We prove a localization formula in equivariant algebraic $K$-theory for an arbitrary complex algebraic group acting with finite stabilizer on a smooth algebraic space. This extends to non-diagonalizable groups the localization formulas H.A. Nielsen in equivariant $K$-theory of vector bundles and R.W. Thomason for higher $K$-theory of equivariant coherent sheaves.
As an application we give a Riemann-Roch formula for quotients of smooth algebraic spaces by proper group actions. This formula extends previous work of B. Toen for stacks with quasi-projective moduli spaces and the authors for quotients by diagonalizable groups.
For a $G$-scheme $X$ with a given equivariant perfect obstruction theory, we prove a virtual equivariant Grothendieck-Riemann-Roch formula, this is an extension of a result of Fantechi-Göttsche to the equivariant context. We also prove a virtual non-abelian localization theorem for schemes over $\mathbb{C}$ with proper actions.
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…
We give a simple proof of the Riemann-Roch theorem for Deligne-Mumford stacks using the equivariant Riemann-Roch theorem and the localization theorem in equivariant K-theory together with some basic commutative algebra of Artin rings.
The goal of this paper is to prove the Riemann-Roch isomorphism for the higher equivariant K-theory of varieties with action of a linear algebraic group.
We introduce the most general to date version of the permutation-equivariant quantum K-theory, and express its total descendant potential in terms of cohomological Gromov-Witten invariants. This is the higher-genus analogue of adelic characterization from the paper [7] by Givental-Tonita, and is based on the application of Kawasaki's Riemann-Roch formula to moduli spaces of stable maps.
We produce a Grothendieck transformation from bivariant operational $K$-theory to Chow, with a Riemann-Roch formula that generalizes classical Grothendieck-Verdier-Riemann-Roch. We also produce Grothendieck transformations and Riemann-Roch formulas that generalize the classical Adams-Riemann-Roch and equivariant localization theorems. As applications, we exhibit a projective toric variety $X$ whose equivariant $K$-theory of vector bundles does not surject onto its ordinary $K$-theory, and describe the operational $K$-theory of spherical varieties in terms of fixed-point data. In an appendix, V…
We prove an equivariant Riemann-Roch formula for divisors on algebraic curves over perfect fields. By reduction to the known case of curves over algebraically closed fields, we first show a preliminary formula with coefficients in Q. We then prove and shed some further light on a divisibility result that yields a formula with integral coefficients. Moreover, we give variants of the main theorem for equivariant locally free sheaves of higher rank.
We introduce twisted permutation-equivariant GW-invariants, and compute them in terms of untwisted ones. The computation is based on Grothendieck-like RR formula corresponding to Adams' operations from K-theory to itself, and the result can be understood as a "quantum" version of such Adams-RR. As in the case of cohomological quantum RR theorem [3], the result is applied to express the invariants of bundle and super-bundle spaces in terms of those of the base. The bonus feature of permutation-equivariant K-theory is that the twisting classes can be understood as the simpler kappa-classes of Ka…
We use the formalism of traces in higher categories to prove a common generalization of the holomorphic Atiyah-Bott fixed point formula and the Grothendieck-Riemann-Roch theorem. The proof is quite different from the original one proposed by Grothendieck et al.: it relies on the interplay between self dualities of quasi- and ind- coherent sheaves on $X$ and formal deformation theory of Gaitsgory-Rozenblyum. In particular, we give a description of the Todd class in terms of the difference of two formal group structures on the derived loop scheme $\mathcal LX$. The equivariant case is reduced to…
In this article, we consider regular projective arithmetic schemes in the context of Arakelov geometry, any of which is endowed with an action of the diagonalisable group scheme associated to a finite cyclic group and with an equivariant very ample invertible sheaf. For any equivariant morphism between such arithmetic schemes, which is smooth over the generic fibre, we define a direct image map between corresponding higher equivariant arithmetic K-groups and we discuss its transitivity property. Then we use the localization sequence of higher arithmetic K-groups and the higher arithmetic conce…
In this short note, we prove a comparision theorem between Levine-Serpé's equivariant higher Chow groups of an algebraic variety equipped with an action of a finite group and ordinary higher Chow groups of its fixed points. As a consequence, we show that the equivariant motivic spectral sequence degenerates rationally. This yields a Riemann-Roch Theorem for equivariant algebraic $K$-theory.
Let G be a torus and M a G-Hamiltonian manifold with Kostant line bundle L and proper moment map. Let P be the weight lattice of G. We consider a parameter k and the multiplicity $m(λ,k)$ of the quantized representation associated to M and the k-th power of L . We prove that the weighted sum $\sum m(λ,k) f(λ/k)$ of the value of a test function f on points of the lattice $P/k$ has an asymptotic development in terms of the twisted Duistermaat-Heckman distributions associated to the graded Todd class of M. When M is compact, and f polynomial, the asymptotic series is finite and exact.
We extract genus $0$ consequences of the all genera quantum HRR formula proved in Part IX. This includes re-proving and generalizing the adelic characterization of genus $0$ quantum K-theory found in [Givental A., Tonita V., in Symplectic, Poisson, and Noncommutative Geometry, <I>Math. Sci. Res. Inst. Publ.</I>, Vol. 62, Cambridge University Press, New York, 2014, 43-91]. Extending some results of Part VIII, we derive the invariance of a certain variety (the "big J-function"), constructed from the genus $0$ descendant potential of permutation-equivariant quantum K-theory, under the action of c…
For a complete toric variety, we obtain an explicit formula for the localized equivariant Todd class in terms of the combinatorial data -- the fan. This is based on the equivariant Riemann-Roch theorem and the computation of the equivariant cohomology and equivariant homology of toric varieties.
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.
We show a Riemann-Roch theorem for group ring bundles over an arithmetic surface; this is expressed using the higher adeles of Beilinson-Parshin and the tame symbol via a theory of adelic equivariant Chow groups and Chern classes. The theorem is obtained by combining a group ring coefficient version of the local Riemann-Roch formula as in Kapranov-Vasserot with results on K-groups of group rings and an explicit description of group ring bundles over P^1. Our set-up provides an extension of several aspects of the classical Fr"ohlich theory of the Galois module structure of rings of integers of …
In the present paper we extend the Riemann-Roch formalism to structure algebras of moment graphs. We introduce and study the Chern character and pushforwards for twisted fibrations of moment graphs. We prove an analogue of the Riemann-Roch theorem for moment graphs. As an application, we obtain the Riemann-Roch type theorem for equivariant $K$-theory of some Kac-Moody flag varieties.
In this paper we give an explicit formula for the Riemann-Roch map for singular schemes which are quotients of smooth schemes by diagonalizable groups. As an application we obtain a simple proof of a formula for the Todd class of a simplicial toric variety. An equivariant version of this formula was previously obtained for complete simplicial toric varieties by Brion and Vergne using different techniques.
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.
For a proper scheme X with a fixed 1-perfect obstruction theory, we define virtual versions of holomorphic Euler characteristic, chi y-genus, and elliptic genus; they are deformation invariant, and extend the usual definition in the smooth case. We prove virtual versions of the Grothendieck-Riemann-Roch and Hirzebruch-Riemann-Roch theorems. We show that the virtual chi y-genus is a polynomial, and use this to define a virtual topological Euler characteristic. We prove that the virtual elliptic genus satisfies a Jacobi modularity property; we state and prove a localization theorem in the toric …
The aim of this paper is to describe how to obtain residue-type formulas for push-forwards in equivariant cohomology, using the Jeffrey-Kirwan nonabelian localization theorem and the related result of Guillemin and Kalkman. This paper relates the description of equivariant push-forwards for homogeneous spaces of classical Lie groups with the action of the maximal torus with symplectic geometry, using the example of projective spaces and classical Grassmannians.
We study the category of matrix factorizations for an isolated hypersurface singularity. We compute the canonical bilinear form on the Hochschild homology of this category. We find explicit expressions for the Chern character and the boundary-bulk maps and derive an analog of the Hirzebruch-Riemann-Roch formula for the Euler characteristic of the Hom-space between a pair of matrix factorizations. We also establish G-equivariant versions of these results.
Given a finite p-group G acting on a smooth projective curve X over an algebraically closed field k of characteristic p, the dimension of the tangent space of the associated equivariant deformation functor is equal to the dimension of the space of coinvariants of G acting on the space V of global holomorphic quadratic differentials on X. We apply known results about the Galois module structure of Riemann-Roch spaces to compute this dimension when G is cyclic or when the action of G on X is weakly ramified. Moreover we determine certain subrepresentations of V, called p-rank representations.
We define the equivariant Chern-Schwartz-MacPherson class of a possibly singular algebraic variety with a group action over the complex number field (or a field of characteristic 0). In fact, we construct a natural transformation from the equivariant constructible function functor to the equivariant homology functor (in the sense of Totaro-Edidin-Graham), which may be regarded as MacPherson's transformation for (certain) quotient stacks. We discuss on other type Chern/Segre classes and give some applications generalizing orbifold Euler characteristics and Thom polynomials of singularities. The…
For any finite abelian group G, the equivariant Gromov-Witten invariants of C^r/G can be viewed as a certain kind of abelian Hurwitz-Hodge integrals. In this note, we use Tseng's orbifold quantum Riemann-Roch theorem to express this kind of abelian Hurwitz-Hodge integrals as a sum over Feynman graphs, where the weight of each graph is expressed in terms of descendant integrals over moduli spaces of stable curves and representations of G. This expression will play a crucial role in the proof of the remodeling conjecture (arXiv:0709.1453, arXiv:0807.0597) for affine toric Calabi-Yau 3-orbifolds …
A Theorem due to Guillemin and Sternberg about geometric quantization of Hamiltonian actions of compact Lie groups $G$ on compact Kaehler manifolds says that the dimension of the $G$-invariant subspace is equal to the Riemann-Roch number of the symplectically reduced space. Combined with the shifting-trick, this gives explicit formulas for the multiplicities of the various irreducible components. One of the assumptions of the Theorem is that the reduction is regular, so that the reduced space is a smooth symplectic manifold. In this paper, we prove a generalization of this result to the case w…
Based on the methods used by the author to prove the Riemann-Roch formula for algebraic stacks, this paper contains a description of the rationnal G-theory of Deligne-Mumford stacks over general bases. We will use these results to study equivariant K-theory, and also to define new filtrations on K-theory of algebraic stacks.
Suppose $X$ is a compact symplectic manifold acted on by a compact Lie group $K$ (which may be nonabelian) in a Hamiltonian fashion, with moment map $μ: X \to {\rm Lie}(K)^*$ and Marsden-Weinstein reduction $\xred = μ^{-1}(0)/K$. There is then a natural surjective map $κ_0$ from the equivariant cohomology $H^*_K(X) $ of $X$ to the cohomology $H^*(\xred)$. In this paper we prove a formula (Theorem 8.1, the residue formula) for the evaluation on the fundamental class of $\xred$ of any $η_0 \in H^*(\xred)$ whose degree is the dimension of $\xred$, provided that $0$ is a regular value of the momen…
Quillen's localization theorem is well known as a fundamental theorem in the study of algebraic K-theory. In this paper, we present its arithmetic analogue for the equivariant K-theory of arithmetic schemes, which are endowed with an action of certain diagonalisable group scheme. This equivariant arithmetic K-theory is defined by means of a natural extension of Burgos-Wang's simplicial description of Beilinson's regulator map to the equivariant case. As a byproduct of this work, we give an analytic refinement of the Riemann-Roch theorem for higher equivariant algebraic K-theory. And as an appl…
We use the theory of cubic structures to give a fixed point Riemann-Roch formula for the equivariant Euler characteristics of coherent sheaves on projective flat schemes over Z with a tame action of a finite abelian group. This formula supports a conjecture concerning the extent to which such equivariant Euler characteristics may be determined from the restriction of the sheaf to an infinitesimal neighborhood of the fixed point locus. Our results are applied to study the module structure of modular forms having Fourier coefficients in a ring of algebraic integers, as well as the action of diam…
For a smooth quasi-projective scheme $X$ over a field $k$ with an action of a reductive group, we establish a spectral sequence connecting the equivariant and the ordinary higher Chow groups of $X$. For $X$ smooth and projective, we show that this spectral sequence degenerates, leading to an explicit relation between the equivariant and the ordinary higher Chow groups. We obtain several applications to algebraic $K$-theory. We show that for a reductive group $G$ acting on a smooth projective scheme $X$, the forgetful map $K^G_i(X) \to K_i(X)$ induces an isomorphism $K^G_i(X)/{I_G K^G_i(X)} \co…
Given a holomorphic vector bundle $E:EX X$ over a compact Kähler manifold, one introduces twisted GW-invariants of $X$ replacing virtual fundamental cycles of moduli spaces of stable maps $f: Σ\to X$ by their cap-product with a chosen multiplicative characteristic class of $H^0(Σ, f^* E) - H^1(Σ, f^*E)$. Using the formalism of quantized quadratic hamiltonians, we express the descendent potential for the twisted theory in terms of that for $X$. The result (Theorem 1) is a consequence of Mumford's Riemann -- Roch -- Grothendieck formula applied to the universal stable map. When $E$ is concave, a…
Fulton and MacPherson introduced the notion of bivariant theories and Grothendieck transformations related to Riemann-Roch-theorems. But there are many situations, where such a bivariant theory or a corresponding Grothendieck transformation is only partially known: characteristic classes of singular spaces (e.g. Stiefel-Whitney or Chern classes), cohomology operations (e.g. singular Adams Riemann-Roch and Steenrod operations for Chow groups) or equivariant theories (e.g. Lefschetz Riemann-Roch). We introduce in this paper a simpler notion of partial (weak) bivariant theories and partial Grothe…
Suppose $(M,ω)$ is a compact symplectic manifold acted on by a compact Lie group $K$ in a Hamiltonian fashion, with moment map $μ: M \to \Lie(K)^*$ and Marsden-Weinstein reduction $M_{red} = μ^{-1}(0)/K$. In this paper, we assume that $M$ has a $K$-invariant Kähler structure. In an earlier paper, we proved a formula (the residue formula) for $η_0 e^{ω_0}[M_{red}]$ for any $η_0 \in H^*(M_{red})$, where $ω_0$ is the induced symplectic form on $M_{red}$. Here we apply the residue formula in the special case $η_0 = Td(M_{red})$; when $K$ acts freely on $μ^{-1}(0)$ this yields a formula for the Rie…
The subject of this dissertation is the Gysin homomorphism in equivariant cohomology for spaces with torus action. We consider spaces which are quotients of classical semisimple complex linear algebraic groups by a parabolic subgroup with the natural action of a maximal torus, the so-called partial flag varieties. We derive formulas for the Gysin homomorphism for the projection to a point, of the form \[\int_X α= Res_{\mathbf{z}=\infty} \mathcal{Z}(\mathbf{z}, \mathbf{t}) \cdot α(\mathbf{t}),\] for a certain residue operation and a map $\mathcal{Z}(\mathbf{z}, \mathbf{t})$, associated to the r…
In geometric representation theory, one often wishes to describe representations realized on spaces of invariant functions as trace functions of equivariant perverse sheaves. In the case of principal series representations of a connected split reductive group G over a local field, there is a description of families of these representations realized on spaces of functions on G invariant under the translation action of the Iwahori subgroup, or a suitable smaller compact open subgroup, studied by Howe, Bushnell and Kutzko, Roche, and others. In this paper, we construct categories of perverse shea…
For a finite group G acting on a smooth projective variety X, we construct two new G-equivariant rings: first the stringy K-theory of X, and second the stringy cohomology of X. For a smooth Deligne-Mumford stack Y we also construct a new ring called the full orbifold K-theory of Y. For a global quotient Y=[X/G], the ring of G-invariants of the stringy K-theory of X is a subalgebra of the full orbifold K-theory of the the stack Y and is linearly isomorphic to the ``orbifold K-theory'' of Adem-Ruan (and hence Atiyah-Segal), but carries a different, ``quantum,'' product, which respects the natura…
The main goal of this paper is to obtain a formula for the T-equivariant Riemann-Roch number of certain G-spaces which are the finite dimensional models of certain infinite dimensional spaces with Hamiltonian LG-actions, here T is a maximal torus of the semi-simple Lie group G. Unlike its finite dimensional cousin, the formula obtained here only needs to valuate on a finite subgroup of T depending on the 'level' of the LG-action and the dual Coxeter number of G, due to a fundamental cancelation. If the fixed-point sets of elements of the finite subgroup do not intersect the compactification lo…
We prove the LeBrun-Salamon Conjecture in low dimensions. More precisely, we show that a contact Fano manifold X of dimension 2n+1 that has reductive automorphism group of rank at least n-2 is necessarily homogeneous. This implies that any positive quaternion-Kahler manifold of real dimension at most 16 is necessarily a symmetric space, one of the Wolf spaces. A similar result about contact Fano manifolds of dimension at most 9 with reductive automorphism group also holds. The main difficulty in approaching the conjecture is how to recognize a homogeneous space in an abstract variety. We contr…
