We define bivariant algebraic K-theory and bivariant derived Chow on the homotopy category of derived schemes over a smooth base. The orientation on the latter corresponds to virtual Gysin homomorphisms. We then provide a morphism between these two bivariant theories and compare the two orientations. This comparison then yields a homological and cohomological Grothendieck-Riemann-Roch formula for virtual classes.
In this paper we prove a categorification of the Grothendieck-Riemann-Roch theorem. Our result implies in particular a Grothendieck-Riemann-Roch theorem for Toën and Vezzosi's secondary Chern character. As a main application, we establish a comparison between the Toën-Vezzosi Chern character and the classical Chern character, and show that the categorified Chern character recovers the classical de Rham realization.
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.
We prove that, for smooth quasi-projective varieties over a field, the $K$-theory $K(X)$ of vector bundles is the universal cohomology theory where $c_1(L\otimes \bar L)=c_1(L)+c_1(\bar L)-c_1(L)c_1(\bar L)$. Then, we show that Grothendieck's Riemann-Roch theorem is a direct consequence of this universal property, as well as the universal property of the graded $K$-theory $GK^\bullet (X)\otimes \mathbb{Q}$.
We give a short and "classical" proof of Borcherds' theorem that the moduli space of Enriques surfaces is quasi-affine. The use of the Borcherds' product is replaced in our proof by an application of the Grothendieck-Riemann-Roch theorem.
The goal of this series of papers is to give a new non-commutative approach to problems about the density of reductions such as the conjecture of Joshi-Rajan, and the generalization of the conjecture of Serre. In this paper, we prove integral Grothendieck Riemann-Roch which was proved by Papas in the case ch$(k)=0$. As a corollary we prove an integral analogue of Kontsevich's comparison theorem, and we show that if a smooth projective variety $X$ has a full exceptional collection then there is an explicit formula of the motive of $X$ up to bounded torsion.
We show that, in characteristic zero, the obvious integral version of the Grothendieck-Riemann-Roch formula obtained by clearing the denominators of the Todd and Chern characters is true (without having to divide the Chow groups by their torsion subgroups). The proof introduces an alternative to Grothendieck's strategy: we use resolution of singularities and the weak factorization theorem for birational maps.
The classical arithmetic Grothendieck-Riemann-Roch theorem can be applied only to projective morphisms that are smooth over the complex numbers. In this paper we generalize the arithmetic Grothendieck-Riemann-Roch theorem to the case of general projective morphisms between regular arithmetic varieties. To this end we rely on the theory of generalized analytic torsion developed by the authors.
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…
This is the second of series of papers on the study of foliations in the setting of derived algebraic geometry based on the central notion of derived foliation. We introduce sheaf-like coefficients for derived foliations, called quasi-coherent crystals, and construct a certain sheaf of dg-algebras of differential operators along a given derived foliation, with the property that quasi-coherent crystals can be interpreted as modules over this sheaf of differential operators. We use this interpretation in order to introduce the notion of good filtrations on quasi-coherent crystals, and define the…
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…
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…
We prove a Hirzebruch-Riemann-Roch type formula for global matrix factorizations. This is established by an explicit realization of the abstract Hirzebruch-Riemann-Roch type formula of Shklarov. We also show a Grothendieck-Riemann-Roch type theorem.
V. Lunts has recently established Lefschetz fixed point theorems for Fourier-Mukai functors and dg algebras. In the same vein, D. Shklyarov introduced the noncommutative analogue of the Hirzebruch-Riemann-Roch theorem. In this short article, we see how these constructions and computations formally stem from their motivic counterparts.
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…
Let $S$ be a finite dimensional noetherian scheme. For any proper morphism between smooth $S$-schemes, we prove a Riemann-Roch formula relating higher algebraic $K$-theory and motivic cohomology, thus with no projective hypothesis neither on the schemes nor on the morphism. We also prove, without projective assumptions, an arithmetic Riemann-Roch theorem involving Arakelov's higher $K$-theory and motivic cohomology as well as an analogue result for the relative cohomology of a morphism. These results are obtained as corollaries of a motivic statement that is valid for morphisms between oriente…
We study the singular Bott-Chern classes introduced by Bismut, Gillet and Soule. Singular Bott-Chern classes are the main ingredient to define direct images for closed immersions in arithmetic K-theory. In this paper we give an axiomatic definition of a theory of singular Bott-Chern classes, study their properties, and classify all possible theories of this kind. We identify the theory defined by Bismut, Gillet and Soule as the only one that satisfies the additional condition of being homogeneous. We include a proof of the arithmetic Grothendieck-Riemann-Roch theorem for closed immersions that…
Mapping a locally free module to its l-th tensor power gives rise to a natural map from the Grothendieck group of all locally free modules to the Grothendieck group of all locally free representations of the l-th symmetric group. In this paper, we prove some formulas of Riemann-Roch type for the behaviour of this tensor power operation with respect to the push-forward homomorphism associated with a projective morphism between schemes. We furthermore establish analogous formulas for higher K-groups.
We prove the classical Riemann-Roch theorems for the Adams operations $\,ψ^j\,$ on $K$-theory: a statement with coefficients on $\mathbb{Z}[j^{-1}]$, that holds for arbitrary projective morphisms, as well as another one with integral coefficients, that is valid for closed immersions. In presence of rational coefficients, we also analyze the relation between the corresponding Riemann-Roch formula for one Adams operation and the analogous formula for the Chern character. To do so, we complete the elementary exposition of the work of Panin-Smirnov that was initiated by the first author in a previ…
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.
We prove two refinements of the higher Riemann-Roch without denominators: a statement for regular closed immersions between arbitrary finite dimensional noetherian schemes, with no smoothness assumptions, and a statement for the relative cohomology of a proper morphism.
In this paper, we formulate axioms of certain graded cohomology theory for which Chern class maps from higher K-theory are defined, following the method of Gillet [Gi1]. We will not include homotopy invariance nor purity in our axioms. It will turn out that the Grothendieck-Riemann-Roch theorem and the Riemann-Roch theorem without denominators hold for our higher Chern classes. We will also give two applications of our Riemann-Roch results.
We prove the Riemann-Roch theorem for homotopy invariant $K$-theory and projective local complete intersection morphisms between finite dimensional noetherian schemes, without smoothness assumptions. We also prove a new Riemann-Roch theorem for the relative cohomology of a morphism. In order to do so, we construct and characterize Gysin morphisms for regular immersions between cohomologies represented by spectra (examples include homotopy invariant $K$-theory, motivic cohomology, their arithmetic counterparts, real absolute Hodge and Deligne-Beilinson cohomology, rigid syntomic cohomology, mix…
This is the integral text of my thesis. The first part is an expanded version of "Riemann-Roch theorems for Deligne-Mumford stacks", where I deal with Artin stacks over general bases. In the second part, I prove some Riemann-Roch statment for D-modules on Deligne-Mumford stacks, and I also consider the problem of algebraization of analytic stacks.
In this article, we show that the combination of the constructions done in SGA 6 and the A^1-homotopy theory naturally leads to results on higher algebraic K-theory. This applies to the operations on algebraic K-theory, Chern characters and Riemann-Roch theorems.
This is the first of a series of papers. Our final goal is to establish Deligne-Riemann-Roch isomorphisms in various settings. In this paper, we establish a uniqueness theorem for Deligne pairings and prove the degree $1$ part of the Deligne-Riemann-Roch theorem.
Smooth and proper dg-algebras have an Euler class valued in the Hochschild homology of the algebra. This Euler class is worthy of this name since it satisfies many familiar properties including compatibility with the familiar pairing on the Hochschild homology of the algebra and that of its opposite. This compatibility is the Riemann-Roch theorems of Shklyarov and Petit. In this paper we prove a broad generalization of these Riemann-Roch theorems. We generalize from the bicategory of dg-algebras and their bimodules to monoidal bicategories and from Euler class to traces of non identity maps. O…
Suppose $\mathcal{S}$ is a smooth, proper, and tame Deligne-Mumford stack. Toën's Grothendieck-Riemann-Roch theorem requires correction terms, involving components of the inertia stack, to the standard formula for schemes. We give a brief overview of Toën's Grothendieck-Riemann-Roch theorem, and explicitly compute the correction terms in the case of an orbifold surface with stabilizers of types ADE.
The goal of this paper is to prove Riemann-Roch type theorems for Deligne-Mumford algebraic stacks. To this end, we introduce a "cohomology with coefficients in representations" and a Chern character, and we prove a Grothendieck-Riemann-Roch theorem for the Riemann-Roch transformation it defines. As a corollary we obtain an Hirzebruch-Riemann-Roch formula for the Euler characteristic of a coherent sheaf, and some formulas for the different topological Euler characteristics of complex algebraic stacks.
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 establish a Hirzebruch-Riemann-Roch type theorem and Grothendieck-Riemann-Roch type theorem for matrix factorizations on quotient Deligne-Mumford stacks. For this we first construct a Hochschild-Kostant-Rosenberg type isomorphism explicit enough to yield a categorical Chern character formula. We next find an expression of the canonical pairing of Shklyarov under the isomorphism.
Considering quasismooth varieities as global $\CC^*$ quotients, we present a Riemann-Roch formula via general Riemann-Roch formula for quotient stacks. Furthermore, we give a parcing formula for Hilbert series associated to a polarized quasismooth projectively Gorenstein algebraic varieties with orbifold curves and dissident points, which is an extension of the result in \cite{BRZ}.
In this thesis we study toric rank functions for chip firing games and prove special cases of a conjectural Riemann-Roch. The original motivation for an investigation into this area of study came for the adaptation (due to Matt Baker) of Riemann-Roch into a graph theoretic analogue through the use of chip-firing games. Here, we collect known results and present new observations that indicate Riemann--Roch holds for trees and polygons. We also prove an asymptotic case of Riemann--Roch (i.e.~Riemann--Roch for divisors of large degree). Finally, we also provide magma code and computational eviden…
We prove an orbifold Riemann--Roch formula for a polarized 3--fold (X,D). As an application, we construct new families of projective Calabi--Yau threefolds.
Recently, Baker and Norine have proven a Riemann-Roch theorem for finite graphs. We extend their results to metric graphs and thus establish a Riemann-Roch theorem for divisors on (abstract) tropical curves.
We develop a formalism involving Atiyah classes of sheaves on a smooth manifold, Hochschild chain and cochain complexes. As an application we prove a version of the Riemann--Roch theorem.
We propose an algebraic Riemann-Roch formula for moving flat bundles on contant families in characteristic zero with values in the ring of algebraic differential characters. This formula lifts Grothendieck-Riemann-Roch formula in the Chow groups and Bismut-Lott formula in the odd dimensional Betti cohomology with $\C/\Z$ coefficients over the field of complex numbers.
We deduce the Riemann-Roch type formula expressing the microlocal Euler class of a perfect complex of D-modules in terms of the Chern character of the associated symbol complex and the Todd class of the manifold from the Riemann-Roch type theorem for periodic cyclic cocycles of a symplectic deformation quantization. The proof of the latter is contained in the sequel to this paper.
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 quotie…
We explain how the Riemann-Roch theorem for divisors on an abelian variety $A$ is related to the reduced norms of the Wedderburn components of $\operatorname{End}^0(A)$ the $\mathbb{Q}$-endomorphism algebra of $A$. We then describe consequences and examples.
This is the first article in an upcoming series of papers. They have arisen through an attempt to answer open questions of Deligne proposed in "Le determinant de la cohomologie", Contemp. Mathematics 67 (1987). It amounts to functorial and metrized versions of the Grothendieck-Riemann-Roch theorem, as well as a Lefschetz-Riemann-Roch formula in the sense of Thomason, cf. "Lefschetz-Riemann-Roch theorem and coherent trace formula" Ann. Sci. Ec. Norm. Sup. 18 (1985), Theorem 3.5. In this article we treat various preliminary results on virtual categories which are the categories where the Deligne…
We give a new proof of the Adams-Riemann-Roch theorem for a smooth projective morphism $X\to Y$, in the situation where $Y$ is a regular scheme, which is quasi-projective over $\mF_p$. We also partially answer a question of B. Köck.
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 …
We show that the space of theta functions on tropical tori is identified with a convex polyhedron. We also show a Riemann-Roch inequality for tropical abelian surfaces by calculating the self-intersection numbers of divisors.
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 note, we give a proof for a variant of the functorial Deligne-Riemann-Roch theorem in positive characteristic based on ideas appearing in Pink and Rössler's proof of the Adams-Riemann-Roch theorem in positive characteristic (see \cite{Pi}). The method of their proof appearing in \cite{Pi}, which is valid for any positive characteristic and which is completely different from the classical proof, will allow us to prove the functorial Deligne-Riemann-Roch theorem in a much easier and more direct way. Our proof is also partially compatible with Mumford's isomorphism.
If G is a finite subgroup of the automorphism group of a projective curve X and D is a divisor on X stabilized by G, then under the assumption that D is nonspecial, we compute a simplified formula for the trace of the natural representation of G on Riemann-Roch space L(D).
For a divisor $D$ on a tropical variety $X$, we define two amounts in order to estimate the value of $h^{0}(X,D)$, which are described by terms of global sections and computed more easily than $h^{0}(X,D)$. As an application of its estimation, we show that a Riemann-Roch inequality holds for smooth tropical toric surfaces.
We prove a Riemann-Roch theorem for real divisors on edge-weighted graphs over the reals, extending the result of Baker and Norine for integral divisors on graphs with multiple edges.
