$G$-fixed Hilbert schemes on $K3$ surfaces, modular forms, and eta products

Let $X$ be a complex $K3$ surface with an effective action of a group $G$ which preserves the holomorphic symplectic form. Let $$ Z_{X,G}(q) = \sum_{n=0}^{\infty} e\left(\operatorname{Hilb}^{n}(X)^{G} \right)\, q^{n-1} $$ be the generating function for the Euler characteristics of the Hilbert schemes of $G$-invariant length $n$ subschemes. We show that its reciprocal, $Z_{X,G}(q)^{-1}$ is the Fourier expansion of a modular cusp form of weight $\frac{1}{2} e(X/G)$ for the congruence subgroup $\Gamma_{0}(|G|)$. We give an explicit formula for $Z_{X,G}$ in terms of the Dedekind eta function for all 82 possible $(X,G)$. The key intermediate result we prove is of independent interest: it establishes an eta product identity for a certain shifted theta function of the root lattice of a simply laced root system. We extend our results to various refinements of the Euler characteristic, namely the Elliptic genus, the Chi-$y$ genus, and the motivic class.


Introduction
Let X be a complex K3 surface with an effective action of a group G which preserves the holomorphic symplectic form. Mukai showed that such G are precisely the subgroups of the Mathieu group M 23 ⊂ M 24 such that the induced action on the set {1, . . . , 24} has at least five orbits [Muk88]. Xiao classified all possible actions into 82 possible topological types of the quotient X/G [Xia96].
The G-fixed Hilbert scheme 1 of X parameterizes G-invariant length n subschemes Z ⊂ X. It can be identified with the G-fixed point locus in the Hilbert scheme of points: Hilb n (X) G ⊂ Hilb n (X).
We define the corresponding G-fixed partition function of X by Z X,G (q) = ∞ n=0 e Hilb n (X) G q n−1 where e(−) is the topological Euler characteristic.
Throughout this paper we set q = exp (2πiτ) so that we may regard Z X,G as a function of τ ∈ H where H is the upper half-plane.

The Main Results.
Our main result is the following: Theorem 1.1. The function Z X,G (q) −1 is a modular cusp form 2 of weight 1 2 e(X/G) for the congruence subgroup Γ 0 (|G|).
Our theorem specializes in the case where G is the trivial group to a famous result of Göttsche [Göt90]. The case where G is a cyclic group was proved in [BO18]. An analogous result for the case where X is an Abelian surface acted on symplectically by a finite group G has been recently proven by Pietromonaco [Pie20].
We give an explicit formula for Z X,G (q) in terms of the Dedekind eta function (1 − q n ) as follows. Let p 1 , . . . , p r be the singular points of X/G and let G 1 , . . . , G r be the corresponding stabilizer subgroups of G. The singular points are necessarily of ADE type: they are locally given by C 2 /G i where G i ⊂ SU(2). Finite subgroups of SU(2) have an ADE classification and we let ∆ 1 , . . . , ∆ r denote the corresponding ADE root systems. For any finite subgroup G ∆ ⊂ SU(2) with associated root system ∆ we define the local G ∆ -fixed partition function by e Hilb n (C 2 ) G ∆ q n− 1 24 .
The main geometric result we prove is the following.
Theorem 1.2. The local partition function for ∆ of type A n is given by and for type D n and E n by Z ∆ (q) = η 2 (2τ)η(4Eτ) η(τ)η(2Eτ)η(2Fτ)η (2V τ) where (E, F, V ) are given by: is the symmetry group of a polyhedral decomposition of S 2 P 1 into isomorphic regular spherical polygons. Then E, F, and V are the number of edges, faces, and vertices of the polyhedron. The key idea in proving the above theorem is to show that Hilb(C 2 ) G ∆ is deformation equivalent to Hilb(Y ) H where Y = Tot(K P 1 ) is the minimal resolution of C 2 /{±1} (see Section 3).
Using the work of Nakajima, we will also prove in Lemma 4.2 that is a shifted theta function for M ∆ , the root lattice of ∆. Here n is the rank of the root system, k = |G ∆ |, and ζ is dual to the longest root (see Section 4 and Equation (4.1) for details). Theorem 1.2 then yields an eta product identity for the theta function θ ∆ (τ) reminiscent of the MacDonald identities: Theorem 1.4. The shifted theta function θ ∆ (τ) defined above (cf. § 4 and Equation (4.1)) is given by an eta product as follows: The 82 possible collections of ADE root systems ∆ 1 , . . . , ∆ r associated to (X, G) a K3 surface with a symplectic G action, are given in Appendix A, Table 1. We let k = |G|, k i = |G i |, and The global series Z X,G (q) can be expressed as a product of local contributions (and thus via Theorem 1.2 as an explicit eta product) by our next result: Theorem 1.6. With the above notation we have Theorem 1.1 then immediately follows from Theorem 4.1 and Theorem 1.6 using the formulas for the weight and level of an eta product given in [Köh11, § 2.1].
In Appendix A, Table 1 we have listed explicitly the eta product of the modular form Z X,G (q) −1 for all 82 possible cases of (X, G).

Consequences of the Main Results.
Having obtained explicit eta product expressions for Z X,G (q) allows us to make several observational corollaries: Corollary 1.7. If G is a finite subgroup of an elliptic curve E, i.e. G is isomorphic to a product of one or two cyclic groups, then Z X,G (q) −1 is a Hecke eigenform. In Table 1  We remark that in these cases, we may form a Calabi-Yau threefold called a CHL model by taking the free group quotient Then the partition function Z X,G (q) gives the Donaldson-Thomas invariants of (X × E)/G in curve classes which have degree zero over X/G (cf. [BO18]).
Remark 1.8. Hecke eigenforms of weight 3 arise in the arithmetic of K3 surfaces: if X is a K3 surface defined over Q and has ρ(X) = rk NS(X) = 20, then there is a weight 3 Hecke eigenform f X (q) = ∞ n=1 a n q n such that for almost all primes p, a p is the trace of the p-th Frobenius morphism acting on H 2 (X)/N S(X).
There are four cases where Z X,G (q) −1 is a weight three Hecke eigenform and they correspond to the cases where G is Z 7 , Z 8 , Z 2 × Z 6 , or Z 4 × Z 4 (numbered 8, 14, 19, 25 on Table 1). If X admits a symplectic G action for one of these four groups, then we may take X to be defined over Q, have ρ(X) = 20, and then remarkably Indeed, in each of these cases, we may take X to be elliptically fibered over P 1 and have G as its group of sections (thus giving rise to the symplectic G action). Moreover, X is then the universal curve over the modular curve parameterizing (E, G), an elliptic curve E with a subgroup G ⊂ E. We thank Shuai Wang and Noam Elkies for noticing and elucidating this phenomenon.
For any eta product expression of a modular form, one may easily compute the order of vanishing (or pole) at any of the cusps [Köh11, Corollary 2.2]. Performing this computation on the 82 cases yields the following: Corollary 1.9. The modular form Z X,G (q) −1 always vanishes with order 1 at the cusps i∞ and 0. Moreover, Z X,G (q) −1 is holomorphic at all cusps except for the two cases with Xiao number 38 or 69, which have poles at the cusps 1/2 and 1/8 respectively. These are precisely the cases where X/G has two singularities of type E 6 .
Remark 1.10. The integers e Hilb n (X) G should have enumerative significance: they can be interpreted as virtual counts of G-invariant curves, whose quotient is rational, in a complete linear series of dimension n on X. This generalizes the famous Yau-Zaslow formula [YZ96] in the case where G is the trivial group. The precise nature between the virtual count and the actual count is expected to be subtle for the case of general G. This has been recently explored in [Zha19] and also in the case of G acting on an Abelian surface in [Pie20].

Refinements of the Euler Characteristic.
We can extend our results to various refinements of the Euler characteristic, namely the elliptic genus, the χ y genus, and the motivic class. These refinements all stem from the next result which we prove in Section 5. Let [Hilb n (X) G ] bir q n−1 be a formal series whose coefficients we regard as birational equivalence classes of projective hyperkahler manifolds. Such equivalence classes form a semi-ring under disjoint union and Cartesian product.
Theorem 1.11. Let Y be the minimal resolution of X/G, then where k = |G|, ∆(τ) = η(τ) 24 , and we have suppressed the trivial group from the notation in the series Z bir Y (q k ).
A famous theorem of Huybrechts [Huy99,Theorem 4.6] asserts that birational projective hyperkahler manifolds are deformation equivalent. Moreover, combining Huybrechts' theorem with [NS17, Proposition 3.21] it follows that birational projective hyperkahler manifolds are equal in K 0 (Var C ), the Grothendieck group of varieties.
Thus we may specialize the series Z bir X,G (q) to Elliptic genus, motivic class, and χ y genus since these are all well defined on birational equivalence classes of projective hyperkahler manifolds. These specializations are all well known for the series Z bir Y and hence we easily get the following corollaries.
A further specialization of particular interest is the (normalized) χ y genus. Let and we note that χ −y (M) = Ell q,y (M)| q=0 .
Then Z χ X,G (q, y) = y −1 (1 − y) 2 Z X,G (q) φ −2,1 (q k , y) where φ −2,1 is the unique weak Jacobi form of weight −2 and index 1. In particular, is a Jacobi form of index 1 and weight We note that for G cyclic, the series Z X,G (q)/φ −2,1 (q k , y) is the leading coefficient in the expansion of the Donaldson-Thomas partition function of (X × E)/G in the variable tracking the curve class in X (see [BO18, Theorem 0.1]).
We also get a formula for the motivic classes of the G-fixed Hilbert schemes: denotes the motivic class of the G-fixed Hilbert scheme. Then We refer the reader to [GZLMH04] for the meaning of [Y ] in the exponent and the formula for the motivic class of Hilb n (Y ). The above series has further specializations giving formulas for the Hodge polynomials and Poincare polynomials of the G-fixed Hilbert schemes.

Structure of paper.
In Section 2 we express the global partition function in terms of the local partition functions and deduce Theorem 1.6. In Section 3 we prove our main geometric result Theorem 1.2 which gives the eta product expression of the local partition functions. In Section 4 we express the local partition functions in terms of certain theta functions and thus prove our Theorem 1.4 which gives us the new theta function identities. In Section 5 we obtain the enhanced result of Theorem 1.11 on the partition function birational equivalence class of the G-fixed Hilbert schemes. Appendix 5 contains a proof of a root theoretic identity we need and Appendix A contains a table listing the modular form Z −1 X,G in all 82 topological types of symplectic actions on a K3 surface.

Acknowledgements.
The authors warmly thank Jenny Bryan, Noam Elkies, Federico Amadio Guidi, Georg Oberdieck, Ken Ono, Stephen Pietromonaco, Balázs Szendrői, Shuai Wang, and Alex Weekes for helpful comments and/or technical help. We would also like to thank the anonymous referee for helping us to fix and greatly simplify the proof of Proposition 3.1.

The global partition function
As in the introduction, let X be a K3 surface with a symplectic action of a finite group G. Recall that p 1 , . . . , p r ∈ X/G are the singular points of X/G with corresponding stabilizer subgroups G i ⊂ G of order k i and ADE type ∆ i . Let {x 1 i , . . . , x k/k i i } be the orbit of G in X corresponding to the point p i (recall that k = |G|). We may stratify Hilb(X) G according to the orbit types of subscheme as follows.
Let Z ⊂ X be a G-invariant subscheme of length nk whose support lies on free orbits. Then Z determines and is determined by a length n subscheme of On the other hand, suppose Z ⊂ X is a G-invariant subscheme of length nk k i supported on the orbit Then Z determines and is determined by the length n component of Z supported on a formal neighborhood of one of the points, say x 1 i . Choosing a G i -equivariant isomorphism of the formal neighborhood of x 1 i in X with the formal neighborhood of the origin in C 2 , we see that Z determines and is determined by a point in Hilb n 0 (C 2 ) G i , where Hilb n 0 (C 2 ) ⊂ Hilb n (C 2 ) is the punctual Hilbert scheme parameterizing subschemes supported on a formal neighborhood of the origin in C 2 .
By decomposing an arbitrary G-invariant subscheme into components of the above types, we obtain a stratification of Hilb(X) G into strata which are given by products of Hilb((X/G) o ) and Hilb 0 (C 2 ) G 1 , . . . , Hilb 0 (C 2 ) G r . Then using the fact that Euler characteristic is additive under stratifications and multiplicative under products, we arrive at the following equation of generating functions: As in the introduction, let a = e(X/G) − r = e ((X/G) o ). Then by Göttsche's formula [Göt90], We also note that e Hilb n 0 (C 2 ) G i = e Hilb n (C 2 ) G i since the natural C * action on both Hilb n 0 (C 2 ) G i and Hilb n (C 2 ) G i have the same fixed points. Thus we may write Multiplying Equation (2.1) by q −1 and substituting the above formulas, we find that >From the following Euler characteristic calculation, we see that the exponent of q in the above equation is zero: This completes the proof of Theorem 1.6.

The local partition function
Recall that the local partition function is defined by where G ∆ ⊂ SU(2) is the finite subgroup corresponding to the ADE root system ∆. In this section, we prove Theorem 1.2 which provides an explicit formula for Z ∆ (q) in terms of the Dedekind eta function. We regard this as the main geometric result of this paper.

Proof of Theorem 1.2 in the A n case.
We wish to prove The action of Z/(n + 1) on C 2 commutes with the action of C * × C * on C 2 and consequently, the Euler characteristics on the left hand side may be computed by counting the C * × C * -fixed subschemes, namely those given by monomial ideals. Such subschemes of length m have a well known bijection with integer partitions of m, whose generating function is given by the right hand side.

Proof of Theorem 1.2 in the D n and E n cases.
Our proof of Theorem 1.2 in the D n and E n cases uses a trick exploiting the fact that the Hilbert schemes of the stack X = [C 2 /{±1}] and the Hilbert schemes of the space Y = Tot(K P 1 ) can both be realized as moduli spaces of quiver representations of the A 1 Nakajima quiver variety.
Let G ⊂ SU(2) be a subgroup where the corresponding root system ∆ is of D or E type. Then {±1} ⊂ G and let H ⊂ SO(3) be the quotient The induced action of H on P 1 S 2 is by rotations. Indeed, H is the symmetry group of a regular polyhedral decomposition of S 2 which is given by the platonic solids in the E n case and the decomposition into two hemispherical (n − 2)-gons in the D n case. H is generated by rotations of order p, q, r, obtained by rotating about the center of an edge, a face, or a vertex respectively. The group H has the following presentation: We summarize this information below:  We then can compute: The following identity follows easily from the Jacobi triple product formula: Substituting this into the previous equation multiplied by q 1 8 we find e Hilb n (Y ) H q 2n .
We can now compute the summation factor in the above equation by the same method we used to compute the global series in Section 2. Here we use the fact that the singularities of Y /H are all of type A and we have already proven our formula for the local series in the A n case. Indeed, the quotient [Y /H] has three stacky points with stabilizers Z p , Z q , and Z r and the complement of those points (Y /H) o has Euler characteristic −1. Proceeding then by the same argument we used in Section 2 to get Equation (2.1), we obtain Substituting into the previous equation and cancelling the factors of q 1 6 , we have thus proved which completes the proof of Theorem 1.2 in the general case.

The local partition function as a theta function via Nakajima
The local partition functions Z ∆ (q) considered in this paper are obtained from a specialization of the partition functions of the stack [C 2 /G ∆ ]. Using the work of Nakajima [Nak02], the partition function of the Euler characteristics of the Hilbert scheme of points on the stack quotient [C 2 /G ∆ ] was computed explicitly in [GNS18] in terms of the root data of ∆. We use this to express Z ∆ (q) in terms of θ ∆ (τ), a shifted theta function for the root lattice of ∆. As a byproduct we obtain an eta product formula for the associated shifted theta function (Theorem 1.4).
A zero-dimensional substack Z ⊂ [C 2 /G ∆ ] may be regarded as a G ∆ invariant, zero-dimensional subscheme of C 2 . Consequently, we may identify the Hilbert scheme of points on the stack [C 2 /G ∆ ] with the G ∆ fixed locus of the Hilbert scheme of points on C 2 : This Hilbert scheme has components indexed by representations ρ of G ∆ as follows Let {ρ 0 , . . . , ρ n } be the irreducible representations of G ∆ where ρ 0 is the trivial representation. We note that n is also the rank of ∆. We define Recall that our local partition function Z ∆ (q) is defined by We then readily see that The following formula is given explicitly in [GNS18, Theorem 1.3], but its content is already present in the work of Nakajima [Nak02]: Theorem 4.1. Let C ∆ be the Cartan matrix of the root system ∆, then We note that under the specialization q i = q d i , where k = |G| is the order of the group G.
Let M ∆ be the root lattice of ∆ which we identify with Z n via the basis given by α 1 , . . . , α n , the simple positive roots of ∆. Under this identification, the standard Weyl invariant bilinear form is given by and d is identified with the longest root. We define We may then write and θ ∆ (τ) is the shifted theta function: where as throughout this paper we have identified q = exp (2πiτ).
In Appendix 5, we will prove the following formula which for ∆ = A n coincides with the "strange formula" of Freudenthal and de Vries [FdV69]: It follows that A = 0 and we obtain the following: Lemma 4.2. The local series Z ∆ (q) is given by

Proof of Theorem 1.11
Let Z = [X/G] be the quotient stack of X by G and let Y → X/G be the minimal resolution. The Hilbert scheme of zero dimensional substacks of Z is naturally identified with the G-fixed Hilbert scheme of X: We emphasize that Hilb(Z) is itself a scheme, not just a stack, as the objects it parameterizes (substacks V ⊂ Z) do not have automorphisms (see [OS03] or [BCY12, § 2.3]). Components of Hilb(Z) are indexed by the numerical K-theory class of O Z for Z ⊂ Z. The K-theory class of O Z can be written in a basis for K-theory as follows: where p ∈ Z is a generic point and p 1 , . . . , p r ∈ Z are the orbifold points. The local group of Z at p i is G ∆(i) ⊂ SU(2) and has corresponding root system ∆(i) of rank n(i), and has irreducible representations ρ 0 (i), ρ 1 (i), . . . , ρ n(i) (i) where ρ 0 (i) is the trivial representation. We note that we do not need to include [O p i ⊗ ρ 0 (i)] in our basis for K-theory because of the following relation in K-theory which holds for all i: where ρ reg (i) is the regular representation of G ∆(i) .
We abbreviate the data m j (i) appearing in the K-theory class above by the symbol m and we denote by To prove this we will first need the following lemma.

Lemma 5.2.
(1) Let ∆ be a rank n ADE root system, let M ∆ be the corresponding root lattice, and let G ∆ ⊂ SU(2) the corresponding finite subgroup. To any element m ∈ M ∆ there is a unique rigid with K-theory class where p ∈ [C 2 /G] is a generic point.
(2) For every datum m there is a unique rigid substack Z m ⊂ Z with K-theory class where p ∈ Z is a generic point.
Proof. Part (2) is implied by Part (1) since we can take the union of the rigid subschemes supported at the orbifold points p 1 , . . . , p r ∈ Z. So we need only prove the local case.
To prove Part (1) we need to show that component of Hilb([C 2 /G ∆ ]) corresponding to substacks with K-theory class is a single isolated point. This component corresponds to the coefficient of Q 1 2 (m|m) · q m 1 1 · · · q m n n in Theorem 4.1. It follows immediately from the formula in Theorem 4.1 that this coefficient is 1, and thus to prove this component is a single point, we need only prove that it has dimension 0. By Equation (5.1), we have and so the component in question is We define δ = (1, d 1 , . . . , d n ) and µ = (0, m 1 , . . . , m n ) so that our v of interest may be written v = 1 2 (m|m)δ + µ.
The first follows directly from our definitions, and the later two are well known properties of the vector δ.
Using the above we compute the dimension of the Hilbert scheme of interest: We thus can conclude that Hilb v 0 ρ 0 +···+v n ρ n ([C 2 /G ∆ ]) = M(v, w) is a single point which finishes the proof of the lemma. where (recalling that d j (i) = dim ρ j (i)), Then where ζ(i) ∈ M ∆(i) ⊗ Q is as in Section 4. Completing the square and using the formula which follows from Lemma A.1, we get .

It then follows that
(n(i) + 1) where we used Theorem 1.4, Theorem 1.6, and we set The previous equation which showed that A = 0 also shows that B = 24. Then since ∆(τ) = η(τ) 24 , we see that Theorem 1.11 follows.

Appendix A. Another Strange Formula
We recall the notation from Section 4. Let ∆ be an ADE root system of rank n. Let α 1 , . . . α n be a system of positive simple roots and let be the largest root. Let (·|·) be the Weyl invariant bilinear form with (α i |α i ) = 2 and let ζ be the dual vector to d in the sense that The identity of the following lemma coincides with Freudenthal and de Vries's "strange formula" when ∆ is A n .
Lemma A.1. Let k, n, and ζ be as above. Then, Proof. The case of ∆ = A n .-For any ADE root system we have (ρ|α) = 1 for all positive roots where ρ = 1 2 α∈R + α is half the sum of the positive roots. Since for A n , d i = 1, it follows from Equation (A.1) that ζ = ρ, and it follows from Equation (A.2) that k = n + 1 = h is the Coxeter number. The lemma is then (n + 1) 2 − 1 24 = (ρ|ρ) 2h .
The case of ∆ = E 6 , E 7 , E 8 .-These three individual cases are easily checked one by one.

Apppendix B. Table of eta products
The following