of a real quasi-projective variety $X$ vanishes in homological degrees larger than the dimension of $X$ in all weights. As an application we obtain a vanishing of homotopy groups of the mod-2 topological groups of averaged cycles and a characterization in a range of indices of the motivic cohomology of a real variety as homotopy groups of the complex of averaged equidimensional cycles. We also establish an equivariant Poincare dua…
We introduce a Bredon motivic cohomology theory for smooth schemes defined over a field and equipped with an action by a finite group. These cohomology groups are defined for finite dimensional representations as the hypercohomology of complexes of equivariant correspondences in the equivariant Nisnevich topology. We generalize the theory of presheaves with transfers to the equivariant setting and prove a Cancellation Theorem.
For a compact quasi-smooth derived scheme M with (-1)-shifted cotangent bundle N, there are at least two ways to localise the virtual cycle of N to M via torus and cosection localisations, introduced by Jiang-Thomas. We produce virtual cycles on both the projective completion and projectivisation and show the ones on the former push down to Jiang-Thomas cycles and the one on the latter computes the difference. Using similar ideas we give an expression for the difference of the quintic and t-twisted quintic GW invariants of Guo-Janda-Ruan.
We establish an Atiyah-Hirzebruch type spectral sequence relating real morphic cohomology and real semi-topological K-theory and prove it to be compatible with the Atiyah-Hirzebruch spectral sequence relating Bredon cohomology and Atiyah's KR-theory constructed by Dugger. An equivariant and a real version of Suslin's conjecture on morphic cohomology are formulated, proved to come from the complex version of Suslin conjecture and verified for certain real varieties. In conjunction with the spectral sequences constructed here this allows the computation of the real semi-topological K-theory of s…
In this paper we study the proalgebraic completion of mapping class relative to their maps to the symplectic group. The main result is that the natural map from the unipotent (a.k.a. Malcev) completion of the Torelli group to the prounipotent radical of the Sp_g completion of the mapping class group is a non trivial central extension with kernel isomorphic to Q, at least when g \ge 8. The theorem is proved by relating the central extension to the line bundle associated to the archemidean height of the cycle C - C- in the Jacobian of the curve C. We also develop some of the basic theory of rela…
In the present paper we introduce and study the notion of an equivariant pretheory: basic examples include equivariant Chow groups, equivariant K-theory and equivariant algebraic cobordism. To extend this set of examples we define an equivariant (co)homology theory with coefficients in a Rost cycle module and provide a version of Merkurjev's (equivariant K-theory) spectral sequence for such a theory. As an application we generalize the theorem of Karpenko-Merkurjev on G-torsors and rational cycles; to every G-torsor E and a G-equivariant pretheory we associate a graded ring which serves as an …
We study the equivariant cobordism theory of schemes for action of linear algebraic groups. We compare the equivariant cobordism theory for the action of a linear algebraic groups with similar groups for the action of tori and deduce some consequences for the cycle class map of the classifying space of an algebraic groups.
In this paper, we develop the theory of equivariant motivic homotopy theory, both unstable and stable. While our original interest was in the case of profinite group actions on smooth schemes, we discuss our results in as broad a setting as possible so as to be applicable in a variety of contexts, for example to the case of smooth group scheme actions on schemes that are not necessarily smooth. We also discuss how ${\mathbb A}^1$-localization behaves with respect to mod-$\ell$-completion, where $\ell$ is a fixed prime.
Let T be a torus, X a smooth quasi-compact separated scheme equipped with a T-action, and [X/T] the associated quotient stack. Given any localizing A1-homotopy invariant of dg categories E, we prove that the derived completion of E([X/T]) at the augmentation ideal I of the representation ring R(T) of T agrees with the Borel construction associated to the T-action on X. Moreover, for certain localizing A1-homotopy invariants, we extend this result to the case of a linearly reductive group scheme G. As a first application, we obtain an alternative proof of Krishna's completion theorem in algebra…
V.Berkovich, K.Fujiwara and R.Huber have proved independently by different methods that the fiber of the vanishing cycles at a point of the special fiber depends only on the formal completion at this point. We refine this result and prove the invariance under formal completion of the perverse monodromy filtration on the fiber of vanishing cycles. This result is used in an essential way by P.Boyer in his work on some "simple Shimura varieties".
In a previous paper, the authors defined an equivariant version of the so-called Saito duality between the monodromy zeta functions as a sort of Fourier transform between the Burnside rings of an abelian group and of its group of characters. Here a so-called enhanced Burnside ring $\widehat{B}(G)$ of a finite group $G$ is defined. An element of it is represented by a finite $G$-set with a $G$-equivariant transformation and with characters of the isotropy subgroups associated to all points. One gives an enhanced version of the equivariant Saito duality. For a complex analytic $G$-manifold with …
Let $G$ be a reductive group and $U,U^-$ be the unipotent radicals of a pair of opposite parabolic subgroups $P,P^-$. We prove that the DG-categories of $U(\!(t)\!)$-equivariant and $U^-(\!(t)\!)$-equivariant D-modules on the affine Grassmannian $Gr_G$ are canonically dual to each other. We show that the unit object witnessing this duality is given by nearby cycles on the Drinfeld-Gaitsgory-Vinberg interpolation Grassmannian defined in arXiv:1805.07721. We study various properties of the mentioned nearby cycles, in particular compare them with the nearby cycles studied in arXiv:1411.4206 and a…
Let $G$ be an algebraic group and let $X$ be a smooth $G$-variety with two orbits: an open orbit and a a closed orbit of codimension $1$. We give an algebraic description of the category of $G$-equivariant vector bundles on $X$ under a mild technical hypothesis. We deduce simpler classifications in the special cases of line bundles and vector bundles which are generically local systems. We apply our results to the study of admissible representations of semisimple Lie groups. Our main result gives a new set of constraints on the associated cycles of unipotent representations.
Flag domains are open orbits of real semisimple Lie groups in flag manifolds of their complexification. A special class of flag domains constitute the classifying spaces for variations of Hodge structure, namely period domains or more generally Mumford-Tate domains. In this paper we classify all proper, equivariant embeddings of a flag domain of Hodge type, required to satisfy a certain transversality condition, in a period domain and show that any such flag domain must be a Hermitian symmetric space of non-compact type of classical type. In particular, the classification describes also the pr…
Algebraic cycles on complex projective space P(V) are known to have beautiful and surprising properties. Therefore, when V carries a real or quaternionic structure, it is natural to ask for the properties of the groups of real or quaternionic algebraic cycles on P(V). In this paper and its sequel the homotopy structure of these cycle groups is determined. They bear a direct relationship to characteristic classes for the classical groups, and functors in K-theory extend directly to these groups. These groups give rise to E-infinity-ring spaces, and the maps extending the K-theory functors are r…
For $G$ an algebraic group definable over a model of $\operatorname{ACVF}$, or more generally a definable subgroup of an algebraic group, we study the stable completion $\widehat{G}$ of $G$, as introduced by Loeser and the second author. For $G$ connected and stably dominated, assuming $G$ commutative or that the valued field is of equicharacteristic 0, we construct a pro-definable $G$-equivariant strong deformation retraction of $\widehat{G}$ onto the generic type of $G$. For $G=S$ a semiabelian variety, we construct a pro-definable $S$-equivariant strong deformation retraction of $\widehat{S…
We develop the connection between equivariant completions of algebraic homogeneous spaces of reductive groups and lower bounds for the Mabuchi energy of a polarized manifold over the space of Bergman metrics. We provide a new definition of Tian's CM Polarization and discuss its properties.
For a linear algebraic group $G$ over a field $k$, we define an equivariant version of the Voevodsky's motivic cobordism $MGL$. We show that this is an oriented cohomology theory with localization sequence on the category of smooth $G$-schemes and there is a natural transformation from this functor to the functor of equivariant motivic cohomology. We give several applications. In particular, we use this equivariant motivic cobordism to study the cobordism ring of the classifying spaces and the cycle class maps from the algebraic to the singular cohomology of such spaces. This theory of motivic…
Open orbits D of noncompact real forms G_0 acting on flag manifolds of their semisimple complexifications G are considered. The unique orbit C of a maximal compact subgroup K_0 of G_0 in D can be regarded as a point in the (full) cycle space of D. The group theoretical cycle space is defined to be the connected component containing C of the intersection of the G-orbit of C with the full cycle space of D. The main result of the present article is that the group theoretical cycle space is closed in the full cycle space. In particular, if they have the same dimension, then they are equal. This fo…
For each $n\ge5$, we give an $S_n$-equivariant basis for $H_4(\overline{\mathcal{M}_{0,n}},\mathbb{Q})$, as well as for $H_{2(n-5)}(\overline{\mathcal{M}_{0,n}},\mathbb{Q})$. Such a basis exists for $H_2(\overline{\mathcal{M}_{0,n}},\mathbb{Q})$ and for $H_{2(n-4)}(\overline{\mathcal{M}_{0,n}},\mathbb{Q})$, but it is not known whether one exists for $H_{2k}(\overline{\mathcal{M}_{0,n}},\mathbb{Q})$ when $3\le k\le n-6$.
