Riemann–Roch coefficients for Kleinian orbisurfaces

Suppose S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {S}$$\end{document} 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.


Introduction
Let S be a Kleinian orbisurface over an algebraically closed field k.That is, S is a smooth, proper, and tame Deligne-Mumford surface with isolated stacky locus, ADE stabilizers, and projective coarse moduli.The classical Riemann-Roch theorem fails for orbisurfaces as it fails to capture contributions from the stacky locus.The Toën-Hirzebruch-Riemann-Roch theorem provides corrections terms to the classical formula for S coming from the inertia stack [16].These correction terms arise by pulling back to the inertia stack to determine the contributions from the twisted sectors.
The purpose of this paper is to determine the correction terms for S.This has already been accomplished when S has a stacky point of type A, [13,Section 3.3].In [4, Appendix A], the authors compute the correction term for the sheaf O S in all ADE cases.Moreover, they relate the correction term to the exceptional divisor of the minimal resolution of the coarse moduli S of S. This is a manifestation of the famous Bridgeland-King-Reid theorem asserting a derived equivalence between the derived categories of S and S [2].
The remaining cases are the binary tetrahedral (E 6 ), binary octahedral (E 7 ), binary icosahedral (E 8 ), and the binary dihedral groups (D).In Section 2, we give an overview of Toën's Riemann-Roch theorem for stacks.Section 3 introduces Kleinian orbisurfaces and spells out the Riemann-Roch theorem for this case.In Proposition 3.6, we recall a formula for the correction terms for a general ADE singularity, which relies on coefficients determined solely by the character table.In Section 4, we compute explicitly these Riemann-Roch coefficients in each of the ADE cases.In a different language, similar correction terms have been computed in [12,Sec. 5]; however, the present formulation is more natural when studying orbisurfaces from the stacky point of view.

Notation and conventions.
For a smooth quasi-projective scheme X over C, we denote by K(X) the Grothendieck group of coherent sheaves on X, and by H * (X) the singular cohomology with rational coefficients of its associated analytic space.We assume all algebraic stacks to be Deligne-Mumford, which implies that stabilizers are finite groups.We use Vistoli's definition of Chern and Todd classes for Deligne-Mumford stacks [18], and denote by K(X ) and H * (X ) the Grothendieck group and the (rational) singular homology of a smooth Deligne-Mumford stack X as described in [1].

Toën's Grothendieck-Riemann-Roch theorem for Deligne-Mumford stacks
In this section, we give an overview of Toën's Riemann-Roch theorem for stacks [16].The theorem holds in arbitrary characteristic, but for ease of exposition we work over the field of complex numbers1 .We direct the interested reader to other accounts of this result such as [3, Appendix A], [17, Appendix A], and to Edidin's equivariant Riemann-Roch formulation [7].
Our first step is to recall the statements for schemes (see for example [9,Chapter 15]).Let X be a smooth projective scheme, and E a perfect complex of sheaves on X. Denote by K(X) the Grothendieck group of coherent sheaves of X, and by H * (X) its singular cohomology.Then there is a linear map 2(1) where ch denotes the Chern character and Td denotes the Todd class.The Grothendieck-Riemann-Roch theorem states that τ is functorial with respect to proper push forwards, i.e. if f : X → Y is a proper map of smooth quasi-projective schemes, and E is a perfect complex of sheaves on X, The special case of Y = pt yields the Hirzebruch-Riemann-Roch theorem, which asserts For X a Deligne-Mumford stack, the analogous of the operator τ X is a map τ X valued in a suitable extension of scalars of the cohomology of the inertia stack I X of X .Recall the definition of the inertia stack: Definition 2.1 ([15, 8.1.17]).Let X be an algebraic stack.The inertia stack where ∆ is the diagonal embedding.
Remark 2.2.More explicitly, the objects of I X are pairs (x, g) where x is an object of X lying above a scheme T , and g is an automorphism of x in X (T ).Suppose X = [Z/H] where H is a finite group acting on a variety Z.Sections of X are H-torsors equipped with an equivariant map to Z.Then, I X can be canonically identified with where c(H) is the set of conjugacy classes of H, and C G (h) denotes the centralizer of a conjugacy class (h).
We denote by µ ∞ the subgroup of C * containing all roots of unity, and define Λ := Q(µ ∞ ) to be the rational numbers adjoined µ ∞ .As usual, for every Z-module A we denote by A Λ the tensor product A ⊗ Z Λ. Define a map ρ : K(I X ) → K(I X ) Λ as follows.A vector bundle E over I X decomposes as a sum of eigenbundles ⊕ ζ∈µ∞ E (ζ) as in the proof of [16, Théorème 3.15] 3 .Then, let Definition 2.3.Let X be a tame smooth Deligne-Mumford stack with quasi-projective coarse moduli space.Define the weighted Chern character, ch : K(X ) → H * (I X ) Λ , as the composition where σ : I X → X is the projection (onto either factor) and ch is the usual Chern character 4 .
Next, we define the weighted Todd class of X .This is a modification of the usual Todd class of I X .Let N denote the normal bundle of the local immersion σ : I X → X , and define [16,Lemme 4.6].Define the weighted Todd class of X as ( 2) and the Toën map τ Toën's Riemann-Roch theorem for stacks asserts that τ X behaves functorially with respect to proper push forwards: Theorem 2.4 ([16, Théorème 4.10]).Let f : X → Y be a proper morphism of smooth Deligne-Mumford stacks with quasi-projective coarse moduli spaces.Then for all E ∈ K(X ) we have 3 The decomposition, roughly, works as follows.A section f ∈ I X (T ) for some scheme T is the datum of a section x ∈ X (T ) with an automorphism a of x.Then, the bundle f * E on T is equipped with an action of < a >, which is diagonalizable by the tameness assumption.For ζ ∈ µ ∞ , one shows that the ζ-eigenbundle of f * E can be written as f * E (ζ) , for some vector bundle E (ζ) on I X . 4The singular homology of a Deligne-Mumford stack coincides rationally with that of its coarse moduli space.Then, we can regard the Chern character as landing in the Chen-Ruan orbifold cohomology of X , defined in [5] as the singular homology of the coarse moduli space I X of I X .
Moreover, if f : X → pt, we obtain Expanding the expression (3) we have ( 4) 3. Riemann-Roch for kleinian orbisurfaces 3.1.Kleinian orbisurfaces.Our object of study is the stacky resolution of singularities of ADE type (also known as Kleinian singularities).
Definition 3.1.An orbisurface is a smooth, proper, and tame Deligne-Mumford surface S over an algebraically closed field k with projective coarse moduli and isolated stacky locus.
For any orbisurface S, the stacky locus is a finite union of residual gerbes corresponding to finitely many k-points p i ∈ S(k), i.e.
Example 3.3.Let S be a surface with tame Kleinian singularities 5 .Then there exists a Kleinian orbisurface S can and a map π : S can → S such that: • the restriction S can \ π −1 (Sing(S)) → S \ Sing(S) is an isomorphism; • π is universal among all dominant, codimension preserving maps to S. The stack S can is called the canonical stack associated with the surface S, see [8].
We will compute a formula for the correction term δ(E) appearing in (4) in the case of a Kleinian orbisurface.Since δ(E) is computed at each residual gerbe independently, we may and will assume that S has a single stacky point, p, with residual gerbe ι : BG → S. We will see that the correction terms involve coefficients determined solely by the natural action of G on the tangent space T p S. For any subgroup G of GL 2 , we denote by V A 2 the natural representation.

The weighted Todd class.
Let S be a Kleinian orbisurface with a single stacky point p with stabilizer G. Arguing as in Remark 2.2, we see that the inertia stack of S is Here where the union is taken over all conjugacy classes (g) of non-trivial elements g ∈ G. Fix one of the components BC G (g).Its normal bundle in S is identified with in K(BC G (g)) (which is free, abelian, and generated by irreducible representations of C G (g)).
The element g acts diagonally on V , with eigenvalues some roots of unity ξ g and ξ −1 g .Thus, ch(ρ g is the character of V evaluated at g, i.e. the trace of g acting through the representation V .Using (2) we obtain: .
Integrating over the twisted sector: (5) δ(O S ) = If S → S is the projection to the coarse moduli space, then S has an ADE singularity at the image of the stacky point.The integral (5) is computed in [4] to be where C is the fundamental cycle of the minimal resolution of the singularity.

Riemann-Roch coefficients.
In this section, we write an expression of the term δ(E) appearing in (4) in terms of the wieghted Chern character of E and of the character table of the stabilizer group G.
The Grothendieck group of BG is free, Abelian and generated by the irreducible representations of G {ρ i | i = 0, ..., M }.For any perfect complex of sheaves E on S, its derived fiber is a formal linear combination On the component BC G (g), the element g acts on ρ i with eigenvalues denoted ζ (l) i , to which correspond eigenspaces ρ (l) i (the action is diagonalizable by the tameness assumption).Therefore, L ι * E decomposes on BC G (g) into weighted eigenbundles as Denote by χ i := χ ρ i = Tr • ρ i the character of the repesentation ρ i .Definition 3.5.For each i = 0, . . ., M set ( 6) .
We call the T i the Riemann-Roch coefficients of G.
Proposition 3.6.The correction term for a complex of sheaves E on S can be written as In particular, it only depends on the ranks of eigenbundles of Lι * E and the coefficients T i .The latter only depend on the character table of G.
Proof.The weighted Chern character of L ι * E |BC G (g) is given by Thus we have (7) δ(E) = .

Computation of Riemann-Roch Coefficients
Now we obtain formulae for the correction term (7), by explicitly computing the corresponding Riemann-Roch coefficients T i for all Kleinian singularities.This extends the computation done in [13] for singularities of type A, and completes the one started in [4], where the authors only compute δ(O S ).The computations do not depend on the characteristic of the base field, and neither do the results.

Singularities of type A.
In this case, the coefficients T j are computed by Lieblich in [13, Sec.3.3.2],who gives an explicit formula for δ(F ).We recall his result here.We'll make use of the following Lemma: Lemma 4.1 ([13, Lemma 3.3.2.1]).Let ζ be a primitive P -th root of unity and j ≤ P a non-negative integer.Then The K-theory of Bµ N is free Abelian of rank N with {χ j | j = 0, ..., N − 1} as a basis.For any perfect complex of sheaves F on S, we have Let ζ ∈ G be a primitive N -th root of unity.It acts on χ j with multiplication by ζ j .Then, equation ( 6) reads as a consequence of Lemma 4.1.

4.2.
Singularities of type D: Binary dihedral groups.In this case, the group acting is the binary dihedral group G = Dic n , it has order 4n and it gives rise to a singularity of type D n+2 with n ≥ 2. We can present it as The center of G is cyclic of order 2, generated by x 2 = a n .The quotient of G by its center is the dihedral group Dih n with 2n elements.
For the representation theory of Dic n , we point the reader to [11, §13] or to [6, §7.1].The group G has n + 3 conjugacy classes, grouped by cardinality as:

The corresponding centralizers have cardinality 4n, 2n and 4.
There are 4 one-dimensional representations.There are several two-dimensional representations, called dihedral, induced by G → Dih n there are n−1 2 of these if n is odd, and n−2 2 if n is even.The l-th dihedral representation is given by the assignment a → e 2lπi n 0 0 e − 2lπi n ; x = 0 1 1 0 .
The remaining representations are called of quaternionic type since they are induced by an inclusion G ⊂ SL(2, C).They are also two-dimensional, there are n−1 2 if n is odd, or n 2 if n is even.
We assume that G acts on C 2 via the first (l = 1) quaternionic representation, and denote by V this representation.Let T ρ be the Riemann-Roch coefficient corresponding to a representation ρ.For G = Dic n , we have .
First, we compute coefficients for the 4 one-dimensional representations.Let ρ x be the representation where x acts by -1 and a acts trivially, similarly for ρ a , and write ρ x,a = ρ x ⊗ ρ a .Let ζ := e − πi n .By applying Lemma 4.1 we obtain The remaining coefficient is (10) T Now we compute the the sum in (10): Using the (11) whence, applying Lemma 4.1 with P = 2n and j = n, If σ is the l-th quaternionic representation, we have ( 12) Using once again the substitution (11), we obtain: Applying Lemma 4.1 with P = 2n and j = l then yields and finally ( 13) If τ is a dihedral representation, then and we may repeat the argument above applying Lemma 4.1 with P = 2n and j = 2l: As above, let ρ 0 , ρ a , ρ x and ρ xa denote the one-dimensional representations.The two dimensional irreducible representation is denoted V .

Table 1 .
Character Table for 2T From (16) we can compute the Riemann-Roch coefficients which we arrange in Table2.

Table 2 .
Riemann-Roch Coefficients for 2T 4.3.2.The E 7 singularity.The binary octahedral group 2O is of order 48.Its character table is in

Table 3 .
Character Table for 2OHere ρ 2 is the natural representation.The Riemann-Roch coefficients are computed as follows: