The Verlinde traces for SU X (2 , ξ ) and blow-ups

. Given a compact Riemann surface X of genus at least 2 with automorphism group G , we provide formulae that enable us to compute traces of automorphisms of X on the space of global sections of G -linearized line bundles deﬁned on certain blow-ups of projective spaces along the curve X . The method is an adaptation of one used by Thaddeus to compute the dimensions of those spaces. In particular, we can compute the traces of automorphisms of X on the Verlinde spaces corresponding to the moduli space SU X (2 , ξ ) when ξ is a line bundle G -linearized of suitable degree.


Introduction
Let X be a complex, irreducible, smooth, projective curve of genus at least 2 and automorphism group G = Aut(X).Let ξ be a G−linearized line bundle over X.
By the Verlinde traces, we refer to the traces of automorphisms of X on the space H 0 (SU X (r, ξ), O(nΘ)), where SU X (r, ξ) is the moduli space of semi-stable rank r vector bundles with determinant ξ and where O(Θ) is the determinantal line bundle of SU X (r, ξ).In this work, we address the problem of computing the Verlinde traces for the case r = 2 and we give a way to compute them which is alternative to the approach of J. E. Andersen for case of the trace τ (k) G (f ) in [2] pg. 3 (G = SU (2) in our case).Our method and result will be explained in the paragraphs below.Before that, we would like to mention a few things related to this problem.As the reader may be aware the case of the Verlinde traces for the identity of G is already solved.A formula for dim H 0 (SU X (r, ξ), O(nΘ)) was conjectured by E. Verlinde [34] and there are many proofs of it(see for instance the following works concerning this case [7,8,12,15,17,28,[31][32][33]35] and [36]).The action of non-trivial automorphism groups G on the Verlinde spaces had already been considered in the work of Dolgachev ([13] in his proof of Cor.6.3), in the work of the first author [23] (see  there) where some Verlinde traces are computed for the cases ξ = O X , n = 1 and arbitrary rank r by computing them on H 0 (J gX −1 X , O(rΘ J g X −1 X )) * ∼ = H 0 (SU X (r, ξ), O(Θ)) and later in the work of Andersen [2] the Verlinde traces correspond to the traces τ (k) G (f ) which we will briefly refer to at the end of this introduction.
As there are automorphisms of SU X (r, ξ) that are not induced by G (for a description Aut(SU X (r, ξ)) and related results see [9,20,21]) we should also mention that the action of torsion elements of the Jacobian of X acting on the Verlinde spaces of some moduli spaces of vector bundles had also been considered in the works [6,27,29].Explicit formulae for the corresponding Verlinde traces are provided in [6] and [29].Coming back to our problem, in the rank 2 case, we followed the method used by Thaddeus to derive the Verlinde formula in [32].We shall see that his method can be extended to compute the Verlinde traces (see formula (4.9) and Sect.5) by just replacing the use of the Riemann-Roch Theorem for the use of the Atiyah-Singer Holomorphic Lefschetz Theorem (see Sect. 3) and in this work, we derive some formulae (Theorem 8.1 ) required to apply the Holomorphic Lefschetz Theorem in Thaddeus' method.
Let K X be the canonical line bundle of X. Suppose that K X ξ is very ample.Let X → P N be the embedding defined by the complete linear system | K X ξ |.Let π : P N X → P N be the blow-up of P N with center X and let E be the corresponding exceptional divisor.The Picard group of P N X is generated by O(E) and the hyperplane line bundle O(H).For integers m, n let O 1 (m, n) = O((m + n)H − nE) and let V m,n = H 0 ( P N X , O 1 (m, n)).Let ξ have degree d.Thaddeus shows that for d > 2g X − 2 there is a natural isomorphism H 0 (SU X (2, ξ), O(kΘ)) ∼ = V k,k(d/2−1) . (1.1) Under some mild conditions on m, n he finds a formula for the dimension of V m,n (see Theorem 4.1 below).The cases not covered by those conditions can be dealt with easily.Now, when we assume that ξ is G-linearized it induces an action of G on P N and on the blow-up P N X such that the embedding X → P N and the blow-up map π are G-equivariant.The line bundles O 1 (m, n) can be equipped with a linearization induced by that of ξ.That is because O(H) = π * O P N (1) comes equipped with a linearization induced by that of ξ; also since E is a G-invariant divisor O(E) admits a linearization of G with trivial action on H 0 ( P N X , O(E)), and since h 0 ( P N X , O(E)) = 1 we can see that this linearization is unique(with trivial action on H 0 ( P N X , O(E))) because there is a bijection between the G-linearizations of O(E) and the G-invariant divisors linearly equivalent to E by G-invariant rational functions (see [13] Prop.2.1 and [14] Ex. 7.4).So one can consider the problem of computing the traces of elements of G on the spaces V m,n (Thaddeus traces).As we will see in Sect.4, the formula for dimV m,n is a linear combination of Euler Characteristics of sheaves B i,m,n defined over symmetric products S i X of the curve (see Theorem 4.1 below) and these sheaves are naturally G-linearized.By tracking back the proof of Theorem 4.1 one notice that the homomorphisms between the cohomology groups involved are G-equivariant and that the trace of an element h ∈ G on V m,n is in fact obtained by replacing the Euler characteristics of the sheaves B i,m,n by their corresponding Lefschetz numbers N i (h) = L(h, B i,m,n ) (see formula (4.9) below and Sect. 5 for its proof).These Lefschetz numbers L(h, B i,m,n ) can be computed using the Holomorphic Lefschetz Theorem (see Sect. 3, Theorem 3.1 and formula (3.6)) because the sheaves B i,m,n are defined on smooth varieties.To do that we need to know for each component Z of the fixed point set the following data: the generalized Chern Character ch h (i * Z B i,m,n ) (see (3.1)), the stable characteristic classes U (N Z/S i X (ν j )) of the normal bundle N Z/S i X (see (3.2)), the Todd class Td(T Z ) of the tangent bundle T Z and det(Id − h | N ∨ Z/S i X ) (see Theorem 3.1).In this way by equation (3.6), the solution of the problem of computing N i (h) = L(h, B i,m,n ) is reduced to the calculation of the following intersection numbers, namely, the contributions As we pointed out earlier, the main goal of this paper (Theorem 8.1) is the calculation of the generalized Chern Character ch h (i * Z B i,m,n ).The other data required to apply (1.2) have been dealt with in the works [23](Prop.3.2) and [24] (Theorem 2.3).However, in Sect.7, we present a generalization of the formula for the stable characteristic classes U (N Z/S i X (ν j )) that was given in [24].The Theorem 6.1 is required for the proof of Theorem 8.1 and part of its proof is modelled on the proof of Proposition 2.1 in [24].At the end of the paper we illustrate the use of formula (4.9), when X is a hyperelliptic curve of genus 2, by computing the Verlinde traces corresponding to the hyperelliptic involution.
The problem of computing Verlinde traces may be considered interesting in its own right and following the work of Thaddeus on moduli of pairs there have been various constructions of series flips using moduli of pairs, triplets, that one should expect to be able to use to compute Verlinde traces on moduli spaces other than SU X (2, ξ).Some of our motivations are to determine the isotypic decomposition of the Verlinde spaces H 0 (SU X (r, ξ), O(nΘ)) to study the varieties defined as the zero locus of the invariant submodules in, for instance, PH 0 (SU X (r, ξ), O(2Θ)) ∨ ; to study the Cox Ring of Blowups( in this case with the Thaddeus traces).Also, as it can be seen from the work of Andersen, Verlinde traces are used to compute invariants of 3-Manifolds, namely, the Witten-Reshetikhin-Turaev invariants of finite order mapping tori Z (k) G (f ).Andersen considers moduli spaces of semi-stable G-bundles on curves and by applying directly the Lefschetz-Riemann-Roch Theorem for singular varieties (see [5]) to the determinantal line bundle of the moduli spaces he obtained an expression for the traces τ (k) G (f ) because the higher cohomologies of the determinantal line bundle vanish.The formula is given by the equation 8.1 or Theorem 1.3 in [2] up to the correction factor Det(f ) − 1 2 ζ , and depends on data coming from the components of the fixed point set.The moduli spaces SU X (r, O X ) correspond to the case G = SU (r) and a description of the components of fixed point set for this case is given in Theorem 6.10 of [2] (see also Theorem 3.4 of [1]).The contribution of the smooth components to the traces τ (k) G (f ) is also determined in his work.In the rank 2 case, the advantage of our method is that one does not have to deal with singular components of fixed points and one can compute the Verlinde traces if one knows the action of the automorphism on the tangent spaces of the fixed points in the curve.Some results of this work are based on results of the Ph.D Thesis of the second author [30].

Notation
Let p be a positive integer and let ν = exp(2iπ/p).Given a finite cyclic group H = h of order p acting trivially on a variety Z and given an H-linearized vector bundle F on Z there is a decomposition into eigenbundles that is, F (ν j ) is the sub-bundle of F where the action of h on the fibres is multiplication by ν j .Most of the times we will say that F is h-linearized.
For a divisor D on a variety W we write O W (D) (or just O(D)) to denote corresponding line-bundle of D. If F is a sheaf on W we usually write F (D) rather than F ⊗ O(D).Usually F n = F ⊗n , also if K is another sheaf some times we could write If F is locally free then F ∨ is the dual sheaf and we also denote by F the vector bundle whose sheaves of sections is F .If x ∈ W then F x is the fibre of F at the point x.For a vector bundle F over X we denote by π : P(F ) → X the projective bundle of one dimensional subspaces of F thus P(F ) = P roj OX (Sym(F ∨ )) where Sym(E) = ⊕ m S m E is the symmetric algebra of a locally free sheaf E over X.The exponential function exp(x) = ∞ n=0 x n /n! is also be represented by e x and in some formulae we mixed both notations.

The Holomorphic Lefschetz theorem
Consider a finite cyclic group H = h of order p ≥ 1. Suppose that H is acting trivially on a variety Z and also consider an hlinearized vector bundle F on Z.The generalized Chern character of F is given by where ch[F (ν j )] is the Chern character of the eigenbundle F (ν j ).For each vector bundle F (ν j ) with ν j = 1 define the stable characteristic class U (F (ν j )) as where r j is the rank F (ν j ) and the x i 's are the Chern roots of F (ν j ).We now make some observations about fixed point sets of h acting on non singular Varieties (see pg. 537 in [3] and Lemma 4.1 in [16]).Let X be an irreducible, non-singular complex algebraic variety on which h acts as an automorphism of X.Let X h denote the set of fixed points of h in X.One has that X h is non-singular and if x ∈ X h then as an analytic variety X h is (in a neighbourhood of x) the image, under a suitable equivariant exponential map, of the eigenspace T X,x (ν 0 ) of the tangent space T X,x to X at x.As T X h ,x = T X,x (ν 0 ) one has that the linear transformation h | N X h /X,x induced by h on the normal space N X h /X,x to X h at x has no eigenvalue ν 0 and therefore One also notice from what was said above that X h is locally irreducible so a point x ∈ X h belongs to a unique irreducible component of X h and so where and can be computed by the Holomorphic Lefschetz Theorem: Theorem 3.1.(Atiyah-Singer, Theorem 4.6, pg.566, [4]) On the right-hand side of (3.5), one is evaluating a cohomology class u ∈ H * (X h , C) (here u is the integrand).By (3.4) on has that and then the evaluation of u is the sum of the evaluations of the components A particular case of Theorem 3.1 is the Atiyah-Bott formula also known as the Woods Hole Fixed Point Theorem which is obtained when X h is a finite set: where trz h|E x is the trace of h|E x (see [10] pgs.631 and 121 ).

The trace formula
From now on what follows X will denote the curve considered in the introduction, that is, a complex, irreducible, smooth curve of genus g ≥ 2 embedded in P N = PH 1 (ξ −1 ) via the linear system |K X ξ|, where ξ is a line bundle on X of degree d.Let π S i X : X × S i X → S i X and π X : X × S i X → X be the natural projections.Let Δ i ⊂ X × S i X be the universal divisor and let j denote its inclusion into X × S i X.Consider the Thaddeus bundles and One has that W − i is a vector bundle of rank i and that if −d + 2i < 0 then Define where (π S i X ) ! : K(X × S i X) → K(S i X) is the direct image homomorphism between Grothendieck groups given by for a coherent sheaf F over X ×S i X (See [19] pg.436).Also, det : K(S i X) → P ic(S i X) is the group homomorphism between the additive group K(S i X) and the multiplicative group P ic(S i X) given by where a 1 , • • • , a m ∈ Z, the F j s are coherent sheaves and det(F j ) is the usual determinant of coherent sheaves defined by means of a locally free resolution of F j .Let U i → S i X be the bundle Let q i = n − (i − 1)m.For i > 0 consider the Euler characteristic where For i = 0 we define B 0,m,n = S m+n H 0 (X, K X ξ) and N 0 = dimB 0,m,n .It is a convention that B i,m,n = 0 when q i − i < 0.
Theorem 4.1.(See (6.9) in [32]) Let m, n ≥ 0 and suppose that Under our hypothesis (namely, those of Theorem 4.1 together with the one that X has automorphism group G and that ξ is G-linearized so that line bundles O 1 (m, n) are G-linearized) one has that for any h ∈ G (see Sect. 5) where N i (h) stands for the Lefschetz number and for i = 0, . (4.11)The far right-hand side of (4.11) follows from the proof of Molien's Theorem 1.10 in [25].So the Lefschetz numbers N i (h), i > 0, can be computed by means of the Holomorphic Lefschetz Theorem if one (see formula (3.6)) can determine the contribution h of the fixed point set.It will be explained in Sect.6 that the components Z of (S i X) h are parametrized by certain set of h-invariant divisors D on the curve X so in the subsequent sections we will write Z D rather than just Z because although components associated to distinct divisors may be isomorphic the data associated to them may not be the same.So far, the main obstruction to compute ) and it will be determined in Sect.8.In Sect.7, Theorem 7.1 we present a new formula for the stable characteristic classes U(N Z/S i X (ν j )) and in (7.5)

one for det(Id
The Todd class of a component is given in formula (6.8).

The proof of Thaddeus
Here we shall derive formula (4.9) so we are assuming that a finite group G is acting on the curve X and that ξ is a G-linearized line bundle of degree d.Recall also that K X ξ is very ample and X is embedded into PH 1 (X, ξ −1 ) by the complete linear system | K X ξ |.We should remark that ξ plays the role of the line bundle Λ considered in [32].Thaddeus' proof of Theorem 4.1 uses 4 results ((6.2),(6.6),(6.7)and (6.8) in [32] ) which we will state here in terms of Lefschetz numbers ( see Lemmas 5.1-5.4below), their proofs follow from those of Thaddeus so we are essentially sketching his proof.We start by introducing some notation related to the main tool which is a series of flips, depicted in the diagram (5.1) below, that connects the space PH 1 (X, ξ −1 ) and the moduli space SU X (2, ξ).
The spaces M i are constructed as moduli spaces of σ i -stable pairs (E, φ) where E is a rank 2 vector bundle on the curve X such that the definition of the stability condition see (1.1) in [32]).The space M 1 is P All the spaces M i turn out to be smooth, integral, rational projective varieties of dimension d + g − 1 and for i > 0 there is a birational map M i ↔ M 1 which is an isomorphism except on closed sets of codimension at least 2. In fact there are embeddings , whose images correspond to the pairs represented in M i but not in M i−1 and pairs represented in M i−1 but not in M i respectively and there is an isomorphism As for the spaces M i , for i > 1, the arrows between M i−1 , M i and M i make up each of the announced flips, specifically, M i is the blow-up of M i−1 along PW − i and it is also the the blow-up of M i along PW + i , furthermore M i can be obtained by blowing-up M i−1 along PW − i and then blowing down the same exceptional divisor to respectively the following exact sequences (see (3.9) and (3.11) in [32]) we recall that according to our notation here . From the exact sequences (5.2) one notice that and in fact is the fibred product of the projections from PW − i and PW + i to S i X and we represent it in the diagram (5.3) for future reference in the proof Lemma 5.3 below. (5.4) The twisting sheaf of E i happens to be

Now we consider the line bundles
Since M 1 and M i are isomorphic outside closed sets of codimension at least 2 there is (as a consequence of Hartogs' theorem), for i > 0, an isomorphism Let O i (m, n) be the image of O 1 (m, n) under this isomorphism.One also has (another consequence of Hartogs' theorem) that (5.5) Recall that we denoted in the introduction On M i one has (see (5.6) in [32]) When we assume that a group G is acting on the curve X one has by functoriality an action on all the moduli spaces M i and N .This action is compatible with the isomorphisms i because pairs representing points on the left-hand side also represent points on the righthand side.In the introduction, we endowed the line bundles O 1 (m, n) with a G-linearization and because of the isomorphism M 1 ↔ M i outside closed subsets of codimension at least 2 one has that this linearization also induces a linearization on the other line bundles O i (m, n).In this way, the action of earizations is the same.The action of G on the spaces M i also lifts to an action on the spaces M i and one also endows the line bundles . Now, we can state the following lemmas.

Using (5.5), we have
For the remainder of this section, we shall assume that m, n ≥ 0, Proof.This follows from the definitions of N 0 (h) and O 0 (m, n).Namely, we have )).Notice that when i = 0, we have q i − i = m + n ≥ 0 (see definition before (4.7)).Therefore, we have the following and and by the definition of B 0,m,n in Sect. 4 the last is which in turn is the definition of N 0 (h) in (4.11).
We can also restate (6.7) of [32] as the following . But this follows because the higher direct images of O i (m, n) under the map π : M i → M i vanish so the Leray spectral sequence implies and one can work on M i .Recall that q i = n − (i − 1)m.Next, one considers two cases.1) When q i ≤ 0, by definition of B i,m,n (see equation (4.8)), one has (5.6).So we assume now that q i < 0. Since E i is a G-invariant divisor on M i , the exact sequence induced by the embedding is a G-equivariant exact sequence of linearized sheaves so for each j we have an equivariant exact sequence with the sheaf(or rather its pushforward to M i ) L m i (−q i −j, −j).The last corresponds to a the tensor product F 1 ⊗ F 2 of sheaves on E i where F 1 is the pull-back of the sheaf L m i ( defined in (4.3)) under the projection π : The exact sequence (5.8) induces a G-equivariant long exact sequence of cohomology groups from which we can write ( using the above identification): (5.9) Summing (5.9) over j, with 0 < j ≤ −q i , and using (5.6) one arrives to (5.10) When h is the identity Thaddeus proves the vanishing of the right-hand side by showing that all the cohomology groups vanish (all the direct images of L m i (−q i −j, −j) under the projection p 1 : . Therefore the right-hand side of (5.10) vanish for any h ∈ G.
2) Now, we assume q i > 0. Summing (5.9) over j, with −q i < j ≤ 0, and using (5.6) one obtains (5.11) Now, the only non-zero direct image of L m i (−q i + j, +j) under the projection π : E i S i X is the (i-1)-th (see (5.12) below): ).We shall see that the only non-zero term of the Leray spectral sequence is (5.12) By the Leray spectral sequence(the usual one) one has and the rest is a rather straightforward verification.One can write (5.14) Multiplying (5.11) by −1 and using (5.14) we have and if j > q i − i then S qi−j−i (W − i ) = 0 so in the last equation the sum becomes a sum that runs only from j = 0 to j = q i − i, that is Proof.This is the same proof of (6.8) in [32] since one shows that i > b implies q i − i < 0 so B i,m,n = 0. Finally, from Lemmas 5.1-5.4,one has

The Chern classes
Let h be an automorphism of the curve X and assume that h has order p = 1.We shall explain below that the k-dimensional components of fixed points of h in the symmetric product S i X are parametrized by certain kind of h-invariant divisors so we represent such a component by Z D , where D is the corresponding invariant divisor.Let ι D denote the inclusion Z D ⊂ S i X.Consider the decompositions into eigenbundles and For the proof of Theorem 8.1 in Sect.8, we need to know the Chern classes of all these eigenbundles and before we compute them we recall from [23] Section 3, that a k-dimensional component of fixed points of h in S i X is isomorphic to the symmetric product S k Y , here Y is the quotient curve X/ h .These components are parametrized by a set of certain kind of hinvariant divisors A k of degree d k = i − pk, hence the notation Z D .More precisely, define A k as the set of divisors D ∈ (S d k X) h satisfying the following property: if x ∈ X is a point in the support of D then D − p−1 j=0 h j x is not an effective divisor nor the zero divisor.For each D ∈ A k there is an embedding where Then, the Chern classes of our eigenbundles can be expressed in terms of the cohomology classes θ, x and σ i ∈ H 2 (S k Y, Z) (see [22] for details on cohomology of symmetric products), where x represents the class of a divisor ) and θ represents the class of the pull back of the theta divisor class Θ ∈ H 2 (J Y , Z) of the Jacobian J Y of the curve Y under the Abel Jacobi map.We recall some relations of these cohomology classes: (1 + zσ i ).(6.7) The Todd class T d(Z D ) of a k-dimensional component is given by see for instance (7.3) in [32], there σ = θ and η = x.Let D ∈ A k .We will consider as well the following decomposition into eigenbundles on Y here λ s,j,n := f * (ξ s (−nD))(ν j ).We have the following Theorem 6.1.Let Z D be a k-dimensional component of fixed points of h in S i X.Let m j,1 , m j,2 ,m j,n denote the degrees of the bundles λ 1,j,1 ,λ 1,j,2 and λ −1,j,n ,respectively, in formula (6.9).Let g Y be the genus of the quotient curve Y .Then, (a) For the eigenbundles in (6.2), their corresponding Chern characters and classes are given by: (1 + 2x) (1+m j,−2 +2k−gY ) .(6.11) (b) For the eigenbundles in equation (6.1) we have: In the diagrams (6.14), (6.16)and (6.18) below we introduce notation for some morphisms that appear in the proof of Theorem 6.1 and of Lemma 6.1.In the diagram (6.14) ρ S k Y and π S i X are the natural projections, ι D is the embedding (6.3) corresponding to the component Z D , j stands for the embedding of the universal divisor Δ i of S i X, According to (6.3), we have and the diagram (6.14) can be subdivided as in (6.16) below (6.16)We will also consider the universal divisor Δ pk of S pk X and the projections The diagram (6.18) involves the projection ρ S k Y which is decomposed as We will often use the line bundle ξ = ρ * X ξ on X × S k Y .For example, we have In particular, the ) and F has finite fibres one can write Next, we shall write Δ in terms of the universal divisor Δ pk of S pk X and of the divisor D (see formula (6.24) below).We have (by definition of Δ and equation (6.15)) and by the universal property of Δ i applied to the right-hand side square of the diagram (6.16) one has that (Id X × A D ) * (Δ i ) is the relative divisor of degree i inducing A D .So we have (Id X × A D ) * (Δ i ) = Δ pk + p * X D and using this in (6.23), we get Δ = (ρ X ) * D + (Id X × ι) * Δ pk . (6.24) Now, using (6.24) in (6.22) and the fact that ).Now we use the following base change isomorphism induced from the lefthand side square of (6.18) ) and using the decomposition into eigenbundles (6.9) we get Notice that given f : X → Y with X Noetherian, the higher direct images commute with direct sums namely  (6.27) Proof.This is (7.4) in [32] setting ) then (6.28) follows because under our conditions the natural base change morphism induced from diagram (6.14), is an isomorphism ∀ j ≥ 0. That is, from Corollary 2 pg.50 in [26] we see that the higher direct images (R j π S i X ) * ξ −1 (2Δ i ), j ≥ 0, are locally free sheaves and that for any y ∈ S i X the natural maps are isomorphisms (in fact, the only ones that are non-zero are (R 1 ) * ξ −1 (2Δ i ) and φ 1 y ).So the isomorphism in (6.29) follows as a particular case of Corollary 6.9.9.2 in [18] or from Theorem 2.1 in [11].Since (by equation (6.17)) we have for the ν j -eigenbundles Now by equation (6.28) the left-hand side of (6.30 ) by Lemma 6.1 (equation (6.20) taking n=-2, s=-1) .That is (6.31) so that using the Grothendieck-Riemann-Roch Theorem one can compute the Chern characters ch(ι ) so taking M = λ s,j,n we have deg M = deg λ s,j,n and in our case s = −1 and n = −2 so deg M = m j,−2 (m j,n as defined in the statement of Theorem 6.1) and so taking m = 2 Lemma 6.2 tell us that Now we have (again by equation (6.17)) that ), and as we mention in the proof of (6.29) On the other hand, using (6.7) one has the factorization (6.33)where r = −1 − m j,−2 − 2k + g Y .So (6.33) can be seen as the Chern class c(L ⊕(r −gY ) ⊕ (L ⊗ E)), where L is a line bundle with Chern class 1 + 2x and E is a rank g Y vector bundle with c(E) = e 4θ .Now we see, by properties of the Chern character and (6.32), that

proof of b):
Using the definition of i is a locally free sheaf on S i X.One can argue similarly as in the proof of a) to get the following base change isomorphism induced by the diagram (6.14) One can see that Consider the exact sequence of sheaves on Since F has finite fibres we get an exact sequence of sheaves on We continue by making the following observations.(1) Let (by the projection formula) is the non-zero term on the right-hand side of the exact sequence of (6.37).
(2) We have F = j F (ν j ) and then R 0 ) is the non-zero term on the right-hand side of the exact sequence of (6.38).( 5) From (6.38) we have the following identity in the Grothendieck ring similarly to how we arrive to (6.35) this is and consequently R s π S k Y * {F (ν j )} = 0.With ( 3) and ( 6) in (6.39) above we see that ), and using Lemma 6.1 the last is (6.40) Applying Lemma 6.2 to each character in (6.40) with M = λ 1,j,1 for the first one and M = λ 1,j,2 for the second one we have For the calculation of the Chern class we write where ), where L 1 and L 2 are line bundles with Chern classes 1 − x and 1 − 2x respectively and E 1 and E 2 are vector bundles with Chern characters r − θ and r + 4θ respectively.So, . From (6.4), one can assume that the non-zero Chern roots of E 1 and E 2 are −σ 1 , . . ., −σ gY and 4σ 1 , . . ., 4σ gY respectively, so that and using (6.7) the last is In case r, r ≥ g Y one argues similarly by choosing integers s, s so that r + s, r + s ≥ g Y , then adding e −x s + e 2x s to both sides of (6.42) one has which can be seen as ), where F 1 , F 2 have ranks r + s and r + s respectively.So ), and by (6.44) we can compute the right-hand side, namely and we also get

Stable characteristic classes
Theorem 7.1 below is a generalization of Theorem 2.3 in [24] where the case D = 0 is considered.The proof, which we have omitted here, can be done using similar arguments to those in the proof of Theorem 6.1 above.Theorem 7.1.Let Z D be a k-dimensional component of fixed points of h in S i X.Let n j and n j be the degrees of the line bundles λ 0,j,−1 and λ 0,j,0 in formula (6.9) respectively.Then ) In particular, we have where

The generalized Chern character for B i,m ,n
Let Z D be a component of fixed points of the automorphism h.Let E and F be h−linearized vector bundles on Z D .The generalized Chern character of E is given by where ch[E(ν j )] is the Chern character of the eigenbundle E(ν j ).As in the case of the usual Chern character one has that So, from equation (4.8) we have The factors on the right-hand side of (8.1) are given in the following result.
Theorem 8.1.Let Z D be a k-dimensional component of fixed points of h in S i X.Let g Y be the genus of the quotient curve Y .Let d k = i − pk be the degree of D. Let ν l and ν l be the eigenvalues corresponding to the action of h on the line-bundles ι * D L i and ∧ i ι * D W − i respectively.We have the following (c) Let m j,n and m j,n denote respectively the degrees of the bundles λ 1,j,n and λ −1,j,n in formula (6.9) .Then Proof.Parts a) and b) follow from (7.5) in [32] and the restriction rules ι * D θ = pθ and ι * D x = x, where we are using the same notation for the cohomology classes of the symmetric products S i X and S k Y ∼ = Z D as introduced in Sect.6 above (recall that for Thaddeus σ = θ and η = x).For (c), let E be a rank r E vector bundle on Z D and let One has (see proof of (7.6) in [32]) I. Moreno-Mejía and Silva-López JFPTA Let F be an h-linearized vector bundle on Z D and let Since S l F = p−1 j=0 (S l F )(ν j ), a Chern root γ of S l F is a Chern root of (S l F )(ν j ) for some j, say γ = s i=1 β i α i where β i ≥ 0, s i=1 β i = l and α i is a Chern root of F (ν ji ) for some integer j i .Then ν j e γ = m(ν j1 e α1 , . . ., ν js e αs ) where m is the degree l monomial m(x 1 , . . ., x s ) = s i=1 x βi i .So one has that Chern roots α of F (ν j ), j = 0, . . ., p − 1.
from which one sees that P (F (ν j ), ν j t).(8.9) we shall assume that F = ι * D U i and compute Q h (F, t) using (8.9).Using (4.6) and taking Chern class, we have that From equations (6.43) and (6.33), one has the following factorizations (8.12) where So from (8.10), (8.11) and (8.12) we have the Chern roots of F (ν j ) and we use them in (8.6) to compute Let h(z) := 1 1−te −z .Expanding the following around σ i = 0 and using σ 2 i = 0 one has, as the reader may check, that and Now we represent P (F (ν j ), t) as the product G 1 G 2 G 3 of 3 factors defined below which we will modify using (8.14), (8.15) and (8.16): using (6.7) one has that x) .Also notice that In a similar way, we compute and and replacing t by ν j t P (F (ν j ), Now we use (8.21) in (8.9) and recalling that ν = e 2iπ/p one can verify that Therefore, In particular, if l = q i − i, (q i as in equation (4.8)), one has

The of a hyperelliptic curve
Putting all data available to us so far in formula (1.2) the contribution of a component Z D of fixed points in S i X of an automorphism h of order p to the number N i (h) is given by the following (see details in Section 11): where The constants l, l , n j , n j , m j,2 , m j,1 , m p−j,−2 (for their definitions see Theorem 8.1, Theorem 7.1 and Theorem 6.1) appearing in formula (9.1) depend on the particular situation (the curve X, the automorphism h, the line bundle ξ) and in this section, we will compute them for the case where X is a hyperelliptic curve of genus g = g X , the automorphism h is the hyperelliptic involution and ξ = K 2 X (see Lemmas 9.1, 9.2 and 9.3 below).For the involution of a hyperelliptic curve, the contribution C i,ZD (h) to the Lefschetz number N i (h) does not depend on D but only on the dimension of Z D , the dimension of S i X and the genus g X of X.So we write for Z D a k−dimensional component.There are 2g X + 2 fixed points of h in the curve X and there are j=0 λ s,j,n .In Sect. 10 we shall use (9.3) to compute the Verlinde traces of a hyperelliptic curve of genus g X = 2 and to compute the contributions C i,k,gX (h) it remains to compute the constants l, l , n j , n j , m j,2 , m j,1 , m p−j,−2 .From their definitions in Theorem 6.1 and Theorem 7.1 one has: In order to compute the degrees of the line bundles λ s,j,n , consider the virtual representation then the virtual dimensions of its eigenspaces are given by and By Riemann-Roch Theorem we have: . Now h acts as multiplication by -1 on T ∨ X,p l , so det(Id − h|T ∨ X,p l ) = 2 .For fixed points p l , p j ∈ X h the action of h in the fibre O(p j ) p l is multiplication by (−1) δ j,l , where δ j,l is the Kronecker delta.To see this, one first notice that the action of h on the fibre O X (p l ) p l = O p l (p l ) is multiplication by −1 because O p l (p l ) can be identified with the normal bundle of the point embedding p l → X which in this case is T X ,p l since p l has dimension zero.If j = l then when one considers the stalks at p l of the following exact sequence of sheaves induced by the point embedding As we mentioned at the beginning of this section the divisors D are supported at deg D distinct points in X h then if p l is a point in the support of D the action of h on O X (D) p l is multiplication by −1 and therefore the action on O X (−nD) p l is multiplication by (−1) n .If p l is not a point in the support of D then one has that the action of h on O X (−nD) p l is multiplication by (1) n .Then when we apply Atiyah-Bott we get Next we use that where Y is the quotient curve X/ < h >= P 1 .We have that the Euler characteristics of the eigenbundles λ s,j,n are given by In particular, we have the following Lemma 9.1.Let h be the involution of a hyperelliptic curve of genus g X and let In the next two lemmas, we will use the composition ι D of equation (6.3).

Lemma 9.2. The action of h on ∧
Proof.Consider the decomposition into eigenbundles

and the action of the involution h on
)) (see the expansion of ch(E) in [19] pg.432).So from Theorem 6.1 part b) we have where l = O X (ι(p)) = O X (f * p), ι D and ι as defined in (6.3), in particular one has l(D) = O X (ι(p) + D).

A hyperelliptic curve of genus g X = 2
Let X be a hyperelliptic curve of genus g X = 2, let h be its hyperelliptic involution and take ξ = K 2 X .We have the embedding X To compute det(I − t • h | H 0 (X, K X ξ)) we compute the dimensions of the eigenspaces of h | H 0 (X, K X ξ) (this is similar to the calculation of (9.4) and (9.5) in the previous section) and we have dimH 0 (X, K X ξ)(ν 1 ) = 3g X − 2 = 4 and dimH 0 (X, K X ξ)(ν 0 ) = 2g X − 3 = 1.
So we have

Contribution simplification
Here, we derive formula (9.1).Using formula (1.2) and the data obtained in Sections 6-8, the contribution of a k-dimensional component Z D of (S i X) Now, one can remove θ from the for C i,ZD (h) by using the following formula (this is (7.2) from [32] ) It is not hard to verify that from which we obtained (9.1).

18 ) 6 . 1 .
Lemma Let Δ Y be the universal divisor of S k Y .Consider the line bundles ξ = ρ X * ξ on X × S k Y and L s,j,n = π Y * λ s,j,n on Y × S k Y .Then we have for l ≥ 0 and also recall that for Thaddeus σ = θ and η = x.Proof of Theorem 6.1.(a) We first notice that