2.2.1 Hilbert schemes of points on orbifolds

Similar to [Reference Bayer, Craw and Zhang3, Section 5.1], we consider the Grothendieck group $K_c(\mathcal {X})$ of $D^b_c(\mathcal {X})$ , the compactly supported objects in the bounded derived category $D^b(\mathcal {X})$ of coherent sheaves on $\mathcal {X}$ . The numerical Grothendieck group $\mathrm {N}_c(\mathcal {X})$ is $K_c(\mathcal {X})$ modulo the kernel of the Euler pairing

$$ \begin{align*} \chi(-,-) : K(\mathrm{D}^b_{\mathrm{perf}}(\mathcal{X}))\times K_c(\mathcal{X}) \to \mathbb{Z} \end{align*} $$

on $K_c(\mathcal {X})$ . We will focus on $\mathrm {N}_0(\mathcal {X})$ , the subgroup generated by 0-dimensional sheaves. For a given class $\alpha \in \mathrm {N}_0(\mathcal {X})$ , [Reference Olsson and Starr32] define a Hilbert scheme

$$ \begin{align*} \mathrm{Hilb}^\alpha(\mathcal{X}) \end{align*} $$

of substacks $\mathcal {Z}\subset \mathcal {X}$ of class $[\mathcal {O}_{\mathcal {Z}}]=\alpha $ . Even though $\mathcal {X}$ is an orbifold, this turns out to be an algebraic space, as the substacks $\mathcal {Z}$ , which it parametrizes, do not have automorphisms. Although these Hilbert spaces are algebraic spaces and such substacks are more than just collections of points, we will refer to these $\mathrm {Hilb}^\alpha (\mathcal {X})$ as Hilbert schemes of points on $\mathcal {X}$ , writing .

Consider the subset of effective classes

Note that by definition, $\mathrm {Hilb}^0(\mathcal {X})$ is a point, and for $\alpha \neq 0$ the Hilbert scheme $\mathrm {Hilb}^\alpha (\mathcal {X})$ is only nonempty if $\alpha $ is effective.

Example 2.5. Given a finite abelian group G of order r acting on $\mathbb {C}^d$ as a diagonally embedded subgroup $G\subset \mathrm {SL}(d)$ , we consider the global quotient stack $\mathcal {X}=[\mathbb {C}^d/G]$ . Its numerical Grothendieck group of 0-dimensional sheaves is $\mathrm {N}_0(\mathcal {X})=\mathbb {Z}^{\oplus r}$ . We see this as follows.

Take the G-equivariant embedding of the origin into $\mathbb {C}^d$ and view it as a closed embedding

$$ \begin{align*} p:\mathrm{B}{G}\hookrightarrow \mathcal{X}. \end{align*} $$

As $K(\mathrm {B}{G})\cong \mathbb {Z}^{\oplus r}$ spanned by the irreducible representations $\rho _0,\dots ,\rho _{r-1}$ of G, it suffices to show that $p_*$ is an isomorphism. Here and throughout, we assume that the $\rho _i$ are labelled such that $\rho _0$ is the trivial character.

To show that $p_*$ is surjective, take the class $[\mathcal {F}]$ of a $0$ -dimensional coherent sheaf. Any sheaf can be deformed algebraically to be supported at the fixed point, so we may assume $\mathcal {F}$ is supported at the origin. We show it is in the image of $p_*$ by induction on the length of $\mathcal {F}$ . This is trivial for length $0$ . Consider the short exact sequence

$$ \begin{align*} 0\to \mathcal{F}' \to \mathcal{F} \to p_*p^*\mathcal{F} \to 0, \end{align*} $$

where $\mathcal {F}'$ is the kernel of the adjunction. By induction $[\mathcal {F}']$ is in the image of $p_*$ as a sheaf of lower length than $\mathcal {F}$ . Hence, $[\mathcal {F}]=[\mathcal {F}']+[p_*p^*\mathcal {F}]$ is also in the image of $p_*$ .

To show that $p_*$ is injective, it suffices to show that $p_*(\rho _i)\neq 0$ for all i. We take $q:\mathcal {X}\to \mathrm {B}{G}$ given by the G-equivariant morphism to the point. This yields the nonvanishing classes $q^*\rho _j$ . Using $p\circ q=\mathrm {id}$ and the projection formula, we get

$$ \begin{align*} \chi\left(q^*\rho_j,p_*\rho_i\right)=\delta_{ij}, \end{align*} $$

which shows that the class $p_*(\rho _i)$ doesn’t vanish for any i.

2.2.2 Moduli of 0-dimensional sheaves on orbifolds

We additionally consider moduli of 0-dimensional sheaves on orbifolds. For schemes, [Reference Huybrechts and Lehn21, Ex. 4.3.6] shows that the moduli spaces of 0-dimensional sheaves are exactly the symmetric products of the base scheme. Following this, we use moduli of 0-dimensional sheaves on orbifolds as our analogues of symmetric products. The existence of such a moduli space can be collected as follows from the literature.

Proposition 2.6. Let $\mathcal {X}$ be an orbifold. Then there exists an algebraic stack $\mathfrak {M}_0$ of zero-dimensional sheaves on $\mathcal {X}$ , which is locally of finite presentation and has affine diagonal. If $\mathcal {X}$ is equipped with a $\mathsf {T}$ -action, so is $\mathfrak {M}_0$ . Moreover, this stack has a good moduli space

$$ \begin{align*} \mathfrak{M}_0 \to M_0, \end{align*} $$

which also comes equipped with a $\mathsf {T}$ -action, compatible with the good moduli space maps.

Given a class $\alpha \in \mathrm {N}_0(\mathcal {X})$ , we consider the subspace

$$ \begin{align*} M_\alpha(\mathcal{X})\subset M_0(\mathcal{X}) \end{align*} $$

of sheaves on $\mathcal {X}$ of class $\alpha $ . This comes with a Hilbert-Chow morphism

$$ \begin{align*} \mathrm{HC}_\alpha :\mathrm{Hilb}^\alpha(\mathcal{X}) \to M_\alpha(\mathcal{X}),\ [\mathcal{Z}]\mapsto [\mathcal{O}_{\mathcal{Z}}], \end{align*} $$

induced by the morphism of fine moduli stacks $\mathrm {Hilb}(\mathcal {X})\to \mathfrak {M}_0(\mathcal {X}),\ [\mathcal {Z}]\mapsto [\mathcal {O}_{\mathcal {Z}}]$ , composed by the good moduli space morphism of Proposition 2.6. This process preserves the numerical class $\alpha \in \mathrm {N}_0(\mathcal {X})$ .

2.2.3 Coarse spaces

Let $\pi :\mathcal {X}\to X$ be a coarse moduli space of $\mathcal {X}$ . We assume X is connected. Pushforward of sheaves maps $\mathrm {N}_0(\mathcal {X})$ into $\mathrm {N}_0(X)$ . For a connected X, we have $\mathrm {N}_0(X)=\mathbb {Z}$ . We write

$$ \begin{align*} b(\alpha) := \pi_*(\alpha)\in\mathbb{Z} \end{align*} $$

for any $\alpha \in \mathrm {N}_0(\mathcal {X})$ . In fact, $\pi _*$ admits a section $\mathrm {N}_0(X)=\mathbb {Z}\to \mathrm {N}_0(\mathcal {X})$ by taking a point p in the nonstacky locus of $\mathcal {X}$ and mapping $1\in \mathbb {Z}$ to $[p]$ . Hence, we get a splitting

$$ \begin{align*} \mathrm{N}_0(\mathcal{X})\cong \mathbb{Z}\oplus\widetilde{\mathrm{N}}_0(\mathcal{X}), \end{align*} $$

where the projection to the first component is just $b=\pi _*$ . Pushforward of sheaves induces a morphism

$$ \begin{align*} \xi_\alpha:M_\alpha(\mathcal{X}) \to \mathrm{Sym}^{b(\alpha)}(X),\ [\mathcal{F}]\mapsto [\pi_*\mathcal{F}], \end{align*} $$

where we use the identification of moduli of 0-dimensional sheaves with symmetric products from [Reference Huybrechts and Lehn21, Ex. 4.3.6].

2.3.1 Obstruction theory

Definition 2.8. Let M be a Deligne-Mumford stack. An obstruction theory is an object $\mathbb {E}\in \mathrm {D}^-_{\mathrm {QCoh}}(M)$ together with a morphisms

$$ \begin{align*} \phi: \mathbb{E} \to \tau_{\geq -1}\mathbb{L}_{M}, \end{align*} $$

such that $h^0(\phi )$ is an isomorphism and $h^{-1}(\phi )$ is surjective. An obstruction theory is

• • perfect if it is a perfect complex of amplitude $[-1,0]$ ,

• • symmetric if $\mathbb {E}$ is a perfect complex and there is an isomorphism $\Theta : \mathbb {E} \xrightarrow {\sim } \mathbb {E}^\vee [1]$ (or $\kappa $ -symmetric if $\Theta : \mathbb {E} \xrightarrow {\sim } \kappa \otimes \mathbb {E}^\vee [1]$ for some weight $\kappa $ in the equivariant case) satisfying $\Theta ^\vee [1]=\Theta $ .

The dual of a perfect obstruction theory is referred to as the virtual tangent sheaf $\mathbb {T}^{\mathrm {vir}}$ .

Identifying the Hilbert schemes of points $\mathrm {Hilb}^\alpha (\mathcal {X})$ with a moduli space of ideal sheaves of class $[\mathcal {O}_{\mathcal {X}}]-\alpha $ in $\mathcal {X}$ , [Reference Gholampour and Tseng17] gives us a perfect obstruction theory

$$ \begin{align*} \mathbb{E} = \pi_{\mathrm{Hilb},*}\left(\mathcal{H} om(\mathcal{I},\mathcal{I})_0 \right)[2]\to \tau_{\geq -1}\mathbb{L}_{\mathrm{Hilb}^\alpha(\mathcal{X})}, \end{align*} $$

where $\mathcal {I}$ is the universal ideal sheaf and $\mathcal {H} om(\mathcal {I},\mathcal {I})_0$ denotes the traceless part. For a CY3 orbifold $\mathcal {X}$ , possibly equipped with a $\mathsf {T}$ -action, this perfect obstruction theory is ( $\kappa $ -)symmetric by Grothendieck-Verdier duality.

In the case of a global quotient stack $\mathcal {X}=[W/G]$ , where G is a finite group, we can describe the obstruction theory for $\mathrm {Hilb}(\mathcal {X})$ using the one on $\mathrm {Hilb}(W)$ as follows. First note that ideal sheaves on $\mathcal {X}$ are exactly ideal sheaves on W equipped with a G-equivariant structure. These correspond exactly to G-invariant subschemes of W. Hence, we find that $\mathrm {Hilb}(\mathcal {X})$ is the G-fixed part of $\mathrm {Hilb}(W)$

$$ \begin{align*} \mathrm{Hilb}(\mathcal{X})=\mathrm{Hilb}(W)^G. \end{align*} $$

The G-action on W equips $\mathrm {Hilb}(W)$ with a G-action and the obstruction theory $\mathbb {E}_W$ above naturally comes with a G-equivariant structure [Reference Ricolfi34]. We can naturally identify the obstruction theories

(5) $$ \begin{align} \mathbb{E}_{\mathcal{X}}\cong \left.{\mathbb{E}_{W}}\right\vert_{\mathrm{Hilb}(\mathcal{X})}^f, \end{align} $$

where $\left.{\mathbb {E}_{W}}\right\vert _{\mathrm {Hilb}(\mathcal {X})}^f$ is the G-fixed part of the restriction of $\mathbb {E}_{W}$ to $\mathrm {Hilb}(\mathcal {X})$ .

2.3.2 Virtual structure sheaf

Given a perfect obstruction theory $\mathbb {E}$ on a moduli space as above, together with a presentation as a 2-term complex of vector bundles $\mathbb {E} \cong [E^{-1}\to E^0]$ , [Reference Behrend and Fantechi5] construct a virtual fundamental class and a virtual structure sheaf. This turns out to be independent of the specific chosen presentation $E^\bullet $ . We are interested in the virtual structure sheaf. Consider the embedding of the intrinsic normal cone $\mathfrak {C}\hookrightarrow h^1/h^0(E^{\bullet \vee })$ given by the perfect obstruction theory. Then by existence of the resolution $E^\bullet $ , we get an induced cone $C\subset E_1$ . The virtual structure sheaf is

$$ \begin{align*} \mathcal{O}^{\mathrm{vir}} = \bigoplus_{i} \mathcal{T} or ^{i}_{E_{1}}\left(\mathcal{O}_{C},\mathcal{O}_{M}\right)[i]\in \mathrm{D}^{b}(M), \end{align*} $$

where $\mathcal {O}_M$ becomes a sheaf on $E_1$ via the $0$ -section. Note that [Reference Behrend and Fantechi5, Remark 5.4] define this as a graded commutative sheaf of algebras, whereas we define it as an element in the derived category, which is reflected in our notation as shifts, which are implicit in their notation.

2.3.3 Twisted virtual structure sheaf

For computations, it is useful to consider a twisted virtual structure sheaf, that is the virtual structure sheaf tensored by a square root of the determinant of the obstruction theory, so that the K-theory class of the virtual structure sheaf is

$$ \begin{align*} \left[\widehat{\mathcal{O}}^{\mathrm{vir}}\right] = \left[\mathcal{O}^{\mathrm{vir}}\right] \otimes \pm\det(\mathbb{E})^{1/2}\in K(\mathrm{Hilb}^\alpha(\mathcal{X})). \end{align*} $$

This choice and technique were introduced originally by Okounkov in [Reference Okounkov31] and allows the use of the so-called rigidity principle, which we will recall and use in Section 5. In [Reference Okounkov31], a quiver-theoretic description of the moduli space is used to introduce the K-theory class of the desired twist.

However, to apply the factorization machinery that we develop in Section 3, we require the twisted virtual structure sheaf to be given as an explicit complex of sheaves, rather than just a K-theory class. So, following the discussion in [Reference Levine24], we make the following definition of the twisted virtual structure sheaf of Hilbert schemes of points on orbifolds as a complex. This explicitly enters the proof of the factorization property in Proposition 4.2. The shift makes the resulting sheaves factorizable. Otherwise, the diagrams which are required to commute in the definition of a factorizable system might only commute up to a sign.

Definition 2.9. Let $\mathcal {X}$ be a CY3 orbifold, and let $\mathrm {Hilb}(\mathcal {X})$ be its Hilbert scheme of points with universal family $\mathcal {Z}$ . We define the twisted virtual structure sheaf as

$$ \begin{align*} \widehat{\mathcal{O}}^{\mathrm{vir}} = \mathcal{O}^{\mathrm{vir}} \otimes \det\left(p_{\mathcal{Z} *}(\mathcal{O}_{\mathcal{Z}})\right)^{-1}[b(\alpha)] \in \mathrm{CoCh}_{\mathbb{Z}/2}(\mathrm{Coh}(\mathrm{Hilb}(\mathcal{X}))) \end{align*} $$

in the projective case, and

$$ \begin{align*} \widehat{\mathcal{O}}^{\mathrm{vir}} = \mathcal{O}^{\mathrm{vir}} \otimes \det\left(\kappa^{\frac{1}{2}}p_{\mathcal{Z} *}(\mathcal{O}_{\mathcal{Z}})\right)^{-1}[b(\alpha)]\in \mathrm{CoCh}_{\mathbb{Z}/2}(\mathrm{Coh}_{\mathsf{T}}(\mathrm{Hilb}(\mathcal{X}))) \end{align*} $$

in the toric case, where $\kappa $ is the $\mathsf {T}$ -weight of the canonical bundle of $\mathcal {X}$ .

To make the above definition well-defined and for the proof of Lemma 2.12 below, we first need the following relatively standard result.

Lemma 2.11. Let $\mathcal {X}$ be a CY3 orbifold, and let $\mathrm {Hilb}(\mathcal {X})$ be its Hilbert scheme of points with universal family $\mathcal {Z}$ . Take $p_{\mathcal {Z}}$ to be the morphism $\mathcal {Z}\hookrightarrow \mathrm {Hilb}(\mathcal {X})\times \mathcal {X}\to \mathrm {Hilb}(\mathcal {X})$ . Then

$$ \begin{align*} p_{\mathcal{Z} *}(\mathcal{O}_{\mathcal{Z}}) \end{align*} $$

is a locally free sheaf of rank $b(\alpha )$ on the component $\mathrm {Hilb}^\alpha (\mathcal {X})$ .

In the (equivariant) projective case [Reference Levine24] shows that the twist $\det \left(p_{\mathcal {Z} *}(\mathcal {O}_{\mathcal {Z}})\right)^{-1}$ in Definition 2.9 above is a square root of the determinant of the obstruction theory. Note that the computations of determinants (but not necessarily the rank computations) in their proof still work for projective DM stacks using [Reference Nironi28, Cor. 2.10] instead of Grothendieck-Serre duality for proper morphisms of schemes. In the case of toric Calabi–Yau orbifolds, we show that the twist defined above is a square root of $\det (\mathbb {E})$ at $\mathsf {T}$ -fixed points of the Hilbert scheme of points. As $\mathsf {T}$ -equivariant Euler characteristics are defined via localization, this suffices for computational purposes.

Lemma 2.12. Let $\mathcal {X}$ be a toric CY3 orbifold. Consider the Hilbert scheme of points $\mathrm {Hilb} (\mathcal {X})$ with its universal family $\mathcal {Z}$ . Let $\mathbb {E}$ be the $\mathsf {T}$ -equivariant obstruction theory on $\mathrm {Hilb}(\mathcal {X})$ . For any closed point p in the $\mathsf {T}$ -fixed locus $\mathrm {Hilb} (\mathcal {X})$ , there is a $\mathsf {T}$ -equivariant identification of K-theory classes

$$ \begin{align*} \det([\mathbb{E}|_p]) = \bigg[{\kappa^{-\operatorname{\mathrm{rk}}(p_{\mathcal{Z} *}(\mathcal{O}_{\mathcal{Z}}))}\det\left(p_{\mathcal{Z} *}(\mathcal{O}_{\mathcal{Z}})\right)^{-2}}\bigg\vert_{p}\bigg] = \bigg[{\det\left(\kappa^{\frac{1}{2}}p_{\mathcal{Z} *}(\mathcal{O}_{\mathcal{Z}})\right)^{-2}}\bigg\vert_{p}\bigg]. \end{align*} $$

Proof. The proof is similar to the one of [Reference Levine24, Theorem 5.2], except that we work at a fixed point $p=[Z]$ in $\mathrm {Hilb} (\mathcal {X})$ directly making some computations easier in the toric case. From now on we consider only K-theory classes and all computations are in K-theory, omitting $[-]$ from the notation. Using the push-pull formula, we get

$$ \begin{align*} \mathbb{E}|_p &= R\mathrm{Hom} (I_Z,I_Z)_0[2],\\ p_{\mathcal{Z} *}(\mathcal{O}_{\mathcal{Z}})|_{p} &= R\Gamma (\mathcal{O}_Z). \end{align*} $$

We write $\chi (-,-)$ for $R\mathrm {Hom} (-,-)$ and $\chi (-)$ for $R\Gamma (-)$ . Then

$$ \begin{align*} R\mathrm{Hom} (I_Z,I_Z)_0=\chi(I_Z,I_Z)-\chi(\mathcal{O},\mathcal{O}) = \chi(\mathcal{O}_Z,\mathcal{O}_Z)-\chi(\mathcal{O},\mathcal{O}_Z)-\chi(\mathcal{O}_Z,\mathcal{O}). \end{align*} $$

Using that Z is of codimension $\geq 2$ , we get $\det (\chi (\mathcal {O}_Z,\mathcal {O}_Z))=\mathcal {O}$ . It remains to check the equivariant weight of this trivial line bundle. This is a local computation at the fixed point, so we may assume $\mathcal {X}=[\mathbb {C}^3/G]$ by Lemma 2.2, and Z corresponds to some colored partition $\pi $ . We can write the structure sheaf of the fixed point Z as $\mathcal {O}_Z=\sum _{\mathbf {n}\in \pi }\mathbf {t}^{\mathbf {n}} \mathcal {O}_0(\rho _{C(\mathbf {n})})$ , where $\rho _{C(\mathbf {n})}$ is the character determined by the coloring of $\pi $ . Tensoring the standard $\mathsf {T}$ -equivariant resolution of $\mathcal {O}_0$ with an irreducible representation $\rho _{j}$ , we compute the character

$$ \begin{align*} \chi_{\mathsf{T}\times G}(\mathcal{O}_0(\rho_i),\mathcal{O}_0(\rho_j)) = \rho_i^{-1}\rho_j\chi_{\mathsf{T}}(\mathcal{O}_0,\mathcal{O}_0) = \rho_i^{-1}\rho_j\prod_{k=1}^3 (1-t_k). \end{align*} $$

Together with the presentation of $\mathcal {O}_Z$ , this gives us

$$ \begin{align*} \chi_{\mathsf{T}\times G}(\mathcal{O}_Z,\mathcal{O}_Z) = \chi_{\mathsf{T}\times G}(\mathcal{O}_Z)\chi_{\mathsf{T}\times G}(\mathcal{O}_Z)^\vee\prod_{k=1}^3 (1-t_k), \end{align*} $$

so that $\det \left(\chi _{\mathsf {T}\times G}(\mathcal {O}_Z,\mathcal {O}_Z)\right)=1$ . We have $\chi (\mathcal {O},\mathcal {O}_Z)=\chi (\mathcal {O}_Z)$ and $\chi (\mathcal {O}_Z,\mathcal {O})=\chi (\mathcal {O}_Z^\vee ) = -\kappa ^{-1}\chi (\mathcal {O}_Z)^\vee $ by equivariant Serre duality for compactly supported sheaves. Putting together these identifications to compute determinants, we get the desired $\mathsf {T}$ -equivariant identification of K-theory classes

$$ \begin{align*} \det(\mathbb{E}|_p) &= \det(\kappa^{-1}\chi(\mathcal{O}_Z)^\vee-\chi(\mathcal{O}_Z))\\ &= \kappa^{-\operatorname{\mathrm{rk}}(\chi(\mathcal{O}_Z))}\det(\chi(\mathcal{O}_Z))^{-2}\\ &={\kappa^{-\operatorname{\mathrm{rk}}(p_{\mathcal{Z} *}(\mathcal{O}_{\mathcal{Z}}))}\det\left(p_{\mathcal{Z} *}(\mathcal{O}_{\mathcal{Z}})\right)^{-2}}\bigg\vert_{p}\\ &={\det\left(\kappa^{\frac{1}{2}}p_{\mathcal{Z} *}(\mathcal{O}_{\mathcal{Z}})\right)^{-2}}\bigg\vert_{p}.\\[-42pt] \end{align*} $$

2.4.1 Generating series

With the obstruction theory and twisted virtual structure sheaves above, we are interested in computing the generating series of degree $0$ DT invariants

(6) $$ \begin{align} \mathsf{Z}(\mathcal{X}) := 1+ \sum_{\alpha \in \mathrm{C}_0(\mathcal{X})} q^{\alpha} \chi\left(\mathrm{Hilb}^\alpha(\mathcal{X}), \widehat{\mathcal{O}}^{\mathrm{vir}}\right)\in K_{\mathsf{T}}(\mathrm{pt})[\mathrm{C}_0(\mathcal{X})], \end{align} $$

where $q^{\alpha }$ is the standard notation for the semigroup ring element associated to $\alpha \in \mathrm {C}_0(\mathcal {X})$ . Here we restricted from $\mathrm {N}_0(\mathcal {X})$ to $\mathrm {C}_0(\mathcal {X})$ , because all classes $\alpha $ , where $\chi \left(\mathrm {Hilb}^\alpha (\mathcal {X}), \widehat {\mathcal {O}}^{\mathrm {vir}}\right)$ does not vanish in particular has $\mathrm {Hilb}^\alpha (\mathcal {X})\neq \emptyset $ , so that $\alpha $ is effective. This definition works as stated only for proper $\mathcal {X}$ . For a quasi-projective $\mathcal {X}$ with an action of a group $\mathsf {T}$ , so that the fixed loci of $\mathrm {Hilb}^\alpha (\mathcal {X})$ are proper, we take the equivariant Euler characteristic $\chi _{\mathsf {T}}$ , which is defined via localization.

Example 2.13. Continuing Example 2.5 of global quotient orbifolds with $d=3$ , the sum above simplifies as $\mathrm {N}_0([\mathbb {C}^3/G])=\mathbb {Z}^r$ and $\mathrm {C}_0([\mathbb {C}^3/G])=\mathbb {N}^r\setminus \{0\}$ .

$$ \begin{align*} \mathsf{Z}(\mathcal{X}) := 1+ \sum_{\mathbf{n}\in\mathbb{N}^r\setminus \{0\}} q_0^{n_0}\cdots q_{r-1}^{n_{r-1}} \chi\left(\mathrm{Hilb}^{\mathbf{n}}([\mathbb{C}^3/G]), \widehat{\mathcal{O}}^{\mathrm{vir}}\right). \end{align*} $$

2.4.2 Localization

In the case of a projective orbifold $\mathcal {X}$ , the moduli spaces under consideration will also be projective, making $\chi $ well-defined. If $\mathcal {X}$ is not necessarily projective, but is equipped with a $\mathsf {T}$ -action, the moduli spaces have an induced $\mathsf {T}$ -action, and we define $\chi $ via localization as follows. For a noetherian separated scheme M with a $\mathsf {T}$ -action, the localization theorem [Reference Thomason37, Theorem 2.2] tells us that

(7) $$ \begin{align} i_*:K_{\mathsf{T}}\left(M^{\mathsf{T}}\right)_{\mathrm{loc}} \to K_{\mathsf{T}}\left(M\right)_{\mathrm{loc}} \end{align} $$

is an isomorphism, where $i:M^{\mathsf {T}}\hookrightarrow M$ denotes the embedding of the fixed locus, and $K_{\mathsf {T}}(-)_{\mathrm {loc}}$ denotes equivariant K-theory with the equivariant parameters inverted. If the fixed locus $M^{\mathsf {T}}$ is proper, we define for a K-theory class F on M

For a $\mathsf {T}$ -equivariant morphism $\pi :M\to N$ , consider the commutative diagram

By commutativity, we can check $i^N_* \pi ^{\mathsf {T}}_* (i^M_*)^{-1}=\pi _*$ , which implies that $\chi $ defined by localization is still invariant under pushforward if $M^{\mathsf {T}}$ and $N^{\mathsf {T}}$ are proper.

If M is smooth, then we can find an explicit inverse for $i_*$ . This is often also simply referred to as localization. To define this inverse, consider the normal bundle $\mathcal {N}_{M^{\mathsf {T}}/M}$ of $M^{\mathsf {T}}$ in M. Then

$$ \begin{align*} i_*^{-1} = \frac{i^*(-)}{\mathsf{e}(\mathcal{N}_{M^{\mathsf{T}}/M})}, \end{align*} $$

where $\mathsf {e}(-)$ is the K-theoretic Euler class defined by setting for a locally free sheaf $\mathcal {E}$ of nontrivial $\mathsf {T}$ -weight and extending by for two such sheaves. Here and in Section 2.4.3 below we require the Euler class of a locally free sheaf $\mathcal {E}$ of nontrivial $\mathsf {T}$ -weight to be invertible in $K_{\mathsf {T}}(M^{\mathsf {T}})_{\mathrm {loc}}$ .

In the course of this paper, we will evaluate Euler characteristics by localization on coarse spaces X of toric orbifolds $\mathcal {X}$ , symmetric products thereof, components of moduli spaces of zero-dimensional sheaves on $\mathcal {X}$ , and Hilbert schemes of points on $\mathcal {X}$ . For a toric orbifold $\mathcal {X}$ , its coarse space X is a toric variety and hence has isolated fixed points. Consequently, any symmetric product of X also has isolated fixed points. The Hilbert scheme of points on $\mathcal {X}$ also has isolated fixed points [Reference Bryan, Cadman and Young9, Lemma 13]. We describe these in the local case more detail in Section 2.4.4 below. Any Euler characteristic on $M_0(\mathcal {X})$ , which we evaluate will be of a class pushed forward via the Hilbert-Chow morphism making the Euler characteristic well-defined by further pushforward to $\mathrm {Sym}^{b(\alpha )}(X)$ .

2.4.3 Virtual localization

Another case, in which an explicit inverse to $i_*$ is known is virtual localization [Reference Graber and Pandharipande18, Reference Ciocan-Fontanine and Kapranov12]. Given a moduli space M with a $\mathsf {T}$ -action as above, which is additionally equipped with a $\mathsf {T}$ -equivariant perfect obstruction theory $\mathbb {E}$ , we can restrict the perfect obstruction theory to the fixed locus. It splits into a $\mathsf {T}$ -fixed and a $\mathsf {T}$ -moving part

$$ \begin{align*} \mathbb{E}|_{M^{\mathsf{T}}} = \mathbb{E}|_{M^{\mathsf{T}}}^f \oplus \mathbb{E}|_{M^{\mathsf{T}}}^m. \end{align*} $$

Then $\mathbb {E}|_{M^{\mathsf {T}}}^f$ turns out to be an obstruction theory on $M^{\mathsf {T}}$ and is the virtual normal bundle. The virtual localization formula is then

$$ \begin{align*} \widehat{\mathcal{O}}^{\mathrm{vir}}_M = i_*\left(\frac{\widehat{\mathcal{O}}^{\mathrm{vir}}_{M^{\mathsf{T}}}}{\hat{\mathsf{e}}(\mathcal{N}^{\mathrm{vir}})}\right), \end{align*} $$

where is the symmetrized K-theoretic Euler class.

2.4.4 Colored vertex

We consider specifically the case $\mathcal {X}=[\mathbb {C}^3/G]$ , where G is a finite abelian diagonally embedded subgroup of $\mathrm {SL}(3)$ , following the discussions in [Reference Young39, Reference Cao, Kool and Monavari11]. In this case the action of G on $\mathbb {C}^3$ decomposes as a direct sum of three irreducible representations $\rho _x, \rho _y, \rho _z$ , so that it defines a G -coloring

$$ \begin{align*} K : \left(\mathbb{Z}_{\geq 0}\right)^3 \to G^*,\ (i,j,k) \mapsto \rho_x^{\otimes i}\otimes \rho_y^{\otimes j} \otimes \rho_z^{\otimes k}, \end{align*} $$

where $G^*$ is the group formed by the set of irreducible G-representations. We have $G^*\cong G$ as G is finite abelian.

We consider $\mathcal {X}$ equivariantly with the natural action by $\mathsf {T}=(\mathbb {C}^*)^3$ . In this case, the $\mathsf {T}$ -fixed locus of $\mathrm {Hilb}(\mathcal {X})$ consists of isolated fixed points. These are in particular fixed points of $\mathrm {Hilb}(\mathbb {C}^3)$ corresponding to plane partitions, but additionally the boxes of these plane partitions are colored by the irreducible representations of G. Here we say for an irreducible G-representation $\rho \in G^*$ that a box $(i,j,k)$ in a plane partition $\pi $ is $\rho $ -colored if $K(i,j,k)=\rho $ . The number of $\rho $ -colored boxes of a plane partition by is denoted by

$$ \begin{align*} \left| \pi_\rho \right| = \left| \{(i,j,k)\in \pi\ |\ K(i,j,k)=\rho\} \right|. \end{align*} $$

Comparing with Example 2.5, the color vector $(\left| \pi _\rho \right|)_{\rho \in G^*}$ corresponds to the K-theory class of the structure sheaf of the corresponding substack. Throughout, when choosing a basis $\left\{\rho _i\right\}$ for $G^*$ , we always label so that $\rho _0$ is the trivial character. Hence, we also write $\pi _0$ as $\pi ^G$ , which is precisely $b(\alpha )$ of the corresponding K-theory class $\alpha $ . With a chosen basis $\left\{\rho _i\right\}$ for $G^*$ , comparing to Example 2.13, we have $n_i=\left| \pi _i \right|$ .

We study the contribution to the generating series $\mathsf {Z}(\mathcal {X})$ at each such fixed point.

Proposition 2.14. Let $\mathcal {X}=[\mathbb {C}^3/G]$ with G is a finite abelian diagonally embedded subgroup of $\mathrm {SL}(3)$ . For each colored plane partition $\pi $ corresponding to a fixed point of the Hilbert scheme of points $\mathrm {Hilb}(\mathcal {X})$ , there is a subset $W(\pi )$ of the $\mathsf {T}$ -weights of $\left.{\mathbb {E}_{\mathcal {X}}^\vee }\right\vert _{[I_\pi ]}$ , such that we get the local contributions

$$ \begin{align*} \left[\widehat{\mathcal{O}}^{\mathrm{vir}}\right] &= \sum_{\pi} (-1)^{\left| \pi_0 \right|} \left[\mathcal{O}_{[I_\pi]}\right]\hat{a}(\pi)\in K_{\mathsf{T}}(\mathrm{Hilb}(\mathcal{X}))_{\mathrm{loc}},\\ \hat{a}(\pi) &= \prod_{w\in W(\pi)} \frac{(\kappa/w)^{1/2}-(w/\kappa)^{1/2}}{w^{1/2}-w^{-1/2}} = \prod_{w\in W(\pi)} \frac{[\kappa/w]}{[w]}, \end{align*} $$

where $\kappa =t_1t_2t_3$ and we write .

Proof. If $\pi $ is a colored plane partition corresponding to a fixed point, we let $\mathcal {O}_{[I_\pi ]}$ be the skyscraper sheaf at the corresponding fixed point $[I_\pi ]$ . Then the virtual localization formula yields

$$ \begin{align*} \left[\widehat{\mathcal{O}}^{\mathrm{vir}}\right] = \sum_{\pi} (-1)^{\left| \pi_0 \right|}\left[\mathcal{O}_{[I_\pi]}\right]\frac{1}{\hat{\mathsf{e}}\left(\mathbb{E}_{\mathcal{X}}^\vee|_{[I_\pi]}^m\right)}\in K_{\mathsf{T}}(\mathrm{Hilb}(\mathcal{X}))_{\mathrm{loc}}, \end{align*} $$

where the sign comes from our choice of shift in Definition 2.9 of the twist of the virtual structure sheaf. The local contribution $\frac {1}{\hat {\mathsf {e}}\left(\left.{\mathbb {E}_{\mathcal {X}}^\vee }\right\vert _{[I_\pi ]}^m\right)}$ can be understood as follows. First, write $\left.{\mathbb {E}_{\mathcal {X}}^\vee }\right\vert _{[I_\pi ]}$ as a sum of its $\mathsf {T}$ -weights as a class in $K_{\mathsf {T}}(\mathrm {pt})$ . Then, as $\mathcal {X}$ is Calabi–Yau, the obstruction theory $\mathbb {E}_{\mathcal {X}}$ is $\kappa $ -symmetric, which means there is a subset $W(\pi )$ of the $\mathsf {T}$ -weights such that

$$ \begin{align*} {\mathbb{E}_{\mathcal{X}}^\vee}\bigg|_{[I_\pi]} = \sum_{w\in W(\pi)} \bigg(w-\frac{\kappa}{w}\bigg). \end{align*} $$

We have seen in Section 2.3.1 that the obstruction theory on $\mathrm {Hilb}(\mathcal {X})$ is the G-fixed part of the obstruction theory on $\mathrm {Hilb}(\mathbb {C}^3)$ . Then [Reference Maulik, Nekrasov, Okounkov and Pandharipande25, Section 4.5] shows that $\mathbb {E}_{\mathbb {C}^3}^\vee $ has no trivial $\mathsf {T}$ -weights at any fixed point, so $\left.{\mathbb {E}_{\mathcal {X}}^\vee }\right\vert _{[I_\pi ]}$ does not have any trivial $\mathsf {T}$ -weights either. Inserting the above weight decomposition into the localization formula gives us

$$ \begin{align*} \big[\widehat{\mathcal{O}}^{\mathrm{vir}}\big] = \sum_{\pi} \big[\mathcal{O}_{[I_\pi]}\big]\det\big({\mathbb{E}^\vee_{\mathcal{X}}}\big\vert_{[I_\pi]}\big)^{-1/2}\prod_{w\in W(\pi)}\big(\frac{1-w/\kappa}{1-w^{-1}}\big)\in K_{\mathsf{T}}(\mathrm{Hilb}(\mathcal{X}))_{\mathrm{loc}}. \end{align*} $$

The twist at $[I_\pi ]$ can be written in terms of weights as

$$ \begin{align*} \det\bigg({\mathbb{E}^\vee_{\mathcal{X}}}\bigg\vert_{[I_\pi]}\bigg)^{-1/2} = \Bigg(\prod_{w\in W(\pi)}\frac{w^2}{\kappa}\Bigg)^{-1/2} = \prod_{w\in W(\pi)}\frac{(\kappa/w)^{1/2}}{w^{1/2}}. \end{align*} $$

Multiplying by this twist yields exactly the desired formula.

3.1.1 Setup

We develop a theory of factorization for orbifolds. This works equivariantly with respect to a group action on the underlying orbifold, for example, the action of a torus. In the equivariant setting, all Euler characteristics in this section, for example, the ones in Corollary 3.25, are defined by localization as explained in Section 2.4.2.Footnote ¹ All morphisms are equivariant and the equivariant structure of the sheaves is preserved by all constructions of this section, so that the proofs in the equivariant setting are the same. So, for clarity, we suppress equivariance from the notation.

Following [Reference Okounkov31, 5.3], factorization for schemes X is a property of systems of sheaves $\mathcal {F}_n$ on $\mathrm {Sym}^n(X)$ , which allows us to compute $1+\sum _{n\geq 1}\chi (\mathrm {Sym}^n(X),\mathcal {F}_n)q^n$ as the plethystic exponential of a series $\sum _{n\geq 1}\chi (X,\mathcal {G}_n)q^n$ where $\mathcal {G}_i$ is a sequence of sheaves on X. This simplifies the problem as X is easier to work with than $\mathrm {Sym}^n(X)$ . Moreover, in the equivariant case, $\chi (X,\mathcal {G}_n)$ can be computed as the product of a fixed localization contribution with a Laurent polynomial in the equivariant parameters.

We will define a notion of a factorizable system in a more general setup, see Definition 3.5. To motivate this, we consider the following morphisms

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Recall from Section 2.2.3 that $b(\alpha )$ is the pushforward of $\alpha $ to X. The vertical map is the quotient morphism. In the cases we are interested in, we will show that the twisted virtual structure sheaves yield a factorizable system of sheaves $\hat {\mathcal {O}}^{\mathrm {vir}}$ on $\mathrm {Hilb}^{\alpha }(\mathcal {X})$ . By then pushing forward, we obtain a factorizable system on $\mathrm {Sym}^{b(\alpha )}(X)$ . Finally pulling back to the stack $\left[X^{b(\alpha )}/S_{b(\alpha )}\right]$ , we obtain a factorizable system of $S_{b(\alpha )}$ -equivariant sheaves on $X^{b(\alpha )}$ . This third factorizable system will be used to prove the main result of this section, Theorem 3.22 about generating series of K-theory classes of factorizable systems of sheaves.

Throughout this section $\mathcal {X}$ denotes an orbifold in the sense of Definition 2.1 with coarse moduli space X. Before we formulate our notion of factorization, we define the notion of a factorization index set.

The condition for a factorization index set roughly says that an I-indecomposable class can only have one underlying point in the coarse space, possibly together with some stacky contribution. We will use this in the proof of Theorem 3.22 below, to guarantee the existence of splittings of classes $\alpha $ in I whenever $b(\alpha )>1$ . The reader should have the following example in mind, which is the example we use in all applications.

Lemma 3.3. For an orbifold $\mathcal {X}$ , set

$$ \begin{align*} I := \{\alpha\in \mathrm{C}_0(\mathcal{X})\ |\ \mathrm{Hilb}^{\alpha}(\mathcal{X})\neq \emptyset\}. \end{align*} $$

This is a factorization index set. If $\mathcal {X}=\mathcal {X}'\times \mathbb {C}$ , then I is closed under addition.

Proof. The first condition of a factorization index set holds immediately, because, by definition, all classes $\alpha \in I$ can be written as $\alpha =[\mathcal {O}_{\mathcal {Z}}]$ , where $\mathcal {Z}$ is some substack. Take a class $\alpha =[\mathcal {O}_{\mathcal {Z}}]$ in I, which is I-indecomposable. Any morphism $f:\mathcal {O}_{\mathcal {X}}\to \mathcal {O}_{\mathcal {Z}}$ gives us a splitting

$$ \begin{align*} \alpha= [\mathcal{O}_{\mathcal{Z}}] = [\mathrm{im}(f)] + [\mathrm{cok}(f)]. \end{align*} $$

Both $[\mathrm {im}(f)]$ and $[\mathrm {cok}(f)]$ are in I as $\mathrm {im}(f)$ and $\mathrm {cok}(f)$ are sheaves with surjective morphisms from $\mathcal {O}_{\mathcal {X}}$ . By I-indecomposability of $\alpha $ , $\mathrm {im}(f)$ or $\mathrm {cok}(f)$ must vanish. Hence, every nonzero morphism $f\in \mathrm {Hom}_{\mathcal {X}}\left(\mathcal {O}_{\mathcal {X}},\mathcal {O}_{\mathcal {Z}}\right)$ must be surjective. By the adjunction of $i:\mathcal {Z}\hookrightarrow \mathcal {X}$ , which preserves surjections, this means that every nonzero morphism $f\in \mathrm {Hom}_{\mathcal {Z}}\left(\mathcal {O}_{\mathcal {Z}},\mathcal {O}_{\mathcal {Z}}\right)$ must be surjective. Using finite length of $\mathcal {Z}$ and additivity of the length function on the short exact sequence

$$ \begin{align*} 0\to \ker(f) \to \mathcal{O}_{\mathcal{Z}} \xrightarrow{f} \mathcal{O}_{\mathcal{Z}} \to \mathrm{cok}(f) \to 0 \end{align*} $$

associated to f, we see that f is surjective if and only if it is injective. So, every nonzero morphism $f\in \mathrm {Hom}_{\mathcal {Z}}\left(\mathcal {O}_{\mathcal {Z}},\mathcal {O}_{\mathcal {Z}}\right)$ must be an isomorphism, which gives us $\mathrm {Hom}_{\mathcal {Z}}\left(\mathcal {O}_{\mathcal {Z}},\mathcal {O}_{\mathcal {Z}}\right)=\mathbb {C}$ as in [Reference Huybrechts and Lehn21, Cor. 1.2.8]. Therefore,

$$ \begin{align*} b(\alpha)=h^0(\mathcal{O}_{\mathcal{Z}})=\dim \mathrm{Hom}_{\mathcal{Z}}\left(\mathcal{O}_{\mathcal{Z}},\mathcal{O}_{\mathcal{Z}}\right) =1. \end{align*} $$

To prove the second part, take $\alpha _1=[\mathcal {O}_{\mathcal {Z}_1}]$ and $\alpha _2=[\mathcal {O}_{\mathcal {Z}_2}]$ in I. As $\mathcal {X}$ is $\mathcal {X}'\times \mathbb {C}$ , we may move $\mathcal {Z}_1$ and $\mathcal {Z}_2$ apart along the trivial direction without changing the K-theory classes $\alpha _i$ , so that we may assume the $\mathcal {Z}_i$ are disjoint. Then $\left[\mathcal {O}_{\mathcal {Z}_1\sqcup \mathcal {Z}_2}\right]$ has class $\alpha _1+\alpha _2$ and is contained in I.

Figure 1

Possible splittings of the class $(2,1,1)$ in terms of decompositions of colored plane partitions. A splitting into invalid colored plane partitions is not a splitting in I.

3.1.2 General factorizable systems

Now we give a general definition of a factorizable system. In practice, we only use this definition in the various ways explained in Section 3.1.3. Factorizable systems in the below definitions consist of (possibly 2-periodic) complexes of coherent sheaves, that is, elements of $\mathrm {CoCh}(\mathrm {Coh}(-))$ or $\mathrm {CoCh}_{\mathbb {Z}/2}(\mathrm {Coh}(-))$ .

Given complexes of sheaves $\mathcal {F}_1,\dots ,\mathcal {F}_k$ on any scheme, an operation $\odot \in \left\{\otimes ,\oplus \right\}$ , and a permutation $\sigma \in S_k$ , we denote by

$$ \begin{align*} S: \mathcal{F}_1\odot\cdots\odot\mathcal{F}_k \xrightarrow{\sim}\mathcal{F}_{\sigma(1)}\odot\cdots\odot\mathcal{F}_{\sigma(k)} \end{align*} $$

the standard isomorphism.

Definition 3.5. For $\mathcal {X}$ an orbifold, consider the following data:

• • Let I be a factorization index set.

• • Let $\mathbf {e}:I\to E$ be an additive morphism to a subset E of a monoid, in the sense that, for any $\alpha _1+\alpha _2=\alpha $ in I, $\mathbf {e}(\alpha _1)+\mathbf {e}(\alpha _2)$ equals $\mathbf {e}(\alpha )$ and is in particular in E.

• • For any $e\in E$ we have an algebraic space $M_e(\mathcal {X})$ with an action of a finite group $G_e$ .

• • For any $e_1+e_2=e$ in E, we have an embedding $G_{e_1}\times G_{e_2}\subseteq G_{e}$ .

• • For any $e_1+e_2=e$ in E, we have $G_{e_1}\times G_{e_2}$ -equivariant open subschemes $U_{e_1e_2}\subseteq M_{e_1}(\mathcal {X})\times M_{e_2}(\mathcal {X})$ . Taking $\sum _{i=1}^r e_i=e$ in E, set $U_{e_1\dots e_r}\subseteq \prod _i M_{e_i}(\mathcal {X})$ to be the intersection of the pullbacks of all the $U_{e_ie_j}$ . Then we assume that there are $G_{e_1}\times \cdots \times G_{e_r}$ -equivariant flat morphisms $U_{e_1\dots e_r}\to M_{e}(\mathcal {X})$ and for any permutation $\tau $ natural isomorphisms $\tau :U_{e_1\dots e_r}\xrightarrow {\sim }U_{e_{\tau _1}\dots e_{\tau _r}}$ making the diagram

  (9)

  
  commute.

• • Fix $\boxdot $ to be either $\boxtimes $ or $\boxplus $ .

With this data, a collection $\{\mathcal {F}_{\alpha }\}_{\alpha \in I}$ of (possibly 2-periodic) complexes of sheaves on $M_{\mathbf {e}(\alpha )}(\mathcal {X})$ is called factorizable with respect to this data if there exists a collection of morphisms of $G_{\mathbf {e}(\alpha _1)}\times G_{\mathbf {e}(\alpha _2)}$ -equivariant complexes of sheaves for $\alpha _1+\alpha _2=\alpha $ in I

$$ \begin{align*} \phi_{\alpha_1\alpha_2}:{\big(\mathcal{F}_{\alpha_1}\boxdot \mathcal{F}_{\alpha_2}\big)}\big\vert_{U_{\mathbf{e}(\alpha_1)\mathbf{e}(\alpha_2)}}\rightarrow {\mathcal{F}_{\alpha}}\big\vert_{U_{\mathbf{e}(\alpha_1)\mathbf{e}(\alpha_2)}}, \end{align*} $$

such that for any $\alpha \in I$ and $e_1,e_2\in E$ with $e_1+e_2=\mathbf {e}(\alpha )$ , the morphism

(10) $$ \begin{align} \bigoplus_{\substack{\alpha_1+\alpha_2=\alpha\\\mathbf{e}(\alpha_i)=e_i}} \phi_{\alpha_1\alpha_2}: \bigoplus_{\substack{\alpha_1+\alpha_2=\alpha\\\mathbf{e}(\alpha_i)=e_i}} {\big(\mathcal{F}_{\alpha_1}\boxdot \mathcal{F}_{\alpha_2}\big)}\big\vert_{U_{e_1e_2}}\rightarrow {\mathcal{F}_{\alpha}}\big\vert_{U_{e_1e_2}} \end{align} $$

is an isomorphism of $G_{e_1}\times G_{e_2}$ -equivariant complexes of sheaves. Moreover, we require the $\phi _{\alpha _1\alpha _2}$ to satisfy the following requirements.

• • Associativity: Given $\alpha _1,\alpha _2,\alpha _3\in I$ , such that all sums of them are in I, we have

  $$ \begin{align*} \phi_{\alpha_1+\alpha_2,\alpha_3} \circ \left(\phi_{\alpha_1\alpha_2}\boxdot \mathrm{id}_{\mathcal{F}_{\alpha_3}}\right) = \phi_{\alpha_1,\alpha_2+\alpha_3} \circ \left(\mathrm{id}_{\mathcal{F}_{\alpha_1}}\boxdot \phi_{\alpha_2\alpha_3}\right) \end{align*} $$
  as morphisms
  $$ \begin{align*} \big(\mathcal{F}_{\alpha_1}\boxdot \mathcal{F}_{\alpha_2}\boxdot \mathcal{F}_{\alpha_3}\big)\big\vert_{U_{\mathbf{e}(\alpha_1)\mathbf{e}(\alpha_2)\mathbf{e}(\alpha_3)}} \rightarrow \mathcal{F}_{\alpha_1+\alpha_2+\alpha_3}\big\vert_{U_{\mathbf{e}(\alpha_1)\mathbf{e}(\alpha_2)\mathbf{e}(\alpha_3)}}. \end{align*} $$

• • Commutativity: Because the diagram (9) commutes, we get an isomorphism $v:\tau ^*\left(\left.{(-)}\right\vert _{U_{e_2e_1}}\right)\cong \left.{(-)}\right\vert _{U_{e_1e_2}}$ . We require that the diagram

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  commutes, where the first vertical arrow arises by composition of the natural isomorphisms S and v. Note here that, due to our associativity condition above, we only need to impose commutativity in this case with two “factors”.

3.1.3 Examples

We explain the various examples of Definition 3.5, which we use in this paper. For applications, we consider only the factorization index set given by Lemma 3.3, but for the following examples, the specific choice of factorization index set is unimportant.

Note that virtual structure sheaves, as defined in Section 2.3.2 are elements of $\mathrm {D}^b_{\mathbb {Z}/2}(\mathrm {Coh}(-))$ of the form $\bigoplus _i \mathcal {H}^i(\mathcal {E})[i]$ for some $\mathcal {E}\in \mathrm {D}^b_{\mathbb {Z}/2}(\mathrm {Coh}(-))$ . For the purpose of factorization, we can view them as genuine complexes $\bigoplus _i \mathcal {H}^i(\mathcal {E})[i]$ with trivial differential and all factorization morphisms from Section 4.1 are well-defined morphisms of complexes.

Remark 3.6. More generally, we have a functor

$$ \begin{align*} \mathrm{D}^b_{\mathbb{Z}/2}(\mathrm{Coh}(-)) \ni \mathcal{E} \mapsto \bigoplus_i \mathcal{H}^i(\mathcal{E})[i] \in \mathrm{CoCh}_{\mathbb{Z}/2}(\mathrm{Coh}(-)), \end{align*} $$

which preserves the class of $\mathcal {E}$ in $K(\mathrm {Coh}(-))$ . We can then define factorizable systems of objects in $\mathrm {D}^b_{\mathbb {Z}/2}(\mathrm {Coh}(-))$ in an analogous way and use the above functor to prove the main result for such systems.

Example 3.7. We consider factorization for moduli $M_\alpha (\mathcal {X})$ of $0$ -dimensional sheaves from Section 2.2.2. Here, we take $\mathbf {e}=\mathrm {id}$ , $G_\alpha $ the trivial group, and the operation $\boxdot =\boxtimes $ . The open subsets $U_{\alpha _1\dots \alpha _r}$ in $M_{\alpha _1}(\mathcal {X})\times \cdots \times M_{\alpha _r}(\mathcal {X})$ contain the points $\left(\mathcal {F}_1,\dots ,\mathcal {F}_r\right)$ , where the $\mathcal {F}_i$ have pairwise disjoint support. The flat morphisms are induced by the natural direct sum morphism

$$ \begin{align*} \prod_{i=1}^{k}M_{\alpha_i}(\mathcal{X}) \to M_{\sum_i\alpha_i}(\mathcal{X}). \end{align*} $$

Example 3.8. To examine factorization properties of obstruction theories and virtual structure sheaves, we consider factorization for Hilbert schemes of points $\mathrm {Hilb}^\alpha (\mathcal {X})$ . We take $\mathbf {e}=\mathrm {id}$ , $G_\alpha $ the trivial group, and either operation $\boxdot \in \{\boxplus ,\boxtimes \}$ . The open subsets and flat morphisms are defined by fiber product, as shown in the commutative diagram

with cartesian squares.

Passing to the coarse space along $\pi :\mathcal {X}\to X$ , we need to take $\mathbf {e}=b$ from Section 2.2.3.

3.1.4 Pushforward & pullback compatibility

We establish that, under some hypotheses, factorizable systems are preserved under pushforward and pullback.

Lemma 3.11. Let $\mathcal {X}$ be an orbifold and let I be a factorization index set and $\boxdot \in \{\boxplus ,\boxtimes \}$ . Let $\mathbf {e}:I\to E$ , $M_e(\mathcal {X})$ , $G_e$ , $U_{e_1e_2}$ and $\mathbf {e}':I\to E'$ , $N_{e'}(\mathcal {X})$ , $G^{\prime }_{e'}$ , $U_{e^{\prime }_1e^{\prime }_2}$ be two collections of factorization data. Let $f:E\to E'$ be a surjective additive morphism such that $f\circ \mathbf {e}=\mathbf {e}'$ . Let $f^G_e:G_e\to G^{\prime }_{f(e)}$ be a group morphism and $f^M_e:M_e(\mathcal {X})\to N_{f(e)}(\mathcal {X})$ be a morphism that is equivariant with respect to $f^G_e$ . Assume for every sum $e_1+e_2=e$ in E mapping to $e^{\prime }_i=f(e_i), e'=f(e)=e^{\prime }_1+e^{\prime }_2$ in $E'$ under f, the open subschemes and flat morphisms are given by fiber product as follows

(12)

where $\overline {U}_{e^{\prime }_1e^{\prime }_2}$ is defined as the fiber product $U_{e^{\prime }_1e^{\prime }_2}\times _{N_{e'}(\mathcal {X})}M_{e}(\mathcal {X})$ , $U_{e_1 e_2}$ is assumed to be $\left(M_{e_1}(\mathcal {X})\times M_{e_2}(\mathcal {X})\right)\times _{N_{e^{\prime }_1}(\mathcal {X})\times N_{e^{\prime }_2}(\mathcal {X})}U_{e^{\prime }_1e^{\prime }_2}$ , $i_{e_1 e_2}$ is an embedding of connected components, and all $i_{e_1 e_2}$ jointly cover $\overline {U}_{e^{\prime }_1e^{\prime }_2}$ . Then

1. (a) Take $\boxdot =\boxtimes $ . If $\left\{\mathcal {F}_\alpha \right\}_{\alpha \in I}$ is a $G_{\mathbf {e}(\alpha )}$ -equivariant factorizable system of sheaves on $M_{\mathbf {e}(\alpha )}(\mathcal {X})$ , and all $\left(f^M_{\mathbf {e}(\alpha )}\right)_*\mathcal {F}_\alpha $ are complexes of coherent sheaves, then $\left\{\left(f^M_{\mathbf {e}(\alpha )}\right)_*\mathcal {F}_\alpha \right\}_{\alpha \in I}$ is a $G^{\prime }_{f(\mathbf {e}(\alpha ))}$ -equivariant factorizable system of sheaves on $N_{f(\mathbf {e}(\alpha ))}(\mathcal {X})$ .

2. (b) Take $f=\mathrm {id}$ , so that $\mathbf {e}=\mathbf {e}'$ and $i_{e_1e_2}=\mathrm {id}$ by the condition on $i_{e_1e_2}$ above. If $\left\{\mathcal {F}_\alpha \right\}_{\alpha \in I}$ is a $G^{\prime }_{\mathbf {e}(\alpha )}$ -equivariant factorizable system of sheaves on $N_{\mathbf {e}(\alpha )}(X)$ , then $\left\{\left(f^M_{\mathbf {e}(\alpha )}\right)^*\mathcal {F}_\alpha \right\}_{\alpha \in I}$ is a $G_{\mathbf {e}(\alpha )}$ -equivariant factorizable system of sheaves on $M_{\mathbf {e}(\alpha )}(\mathcal {X})$ .

Proof. The proofs of both statements are analogous, so here, we only show the more complicated case of pushing forward with $\boxdot =\boxtimes $ . For the operation $\boxtimes $ , we have $\left(f^M_{e_1}\times f^M_{e_2}\right)_*$ commutes with $\boxtimes $ as follows

(13) $$ \begin{align} \left(f^M_{e_1}\times f^M_{e_2}\right)_*\left(-\boxtimes -\right) \cong \left(f^M_{e_1}\right)_*(-)\boxtimes\left(f^M_{e_2}\right)_*(-), \end{align} $$

by using the pullbacks in the definition of $\boxtimes $ .

As pushforward preserves isomorphisms and commutative diagrams, the only statement to check is how pushing forward along $f^M_{\mathbf {e}(\alpha )}$ interacts with the direct sum splitting in (10). Consider the base change $f^U_{e_1e_2}$ of the morphism $f^M_{e_1}\times f^M_{e_2}$ . We push the factorization morphisms $\phi _{\alpha _1\alpha _2}$ forward along $f^U_{\mathbf {e}(\alpha _1)\mathbf {e}(\alpha _2)}$ . Take a given $\alpha $ and $e^{\prime }_1+e^{\prime }_2=e'$ in $E'$ with $e'=\mathbf {e}'(\alpha )$ . Take $e_i$ such that $f(e_i)=e^{\prime }_i$ . By assumption, on the components $U_{e_1e_2}$ , we have the isomorphisms

$$ \begin{align*} \bigoplus_{\substack{\alpha_1+\alpha_2=\alpha\\\mathbf{e}(\alpha_i)=e_i}} \phi_{\alpha_1\alpha_2}: \bigoplus_{\substack{\alpha_1+\alpha_2=\alpha\\\mathbf{e}(\alpha_i)=e_i}} {\big(\mathcal{F}_{\alpha_1}\boxtimes \mathcal{F}_{\alpha_2}\big)}\big\vert_{U_{e_1e_2}} \rightarrow {\mathcal{F}_{\alpha}}\big\vert_{U_{e_1e_2}}. \end{align*} $$

Now, by assumption, $\overline {U}_{e^{\prime }_1e^{\prime }_2}$ consists of various components $U_{e_1 e_2}$ , one for each splitting $e_1+e_2=e$ with $f(e_i)=e^{\prime }_i$ . So, the above isomorphisms on these components glue to give isomorphisms

$$ \begin{align*} \bigoplus_{\substack{e_1+e_2=e\\ f(e_i)=e^{\prime}_i}}\bigoplus_{\substack{\alpha_1+\alpha_2=\alpha\\\mathbf{e}(\alpha_i)=e_i}} i_{e_1 e_2 *}\phi_{\alpha_1 \alpha_2}: \bigoplus_{\substack{e_1+e_2=e\\ f(e_i)=e^{\prime}_i}}\bigoplus_{\substack{\alpha_1+\alpha_2=\alpha\\\mathbf{e}(\alpha_i)=e_i}} i_{e_1 e_2 *}\bigg({\big(\mathcal{F}_{\alpha_1}\boxtimes \mathcal{F}_{\alpha_2}\big)}\bigg\vert_{U_{e_1 e_2}}\bigg) \rightarrow {\mathcal{F}_{\alpha}}\big\vert_{\overline{U}_{e^{\prime}_1e^{\prime}_2}}. \end{align*} $$

Reorder the double direct sum to a direct sum over $\alpha _i$ with $\mathbf {e}(\alpha _i)=e_i$ , and use the uniquely determined $e_i=\mathbf {e}(\alpha _i)$ . We also push forward along $f^{\overline {U}}_{e^{\prime }_1e^{\prime }_2}$ to obtain isomorphisms

$$ \begin{align*} \bigoplus_{\substack{\alpha_1+\alpha_2=\alpha\\\mathbf{e}'(\alpha_i)=e^{\prime}_i}} \bigg(f^{U}_{\mathbf{e}(\alpha_1)\mathbf{e}(\alpha_2)}\bigg)_*\phi_{\alpha_1 \alpha_2} \end{align*} $$

between

$$ \begin{align*} \bigoplus_{\substack{\alpha_1+\alpha_2=\alpha\\\mathbf{e}'(\alpha_i)=e^{\prime}_i}}\bigg( \bigg(f^M_{\mathbf{e}(\alpha_1)}\times f^M_{\mathbf{e}(\alpha_2)}\bigg)_*\bigg(\mathcal{F}_{\alpha_1}\boxtimes \mathcal{F}_{\alpha_2}\bigg)\bigg)\bigg\vert_{U_{e^{\prime}_1 e^{\prime}_2}} \rightarrow {\bigg(\bigg(f^M_{\mathbf{e}(\alpha)}\bigg)_*\mathcal{F}_{\alpha}\bigg)}\bigg\vert_{U_{e^{\prime}_1e^{\prime}_2}}, \end{align*} $$

where we use the push-pull formula along the cartesian squares above. Using the canonical isomorphism (13) on the left-hand side yields the desired factorization isomorphisms.

We use this Lemma for pushing forward and pulling back along the morphisms in the diagram (8) as follows.

Example 3.12. For an orbifold $\mathcal {X}$ with coarse moduli space $\pi :\mathcal {X}\to X$ , we first consider the morphism

A quasi-projective orbifold (in our sense) $\mathcal {X}$ admits an open embedding into a projective finite type DM-stack $\overline {\mathcal {X}}$ by [Reference Kresch, Abramovich, Bertram, Katzarkov, Pandharipande and Thaddeus23, Sec. 5], which in particular has projective coarse moduli space $\overline {X}$ . As coarse moduli are Zariski-local, we get a cartesian diagram

which in turn yields the cartesian diagram

By [Reference Olsson and Starr32, Theorem 1.5], the Hilbert scheme of points $\mathrm {Hilb}\left(\overline {\mathcal {X}}\right)$ is projective, making $\eta $ proper.

Hence, $\eta _\alpha $ preserves coherence and $\mathrm {Hilb}^\alpha (\mathcal {X})$ and $\mathrm {Sym}^{b(\alpha )}(X)$ come with factorization data from Examples 3.8 and 3.9 respectively.

Taking $f=b$ , we argue that the conditions of Lemma 3.11 are satisfied, so that we may push forward factorizable systems from $\mathrm {Hilb}^\alpha (\mathcal {X})$ to $\mathrm {Sym}^{b(\alpha )}(X)$ . The only thing to check are the conditions on the open subschemes in diagram (12). By Example 3.8 the open subschemes for the Hilbert scheme are defined via pullback from the ones for moduli of $0$ -dimensional sheaves in Example 3.7, so we may consider these.

Recall the fixed isomorphism $\mathrm {N}_0(\mathcal {X})\cong \mathbb {Z}\oplus \widetilde {\mathrm {N}}_0(\mathcal {X})$ from Section 2.2.3. Write $\alpha =(b(\alpha ),\tilde {\alpha })$ for classes on $\mathcal {X}$ , and consider factorizable systems on the moduli $M_{(b,\tilde {\alpha })}(\mathcal {X})$ as in Examples 3.7. For given $\alpha \in I$ and $b_1,b_2$ , the loci $U_{(b_1,\tilde {\alpha }_1),(b_2,\tilde {\alpha }_2)}$ in $M_{(b_1,\tilde {\alpha }_1)}(\mathcal {X})\times M_{(b_2,\tilde {\alpha }_2)}(\mathcal {X})$ are exactly the subschemes of two $0$ -dimensional sheaves with disjoint support of classes $(b_i,\tilde {\alpha }_i)$ . The coarse moduli space $\pi :\mathcal {X} \to X$ is bijective on geometric points, so the supports of two $0$ -dimensional sheaves on $\mathcal {X}$ are disjoint if and only if the supports of their pushforwards to X are disjoint. Hence, $U_{(b_1,\tilde {\alpha }_1),(b_2,\tilde {\alpha }_2)}$ are exactly the fiber product in the left cartesian square in (12). Analogously, the fiber product in right cartesian square in (12) is the filled in by the disjoint union of the $U_{(b_1,\tilde {\alpha }_1),(b_2,\tilde {\alpha }_2)}$ , varying $\tilde {\alpha }_1$ and $\tilde {\alpha }_2$ , as required.

3.1.5 Extension to more general morphisms

We note here, that the above theory of factorization extends to the following setting. Take a morphism

$$ \begin{align*} f:\mathcal{Y} \to X, \end{align*} $$

from an orbifold $\mathcal {Y}$ to a connected scheme X. Then $\mathrm {N}_0(X)\cong \mathbb {Z}$ . Following the notation of Section 2.2.1, take

Denote the effective classes in $\mathrm {N}_{\mathrm {exc}}(\mathcal {Y})$ by $\mathrm {C}_{\mathrm {exc}}(\mathcal {Y})$ . We assume there is a splitting of $f_*:\mathrm {N}_{\mathrm {exc}}(\mathcal {Y})\to \mathrm {N}_0(X)$ , such that

$$ \begin{align*} \mathrm{N}_{\mathrm{exc}}(\mathcal{Y}) \cong \mathbb{Z}\oplus \widetilde{\mathrm{N}}_{\mathrm{exc}}(\mathcal{Y}), \end{align*} $$

where the first projection is just $f_*$ and is denoted by $b(-)$ . Then we can analogously define a factorization index set to be a subset $I\subseteq \mathrm {C}_{\mathrm {exc}}(\mathcal {Y})$ satisfying the same two conditions of Definition 3.1. Similarly, we can define factorizable systems on $\mathrm {Sym}^{b(\alpha )}(X)$ and on $\left[X^{b(\alpha )}/S_{b(\alpha )}\right]$ as in Examples 3.9 and 3.10.

As the proof of the main factorization Theorem 3.22 below works entirely on $\mathrm {Sym}^{b(\alpha )}(X)$ and on $\left[X^{b(\alpha )}/S_{b(\alpha )}\right]$ , we get an analogous result in this more general setting. For example, this applies to the following interesting situations:

• • For f the coarse moduli space of an orbifold, this recovers the above notion of factorization for orbifolds.

• • For f the crepant resolution of an orbifold, this allows us to factorize generating series indexed by exceptional classes.

• • For f a fibration, this allows us to factorize generating series indexed by fiber classes.

We plan to explore these applications in future work.

3.2.1 Plethystic exponentials

For an orbifold $\mathcal {X}$ with coarse moduli space $\pi :\mathcal {X}\to X$ , we define two notions of a partition of $\alpha $ , which will come up in the statement and proof of Theorem 3.22. One should think of the difference between partitions of an integer n and the set $[n]$ . We denote the set of partitions of the integer n by $P(n)$ and the set of partitions of the set $[n]$ by $P[n]$ .

Definition 3.15. Given $\alpha $ , we write $\alpha =(b(\alpha ),\tilde {\alpha })$ under the splitting $\mathrm {N}_0(\mathcal {X})\cong \mathbb {Z}\oplus \widetilde {\mathrm {N}}_0(\mathcal {X})$ from Section 2.2.3. Denote by $[\alpha ]$ the tuple $([b(\alpha )],\tilde {\alpha })$ . Given a factorization index set I, an I-partition of $[\alpha ]$ is a collection

$$ \begin{align*} \{(B_1,\tilde{\alpha}_1), \dots, (B_k,\tilde{\alpha}_k)\}, \end{align*} $$

where $B_i\subseteq [b(\alpha )]$ , such that $\{B_1,\dots ,B_k\}\in P[b(\alpha )]$ , and $\tilde {\alpha }_i\in \widetilde {\mathrm {N}}_0(\mathcal {X})$ classes such that the collection $\{(\left| B_1 \right|,\tilde {\alpha }_1),\dots ,(\left| B_k \right|,\tilde {\alpha }_k)\}$ is an I-partition of $\alpha $ in the above sense. Here, $P[b(\alpha )]$ denotes the set of partitions of the set $[b(\alpha )]$ .

We denote the set of I-partitions of $[\alpha ]$ by $P_I[\alpha ]$ or just $P[\alpha ]$ if the factorization index set is clear. The length of a partition $\Xi =\{(B_1,\tilde {\alpha }_1), \dots , (B_k,\tilde {\alpha }_k)\}$ is k.

Given an element $\Xi =\{(B_1,\tilde {\alpha }_1), \dots , (B_k,\tilde {\alpha }_k)\}$ of $P_I[\alpha ]$ , we denote by $B(\Xi )$ the underlying partition $\{B_1, \dots , B_k\}$ of $[b(\alpha )]$ .

Note that an element $\Xi =\{(B_1,\tilde {\alpha }_1), \dots , (B_k,\tilde {\alpha }_k)\}$ of $P_I[\alpha ]$ yields an element $\left| \Xi \right|=\{(\left| B_1 \right|,\tilde {\alpha }_1),\dots ,(\left| B_k \right|,\tilde {\alpha }_k)\}$ of $P_I(\alpha )$ by definition. We write $P_I^\lambda [\alpha ]=\{\Xi \in P_I[\alpha ]\ |\ \left| \Xi \right|=\lambda \}$ .

Now we can define a K-theoretic version of the plethystic exponential. Note that throughout this paper, we consider the K-theory class of a (possibly 2-periodic) complex of sheaves to be the class in $K(\mathrm {Coh}(-))$ , rather than one in $K(\mathrm {CoCh}^{(\mathbb {Z}/2)}\mathrm {Coh}(-))$ .

Definition 3.16. For $\pi :\mathcal {X}\to X$ an orbifold with coarse moduli space and I a factorization index set, let $\Xi =\{(B_1,\tilde {\alpha }_1), \dots , (B_k,\tilde {\alpha }_k)\}$ be a partition of $[\alpha ]$ . For each $\beta \in I$ , let $\mathcal {G}_{\beta }$ be an $S_{b(\beta )}$ -equivariant complex of sheaves on $X^{b(\beta )}$ . Assume $\Xi $ is ordered such that $\min (B_1) \leq \dots \leq \min (B_k)$ . We define

$$ \begin{align*} \mathcal{G}_{\Xi} := \mathcal{G}_{(\left| B_1 \right|,\tilde{\alpha}_1)} \boxtimes \dots \boxtimes \mathcal{G}_{(\left| B_k \right|,\tilde{\alpha}_k)}, \end{align*} $$

as a complex of sheaves on $X^{b(\alpha )}$ , where each pullback is along the map $\mathrm {pr}_{b_i^1,\dots ,b_i^{l_i}}:X^{b(\alpha )} \to X^{\left| B_i \right|}$ specified by the subset $B_i=\{b_i^1,\dots ,b_i^{l_i}\}$ , where we order $B_i$ such that $b_i^1<\dots <b_i^{l_i}$ .

For a partition $\lambda \in P(\alpha )$ of $\alpha $ , we define the complex of sheaves on $X^{b(\alpha )}$

$$ \begin{align*} \mathcal{G}_\lambda := \bigoplus_{\Xi\in P^\lambda[\alpha]} \mathcal{G}_{\Xi}, \end{align*} $$

which inherits a natural $S_{b(\alpha )}$ -equivariant structure, as $S_{b(\alpha )}$ permutes the various projections $X^{b(\alpha )} \to X^{\left| B_i \right|}$ by permuting the partitions $\{B_i\}_i$ of $[b(\alpha )]$ realizing the partition $\{\left| B_i \right|\}_i$ of $b(\alpha )$ . For compatibility with constructions in Section 3.3.3, the equivariant structure involves an additional sign. We will explain this sign and details about the equivariant structure in more generality in Definition 3.31.

Analogously, we make the following definition on the symmetric product.

Definition 3.17. For each $\beta \in I$ , let $\mathcal {G}_{\beta }$ be a complex of sheaves on $\mathrm {Sym}^{b(\alpha )}(X)$ . Let $\lambda =\{\alpha _1,\dots ,\alpha _k\}$ in $P(\alpha )$ be a partition of $\alpha $ . Take a partition $\Xi =\{(B_1,\tilde {\alpha }_1), \dots , (B_k,\tilde {\alpha }_k)\}$ in $P^\lambda [\alpha ]$ . Assume $\Xi $ is ordered such that $\min (B_1) \leq \dots \leq \min (B_k)$ . Set

$$ \begin{align*} \mathcal{G}_{\Xi} := \mathcal{G}_{(\left| B_1 \right|,\tilde{\alpha}_1)} \boxtimes \dots \boxtimes \mathcal{G}_{(\left| B_k \right|,\tilde{\alpha}_k)}, \end{align*} $$

as a complex of sheaves on $\prod _{i=1}^k\mathrm {Sym}^{\left| B_i \right|}(X)$ . Take $S^B_\lambda $ to be the group $\prod _i S_{l_i}$ , where $l_i$ is the number of occurrences of $\alpha _i$ in $\lambda $ . It acts on $\prod _{i=1}^k\mathrm {Sym}^{\left| B_i \right|}(X)$ by permuting the blocks $\mathrm {Sym}^{\left| B_i \right|}(X)$ with the same $\alpha _i$ . $\mathcal {G}_{\Xi }$ carries a natural $S^B_\lambda $ -equivariant structure. We have the composition of morphisms

$$ \begin{align*} \vartheta: \left[\left(\prod_{i=1}^k\mathrm{Sym}^{\left| B_i \right|}(X)\right)/S^B_\lambda\right] \to \prod_{\alpha_i\in \lambda} \mathrm{Sym}^{l_i}\left(\mathrm{Sym}^{b(\alpha_i)}(X)\right) \to \mathrm{Sym}^{b(\alpha)}(X), \end{align*} $$

where the product in the second term goes over every $\alpha _i$ in $\lambda $ viewed as a set instead of a multiset, the first morphism is the coarse space, and the second morphism adds up the different points in X. Set

We will see in Lemma 3.18(b) below that the construction of $\mathcal {G}_\lambda $ is independent of the choice of $\Xi $ .

We now show some important compatibilities and properties of the previous two constructions.

Proof. Note first that the coarse space p satisfies that the adjunction $p_*p^*\cong \mathrm {id}$ is a natural isomorphism. Assuming (a), we show (b), (c), and (d). For (b), we see

$$ \begin{align*} p_*\left((p^*\mathcal{G})_\lambda\right) \cong \left(p_*p^*\mathcal{G}\right)_\lambda \cong \mathcal{G}_\lambda, \end{align*} $$

where we first use (a) and then the adjunction isomorphism. For (c), we see

$$ \begin{align*} p_*\left((p^*p_*\mathcal{E})_\lambda\right) \cong \left(p_*p^*p_*\mathcal{E}\right)_\lambda \cong \left(p_*\mathcal{E}\right)_\lambda \cong p_*\left(\mathcal{E}_\lambda\right), \end{align*} $$

where we first use (a), then the adjunction isomorphism, and then (a) again. For (d), note that the construction of $\mathcal {G}_\lambda $ in Definition 3.17 involves only pullbacks along flat morphisms and a pushforward along $\vartheta $ , which is the composition of a finite morphism and a proper coarse moduli space, making $\vartheta _*$ exact. Hence, the operation $\left\{\mathcal {G}_\alpha \right\}\mapsto \mathcal {G}_\lambda $ descends to K-theory.

It remains to prove (a). Take $\Xi =\{(B_1,\tilde {\alpha }_1), \dots , (B_k,\tilde {\alpha }_k)\}$ in $P^\lambda [\alpha ]$ and assume $\Xi $ is ordered such that $\min (B_1) \leq \dots \leq \min (B_k)$ . Consider $\mathcal {E}_\Xi $ on $X^{b(\alpha )}$ . It comes with a natural $\prod _i S_{b(\alpha _i)}$ -equivariant structure, where $\prod _i S_{b(\alpha _i)}$ acts on $X^{b(\alpha )}$ by permuting the indices within the blocks indexed by the $B_i$ . As in Definition 3.17 $S^B_\lambda $ acts on $X^{b(\alpha )}$ by permuting entire blocks with indices $B_i$ and $B_j$ with $\alpha _i=\alpha _j$ . Define the subgroup

$$ \begin{align*} S_\Xi \subseteq S_{b(\alpha)} \end{align*} $$

to be the subgroup of $S_{b(\alpha )}$ fixing the partition $\Xi $ . $\mathcal {E}_\Xi $ comes with a natural $S_\Xi $ -equivariant structure. We can identify this group as

$$ \begin{align*} S_\Xi = \left(\textstyle\prod_i S_{b(\alpha_i)}\right)\rtimes S^B_\lambda. \end{align*} $$

By Proposition A.1, the inclusion of $S_\Xi $ in $S_{b(\alpha )}$ gives us a morphism $q_\Xi :\left[X^{b(\alpha )}/S_\Xi \right]\to \left[X^{b(\alpha )}/S_{b(\alpha )}\right]$ . We have a bijection $S_{b(\alpha )}/S_\Xi \xrightarrow {\sim }P^\lambda (\alpha )$ given by $[\sigma ]\mapsto \sigma (\Xi )$ , which together with Proposition A.2 gives us an isomorphism

(14) $$ \begin{align} \left(q_\Xi\right)_* \mathcal{E}_\Xi \cong \mathcal{E}_\lambda. \end{align} $$

Details about the equivariant structures are explained in Remark 3.32 after the equivariant structure of $\bar {\mathcal {G}}_\lambda $ is introduced in detail in Definition 3.31. Consider the commutative diagram

This gives us

$$ \begin{align*} p_*\left(\mathcal{E}_\lambda\right) \cong p_*\left(q_\Xi\right)_*\left(\mathcal{E}_\Xi\right)\cong \vartheta_*\left(\textstyle\prod_i p\right)_*\left(\mathcal{E}_\Xi\right). \end{align*} $$

But by construction $\left(\textstyle \prod _i p\right)_*\left(\mathcal {E}_\Xi \right)$ is the same as $\left(p_*\mathcal {E}\right)_\Xi $ , finishing the proof of (a).

Using the pieces in Definitions 3.16 and 3.17, together with Lemma 3.18(d), we can now define plethystic exponentials in K-theory.

Definition 3.19. Given a factorization index set I, for a system $\mathcal {G}_{\alpha }$ of $S_{b(\alpha )}$ -equivariant complexes of sheaves on $X^{b(\alpha )}$ , we define the plethystic exponential to be

$$ \begin{align*} \mathrm{PExp}^I\left(\sum_{\alpha\in I}q^{\alpha} [\mathcal{G}_{\alpha}]\right) := 1+ \sum_{\alpha\in I} q^{\alpha} \sum_{\lambda\in P(\alpha)} [\mathcal{G}_\lambda], \end{align*} $$

where $1$ denotes the unit in the K-theory of $X^0=\mathrm {pt}$ .

For a system $\mathcal {G}_{\alpha }$ of complexes of sheaves on $\mathrm {Sym}^{b(\alpha )}(X)$ , we define the plethystic exponential to be

$$ \begin{align*} \mathrm{PExp}^I\left(\sum_{\alpha\in I}q^{\alpha} [\mathcal{G}_{\alpha}]\right) := p_\ast\mathrm{PExp}^I\left(\sum_{\alpha\in I}q^{\alpha} [p^*\mathcal{G}_{\alpha}]\right), \end{align*} $$

where $p:[X^{b(\alpha )}/S_{b(\alpha )}] \to \mathrm {Sym}^{b(\alpha )}(X)$ denotes the quotient morphism. Here we use Lemma 3.18(d) for well-definedness in K-theory.

For a system $\mathcal {G}_{\alpha }$ of complexes of sheaves on X, we define the plethystic exponential to be

$$ \begin{align*} \mathrm{PExp}^I\left(\sum_{\alpha\in I}q^{\alpha} [\mathcal{G}_{\alpha}]\right) := \mathrm{PExp}^I\left(\sum_{\alpha\in I}q^{\alpha} [i_\ast\mathcal{G}_{\alpha}]\right), \end{align*} $$

where $i:X \to \mathrm {Sym}^{b(\alpha )}(X)$ denotes the diagonal embedding.

The plethystic exponential defined above can in general depend on I, which is reflected in the notation $\mathrm {PExp}^I$ . Recall that [Reference Okounkov31] defines the plethystic exponential instead using the operation $\mathsf {S}^\bullet $ , which is the morphism in K-theory induced by the symmetric product of vector spaces. We will now show the relation between our definition and the one in [Reference Okounkov31]. If the factorization index set I is closed under addition, as is the case for schemes, the definitions agree, and, by abuse of notation, we will sometimes write $\mathrm {PExp}$ for the function $\mathsf {S}^\bullet $ below.

Lemma 3.20. For a system $\mathcal {G}_{\alpha }$ of complexes of sheaves on $\mathrm {Sym}^{b(\alpha )}(X)$ , we have

$$ \begin{align*} \chi\left(\mathrm{PExp}^I\left(\sum_{\alpha\in I}q^\alpha [\mathcal{G}_\alpha]\right)\right)= \left.{\mathsf{S}^\bullet\left(\sum_{\alpha\in I}q^\alpha \chi\left([\mathcal{G}_\alpha]\right)\right)}\right\vert_{I}, \end{align*} $$

where $\left.{-}\right\vert _{I}$ restricts the right-hand side to all terms $(\cdots )q^\alpha $ with $\alpha \in I$ . For I closed under addition, for example, in the scheme case of Lemma 3.3, the restriction $\left.{-}\right\vert _{I}$ is not necessary. In particular, in this case $\chi \left(\mathrm {PExp}^I\left(-\right)\right)$ is multiplicative under addition of the argument.

Corollary 3.21. In the $\mathsf {T}=(\mathbb {C}^*)^3$ -equivariant case, the identification in Lemma 3.20 above gives us the explicit computational formula

$$ \begin{align*} \chi_{\mathsf{T}}\left(\mathrm{PExp}^I\left(\sum_{\alpha\in I}q^\alpha [\mathcal{G}_\alpha]\right)\right)= \left.{\exp\left(\sum_{n>0}\frac{1}{n}\sum_{\alpha\in I}q^{n\alpha}\chi_{\mathsf{T}}([\mathcal{G}_\alpha],t_1^n,t_2^n,t_3^n)\right)}\right\vert_{I}, \end{align*} $$

where $t_1,t_2,t_3$ is a basis for the characters of $\mathsf {T}$ .

Proof of Lemma 3.20

Take $\alpha \in I$ , $\lambda =\{\alpha _1,\dots ,\alpha _k\}$ in $P(\alpha )$ , and $\Xi \in P^\lambda [\alpha ]$ . By invariance of Euler characteristics under pushforward and Lemma 3.18 (b), we obtain the identity

(15) $$ \begin{align} \chi\left(\mathrm{PExp}^I\left(\sum_{\alpha\in I}q^\alpha [\mathcal{G}_\alpha]\right)\right)= 1+ \sum_{\alpha\in I} q^{\alpha} \sum_{\lambda\in P(\alpha)} \chi\left(\left[\left(\textstyle\prod_i \mathrm{Sym}^{b(\alpha_i)}(X)\right)/S^B_\lambda\right],\mathcal{G}_\Xi\right), \end{align} $$

where we have chosen a representative $\Xi $ for every $\lambda $ .

The coefficient of $q^\alpha $ in $\mathsf {S}^\bullet \left(\sum _{\alpha \in I}q^\alpha \chi \left([\mathcal {G}_\alpha ]\right)\right)$ is

$$ \begin{align*} \sum_{\lambda\in P(\alpha)}\left(\textstyle\prod_i \chi\left(\mathcal{G}_{\alpha_i}\right)\right)^{S_\lambda^B}, \end{align*} $$

where we have again chosen a representative $\Xi \in P^\lambda (\alpha )$ which determines the order of the product. As above, $S_\lambda ^B$ acts by permuting the various multiples of the same factors and we take the $S_\lambda ^B$ -invariant part. Evaluating the Euler characteristics in the right-hand side of (15) is the same as taking the $S_\lambda ^B$ -invariant part of the pushforward of the class of $\mathcal {G}_\Xi $ to $BS_\lambda ^B$ . Summing over $\lambda \in P(\alpha )$ , this yields exactly the terms of $\mathsf {S}^\bullet \left(\sum _{\alpha \in I}q^\alpha \chi \left([\mathcal {G}_\alpha ]\right)\right)$ above, proving the first identification.

We can now use that both sides behave the same under addition of arguments. So, we may assume all $\chi \left([\mathcal {G}_\alpha ]\right)$ are simply characters $\mathbf {t}^{\boldsymbol {\mu }_\alpha }$ . Then we have by definition

$$ \begin{align*} \chi\left(\mathrm{PExp}^I\left(\sum_{\alpha\in I}q^\alpha [\mathcal{G}_\alpha]\right)\right) = 1+ \sum_{\alpha\in I} q^\alpha \sum_{\lambda\in P(\alpha)} \mathbf{t}^{\sum_{\alpha_i\in \lambda}\boldsymbol{\mu}_{\alpha_i}}. \end{align*} $$

We can rewrite this sum as

$$ \begin{align*} \left.{\prod_{\alpha\in I} \left(\sum_{k\geq 0} \left(q^\alpha \mathbf{t}^{\boldsymbol{\mu}_{\alpha}}\right)^k\right)}\right\vert_{I} = \left.{\prod_{\alpha\in I} \left(1-q^\alpha \mathbf{t}^{\boldsymbol{\mu}_\alpha}\right)^{-1}}\right\vert_{I}, \end{align*} $$

where we identified the geometric series. The formal power series of the logarithm now allows us to rewrite this in the desired form.

$$ \begin{align*} \chi\left(\mathrm{PExp}^I\left(\sum_{\alpha\in I}q^\alpha [\mathcal{G}_\alpha]\right)\right) = \left.{\exp\left(\sum_{n>0}\frac{1}{n}\sum_{\alpha\in I}q^{n\alpha} \mathbf{t}^{n \boldsymbol{\mu}_\alpha}\right)}\right\vert_{I}. \end{align*} $$

3.2.2 Factorization theorem

Now we can state the main result of this section.

Theorem 3.22. Let $\mathcal {X}$ be an orbifold with coarse moduli space $\pi :\mathcal {X}\to X$ and let I be a factorization index semigroup. For a factorizable system $\mathcal {F}_{\alpha }$ on $\mathrm {Sym}^{b(\alpha )}(X)$ , there exist classes $[\mathcal {G}_{\alpha }]$ on X such that

$$ \begin{align*} 1 + \sum_{\alpha\in I} q^{\alpha} [\mathcal{F}_{\alpha}] = \mathrm{PExp}^I\left(\sum_{\alpha\in I}q^{\alpha} [\mathcal{G}_{\alpha}]\right) \end{align*} $$

in $\bigoplus _{\alpha \in I}q^\alpha K(\mathrm {Sym}^{b(\alpha )}(X))$ .

Let U be the nonstacky locus, and let S be its complement in X. If the system is given such that for any $\alpha \in I\setminus \Delta _I$ , $\left.{\mathcal {F}_{\alpha }}\right\vert _{\mathrm {Sym}^{b(\alpha )}(U)}=0$ , then the classes $[\mathcal {G}_{\alpha }]$ can be chosen so that they are supported on $S\subset X$ for $\alpha \in I\setminus \Delta _I$ .

The second case of the theorem is fulfilled for pushforwards of virtual structure sheaves from the Hilbert scheme, which is our main application.

By applying Euler characteristics on both sides, we obtain the following computational formula, which can be combined with Lemma 3.20 to compute the left-hand side in simpler terms.

Corollary 3.25. In the setting of Theorem 3.22 above, for a factorizable system $\mathcal {F}_{\alpha }$ on $\mathrm {Sym}^{b(\alpha )}(X)$ , there exist classes $[\mathcal {G}_{\alpha }]$ on X such that

$$ \begin{align*} 1 + \sum_{\alpha\in I} q^{\alpha} \chi\left(\mathrm{Sym}^{b(\alpha)}(X),\mathcal{F}_{\alpha}\right) = \mathrm{PExp}^I\left(\sum_{\alpha\in I}q^{\alpha} \chi(X,\mathcal{G}_{\alpha})\right), \end{align*} $$

where the right-hand side can be computed using Lemma 3.20.

As in Theorem 3.22 above, if the system is given such that for any $\alpha \in I\setminus \Delta _I$ , $\left.{\mathcal {F}_{\alpha }}\right\vert _{\mathrm {Sym}^{b(\alpha )}(U)}$ vanishes, then the classes $[\mathcal {G}_{\alpha }]$ can be chosen so that they are supported on $S\subset X$ for $\alpha \in I\setminus \Delta _I$ .

3.3.1 K-theory class

The (2-periodic) complexes of sheaves of the given factorizable system $\mathcal {F}_\alpha $ factor into simpler parts according to partitions of $\alpha $ . Roughly, the $\mathcal {G}_\alpha $ are defined as (double) complexes tracking the possible splittings of $\alpha $ into smaller parts by systems of subordinate partitions. For now, we will only define the complexes $\mathcal {G}_\alpha $ with trivial differential and examine their K-theory classes. Later in Section 3.3.3, we will introduce a nontrivial differential and use that K-theory classes are independent of the differential.

To keep track of the different ways to split $[\alpha ]$ into smaller parts by different partitions, we use the following notion of an $\alpha $ -index tree.

For several constructions we need a specified ordering on the index tree or on a given partition of $[\alpha ]$ .

Definition 3.29. The standard ordering of an $\alpha $ -index tree T, which we denote by $\leq _T$ , is defined recursively as follows. Take the root and make it the unique maximal node. Order the root children $\{v_1,\dots ,v_k\}$ by

$$ \begin{align*} v_i \leq_T v_j :\Leftrightarrow \min(B_{v_i}) \leq \min(B_{v_j}). \end{align*} $$

For each root child $v_i$ , order all its descendants recursively as above (pretending that $v_i$ is the root node). Finally for any pair of root children $v_i \leq _T v_j$ order all descendants of $v_i$ to be smaller than all descendants of $v_j$ .

Example 3.30. Note that given a partition $\Xi =\{(B_1,\tilde {\alpha }_1), \dots , (B_k,\tilde {\alpha }_k)\}$ of $[\alpha ]$ , the standard ordering on the $\alpha $ -index tree

defines an ordering of the k subsets of the partition. From now on, when we specify a partition $\Xi =\{(B_1,\tilde {\alpha }_1), \dots , (B_k,\tilde {\alpha }_k)\}$ of $[\alpha ]$ , we will assume, the $(B_i,\tilde {\alpha }_i)$ are ordered by this standard ordering.

Definition 3.31. Given an ordered index tree $(T,\leq )$ with leaf partition $\Xi =\{(B_1,\tilde {\alpha }_1), \dots , (B_k,\tilde {\alpha }_k)\}$ of $[\alpha ]$ ordered by $\leq $ , and a system of $S_{b(\beta )}$ -equivariant (2-periodic) complexes of sheaves $\mathcal {G}_{\beta }$ on $X^{b(\beta )}$ , we define

$$ \begin{align*} \mathcal{G}_{T,\leq} := \mathcal{G}_{(\left| B_1 \right|,\tilde{\alpha}_1)} \boxtimes \dots \boxtimes \mathcal{G}_{(\left| B_k \right|,\tilde{\alpha}_k)} \end{align*} $$

as a complex of sheaves on $X^{b(\alpha )}$ , where each pullback is along the map $\mathrm {pr}_{b_i^1,\dots ,b_i^{l_i}}:X^{b(\alpha )} \to X^{\left| B_i \right|}$ specified by the subset $B_i=\{b_i^1,\dots ,b_i^{l_i}\}$ , where we order $B_i$ such that $b_i^1<\dots <b_i^{l_i}$ .

Given a partition $\Xi =\{(B_1,\tilde {\alpha }_1), \dots , (B_k,\tilde {\alpha }_k)\}$ of $[\alpha ]$ , and a system of $S_{b(\beta )}$ -equivariant (2-periodic) complexes of sheaves $\mathcal {G}_{\beta }$ on $X^{b(\beta )}$ , we define

$$ \begin{align*} \mathcal{G}_{\Xi} := \mathcal{G}_{T,\leq} \end{align*} $$

as a complex of sheaves on $X^{b(\alpha )}$ , where $(T,\leq )$ is the standard ordered index tree associated to the partition $\Xi $ from Example 3.30.

For a partition $\lambda $ of $\alpha $ , we define

$$ \begin{align*} \mathcal{G}_\lambda := \bigoplus_{\Xi\in P^\lambda[\alpha]} \mathcal{G}_{\Xi}, \end{align*} $$

which inherits a natural $S_{b(\alpha )}$ -equivariant structure as follows. An element $\sigma $ of $S_{b(\alpha )}$ permutes the various projections $X^{b(\alpha )} \to X^{\left| B_i \right|}$ by acting on a partition $\Xi =\{(B_1,\tilde {\alpha }_1), \dots , (B_k,\tilde {\alpha }_k)\}$ of $[\alpha ]$ by $\sigma \left(\Xi \right)= \{(\sigma (B_1),\tilde {\alpha }_1), \dots , (\sigma (B_k),\tilde {\alpha }_k)\}$ . This induces a permutations $\sigma (T,\leq )=(\sigma (T),\sigma (\leq ))$ of the standard ordered index trees $(T,\leq )$ associated to the partitions $\Xi $ , which permutes the labels while preserving the ordering. The $S_{b(\beta )}$ -equivariant structure of the $\mathcal {G}_{\beta }$ on $X^{b(\beta )}$ then induces canonical isomorphisms

(16) $$ \begin{align} \sigma^*\left(\mathcal{G}_{T,\leq}\right) \xrightarrow{\sim} \mathcal{G}_{\sigma\left(T,\leq\right)}. \end{align} $$

The $S_{b(\alpha )}$ -equivariant structure on $\mathcal {G}_\lambda $ is defined as the sum over $\Xi \in P^\lambda [\alpha ]$ of the compositions

(17) $$ \begin{align} \sigma^*\left(\mathcal{G}_{T,\leq}\right) \xrightarrow{\sim} \mathcal{G}_{\sigma(T),\sigma(\leq)} \xrightarrow{s(\sigma(\leq),\leq)\cdot S}\mathcal{G}_{\sigma(T),\leq}, \end{align} $$

where S is the standard reordering isomorphism to obtain the standard ordering $\leq $ on $\sigma (T)$ .

Note that the above definition of $\mathcal {G}_{\Xi }$ and $\mathcal {G}_{\lambda }$ agrees with the one of Definition 3.16. Now we can define the following complex, which is the central object in the proof of Theorem 3.22. While it is defined with trivial differential for the current study of its K-theory class, we will later equip it with a nontrivial differential.

Definition 3.33. Let I be a factorization index semigroup, and let $\mathcal {F}_{\beta }$ be a factorizable system of $S_{b(\beta )}$ -equivariant (2-periodic) complexes of sheaves on $X^{b(\beta )}$ . For $\alpha \in I$ , $k\geq 0$ , we define the complex of sheaves on $X^{b(\alpha )}$

$$ \begin{align*} \mathcal{G}_{\alpha}^{-k} := \bigoplus_{T\in\mathcal{T}(\alpha,k)} \mathcal{F}_{T,\leq_T}{\color{black} {\in \mathrm{CoCh}_{\mathbb{Z}/2}(\mathrm{Coh}(X^{b(\alpha)}))}}, \end{align*} $$

where we recall that $\mathcal {T}(\alpha ,k)$ is the set of $\alpha $ -index trees with exactly k nonleaf nodes, and $\leq _T$ is the standard ordering of a given index tree. We define the (double) complex

$$ \begin{align*} \mathcal{G}_{\alpha}:=(\mathcal{G}_{\alpha}^\bullet,0) \end{align*} $$

with trivial differential.

We equip this complex with an $S_{b(\alpha )}$ -equivariant structure as follows.

Definition 3.34. The $S_{b(\alpha )}$ -equivariant structure on $\mathcal {G}_\alpha $ is induced as in Definition 3.31. An element $\sigma $ of $S_{b(\alpha )}$ permutes the partitions $\Xi =\{(B_1,\tilde {\alpha }_1), \dots , (B_k,\tilde {\alpha }_k)\}$ of $[\alpha ]$ by $\sigma \left(\Xi \right)= \{(\sigma (B_1),\tilde {\alpha }_1), \dots , (\sigma (B_k),\tilde {\alpha }_k)\}$ . This induces permutations $\sigma (T,\leq )=(\sigma (T),\sigma (\leq ))$ of the index trees $(T,\leq )$ by permuting the labels while preserving the ordering. The $S_{b(\alpha )}$ -equivariant structure on $\mathcal {G}_\alpha $ is defined as the sum over $T\in \mathcal {T}(\alpha ,k)$ of the compositions

$$ \begin{align*} \sigma^*\left(\mathcal{G}_{T,\leq}\right) \xrightarrow{\sim} \mathcal{G}_{\sigma(T),\sigma(\leq)} \xrightarrow{s(\sigma(\leq),\leq)\cdot S}\mathcal{G}_{\sigma(T),\leq}, \end{align*} $$

where the first isomorphism is the canonical isomorphism induced by the $S_{b(\beta )}$ -equivariant structure of the $\mathcal {F}_{\beta }$ on $X^{b(\beta )}$ , and S is the standard reordering isomorphism to obtain the standard ordering $\leq $ on $\sigma (T)$ . The sign $s(\sigma (\leq ),\leq )$ makes the complex a complex of $S_{b(\alpha )}$ -equivariant sheaves.

For the proof of Theorem 3.22, we need to express the K-theory class of the (2-periodic) complexes of sheaves $\mathcal {F}_{\beta }$ of the factorizable system in simpler terms. The next lemma allows us to do exactly that by establishing a relation between the K-theory classes of $\mathcal {F}_{\beta }$ and the complexes $\mathcal {G}_{\alpha }$ defined above.

Note here, that $\mathcal {G}_{\alpha }$ is a double complex. Its K-theory class is the K-theory class of its total complex, which is the same as a signed sum of the classes of the complexes $\mathcal {G}_{\alpha }^k$ .

Lemma 3.35. In $K_{S_{b(\alpha )}}(X^{b(\alpha )})$ we have the identity

$$ \begin{align*} [\mathcal{G}_{\alpha}] = [\mathcal{F}_{\alpha}] - \sum_{\lambda\in P(\alpha)\ non-trivial} [\mathcal{G}_{\lambda}], \end{align*} $$

where the last term is equipped with its standard equivariant structure, defined as in Definition 3.16.

Example 3.36. Before we prove this Lemma in general, we compute it in a simple setting, which highlights that the complexes $\mathcal {G}_\alpha $ were built exactly to satisfy this Lemma. Take $\mathcal {X}=[\mathbb {C}^2/\mu _2]\times \mathbb {C}$ where $\mu _2$ acts diagonally with $\pm 1$ . By Examples 2.5 and 2.7, we have $\mathrm {N}_0(\mathcal {X})=\mathbb {Z}^2$ with $b(-)$ being the projection to the first coordinate. Take the factorization index semigroup I of Lemma 3.3.

For $\alpha =(2,2)$ , we have the partitions

$$ \begin{align*} P_I(\alpha)=\left\{((2,2)),((1,1),(1,1)),((1,2),(1,0))\right\}. \end{align*} $$

By definition of I, the classes $(1,0)$ , $(1,1)$ , and $(1,2)$ allow no further partitions, so that there are no nontrivial index trees and

$$ \begin{align*} \mathcal{G}_{(1,j)}=\mathcal{F}_{(1,j)} \end{align*} $$

for $j=0,1,2$ . Hence the RHS of Lemma 3.35 is just

(18) $$ \begin{align} [\mathcal{F}_{(2,2)}]- [\mathcal{F}_{(1,1)}^{\boxtimes 2}] - [\mathcal{F}_{(1,0)}\boxtimes \mathcal{F}_{(1,2)}]. \end{align} $$

On the other hand, we have

so that we can compute the complex on the LHS

$$ \begin{align*} \mathcal{G}_{(2,2)} = \left[\mathcal{F}_{(1,1)}^{\boxtimes 2}\oplus \left(\mathcal{F}_{(1,0)}\boxtimes \mathcal{F}_{(1,2)}\right) \xrightarrow{0} \mathcal{F}_{(2,2)}\right], \end{align*} $$

where the second term is in degree $0$ . Its K-theory class exactly matches the desired (18) on the RHS.

Proof of Lemma 3.35

By definition $\mathcal {G}_\lambda = \bigoplus _{\Xi \in P^\lambda [\alpha ]} \mathcal {G}_{\Xi }$ , so our strategy will be to find a filtration of $\mathcal {G}_{\alpha }$ that allows us to split its K-theory class into pieces of the form $-[\mathcal {G}_{\Xi }]$ .

Consider the filtration $F_l\mathcal {G}_{\alpha }$ of subcomplexes of $\mathcal {G}_{\alpha }$ consisting of sums over all trees with $\leq l$ root children. Let $Q_l\mathcal {G}_{\alpha }$ be the quotient complexes of the filtration, so the subcomplexes exactly consisting of sums over all trees with exactly l root children. We get

$$ \begin{align*} [\mathcal{G}_{\alpha}] = [\mathcal{F}_{\alpha}] + \sum_{l> 0}[Q_l\mathcal{G}_{\alpha}]. \end{align*} $$

Note that the $S_{b(\alpha )}$ -equivariant structure is preserved by the above filtration, so this equality holds in $K_{S_{b(\alpha )}}(X^{b(\alpha )})$ .

We can split this further into parts with fixed partition at the level of root children. Given $\Xi \in P[\alpha ]$ of length l, we let $Q_l^{\Xi }\mathcal {G}_{\alpha }$ be the subcomplex consisting of the sums over all trees with partition $\Xi $ at the level of root children. For a partition $\lambda \in P(\alpha )$ , define

$$ \begin{align*} Q_l^{\lambda}\mathcal{G}_{\alpha} := \bigoplus_{\Xi\in P^\lambda[\alpha]} Q_l^{\Xi}\mathcal{G}_{\alpha}, \end{align*} $$

with its natural $S_{b(\alpha )}$ -equivariant structure defined analogously to Definition 3.31. By definition of the $S_{b(\alpha )}$ -equivariant structure of $\mathcal {G}_{\alpha }$ , we get an equality

(19) $$\begin{align} [\mathcal{G}_{\alpha}] =& \ [\mathcal{F}_{\alpha}] + \sum_{l> 0}\sum_{\lambda\in P_l(\alpha)} [Q_l^\lambda\mathcal{G}_{\alpha}]\nonumber\\ =& \ [\mathcal{F}_{\alpha}] + \sum_{\lambda\in P(\alpha)\ non-trivial} [Q_{l(\lambda)}^\lambda\mathcal{G}_{\alpha}] \end{align}$$

in $K_{S_{b(\alpha )}}\left(X^{b(\alpha )}\right)$ .

Now we examine the pieces $Q_l^{\Xi }\mathcal {G}_{\alpha }$ further. As the differential of $\mathcal {G}_\alpha $ is defined as $0$ for now, we can ignore it in this proof. Note that these have no $S_{b(\alpha )}$ -equivariant structure, but the $S_{b(\alpha )}$ -equivariant structure of $Q_l^{\lambda }\mathcal {G}_{\alpha }$ is induced in the same way as the one of $\mathcal {G}_{\lambda }$ . We want to show

$$ \begin{align*} Q_{l(\Xi)}^{\Xi}\mathcal{G}_{\alpha}[1] = \mathcal{G}_{\Xi}. \end{align*} $$

Let $l=l(\Xi )$ . The complex $Q_{l}^{\Xi }\mathcal {G}_{\alpha }$ is given by

$$ \begin{align*} Q_{l}^{\Xi}\mathcal{G}_{\alpha}=\bigoplus_{T\in \mathcal{T}_{\Xi}(\alpha,\bullet)} \mathcal{F}_{T,\leq_T}. \end{align*} $$

Let $\Xi =\{(B_1,\tilde {\alpha }_1),\dots ,(B_l,\tilde {\alpha }_l)\}$ be ordered by the standard ordering. Then we have

$$ \begin{align*} \bigoplus_{T\in \mathcal{T}_{\Xi}(\alpha,k)} \mathcal{F}_{T,\leq_T} = \bigoplus_{k_1+\cdots+k_l\leq k-1}\bigoplus_{\substack{T_i\in \mathcal{T}((\left| B_i \right|,\tilde{\alpha}_i),k_i)\\1\leq i\leq l}} \mathcal{F}_{T_1,\leq_{T_1}}\boxtimes\cdots\boxtimes \mathcal{F}_{T_l,\leq_{T_l}} \end{align*} $$

Here we take $k_1+\cdots +k_l\leq k-1$ as we remove the root node in the process of passing from left to right. As a complex this is the same as shifting, so we obtain

$$ \begin{align*} Q_{l}^{\Xi}\mathcal{G}_{\alpha}[1] = \displaystyle\bigoplus_{\substack{k_1+\cdots+k_l\leq\bullet\\T_i\in \mathcal{T}\left(\left(\left| B_i \right|,\tilde{\alpha}_i\right),k_i\right)\\1\leq i\leq l}} \mathcal{F}_{T_1,\leq_{T_1}}\boxtimes\cdots\boxtimes \mathcal{F}_{T_l,\leq_{T_l}} \end{align*} $$

We can distribute to get

$$ \begin{align*} Q_{l}^{\Xi}\mathcal{G}_{\alpha}[1] = \displaystyle\left(\bigoplus_{\substack{k_1,\ T_1\in \mathcal{T}\left(\left(\left| B_1 \right|,\tilde{\alpha}_1\right),k_1\right)}} \mathcal{F}_{T_1,\leq_{T_1}}\right)\boxtimes\cdots\boxtimes \left(\bigoplus_{\substack{k_l,\ T_l\in \mathcal{T}\left(\left(\left| B_l \right|,\tilde{\alpha}_l\right),k_l\right)}} \mathcal{F}_{T_l,\leq_{T_l}}\right), \end{align*} $$

where the degree k part on the right-hand side is exactly given by the terms where $k_1+\cdots +k_l\leq k$ . But this is exactly

$$ \begin{align*} Q_{l}^{\Xi}\mathcal{G}_{\alpha}[1] = \mathcal{G}_{\left(\left| B_1 \right|,\tilde{\alpha}_1\right)}\boxtimes\cdots\boxtimes \mathcal{G}_{\left(\left| B_l \right|,\tilde{\alpha}_l\right)}= \mathcal{G}_{\Xi} \end{align*} $$

identified as complexes. By summing over partitions $\Xi $ with $\left| \Xi \right|=\lambda $ we get an $S_{b(\alpha )}$ -equivariant isomorphism of complexes of sheaves

$$ \begin{align*} Q_{l}^\lambda\mathcal{G}_{\alpha}[1] = \mathcal{G}_{\lambda}, \end{align*} $$

since the $S_{b(\alpha )}$ -equivariant structure is induced in the same way on both sides. In particular, we get $[Q_{l}^\lambda \mathcal {G}_{\alpha }] = -[\mathcal {G}_{\lambda }]$ . Inserting this into (20) gives the desired equality in $K_{S_{b(\alpha )}}(X^{b(\alpha )})$ .

The previous Lemma 3.35 tells us that in order to understand the generating series of K-theory classes of a factorizable system, we can express this in terms of the K-theory classes of the $\mathcal {G}_{\alpha }$ . To prove Theorem 3.22, we equip $\mathcal {G}_\alpha $ with a differential and study the support of the resulting complex. Note that modifying the differential leaves the K-theory class unchanged, and hence the above Lemma 3.35 still holds for the resulting complex.

3.3.2 Level systems

Recall that we have a fixed orbifold with a coarse moduli space $\pi :\mathcal {X}\to X$ and a factorization index semigroup I. In this section, we want to equip the complexes $\mathcal {G}_\alpha $ of Definition 3.33 with a nontrivial differential. The terms of $\mathcal {G}_\alpha $ are $\mathcal {F}_\Xi $ for $[\alpha ]$ -partitions $\Xi $ associated to leaves of various $\alpha $ -index trees. We want to construct the differential of $\mathcal {G}_\alpha $ from the factorization morphisms $\phi $ by splitting $[\alpha ]$ -partitions into smaller parts. To do this, we need to extend our morphisms to be defined over the whole domain. We now make this precise.

Instead of using the entire factorizable systems, we use a part of it to define a system in the following sense.

Definition 3.38. We consider the scheme $X^{b(\alpha )}$ together with its $S_{b(\alpha )}$ -action. For each element B of $P[b(\alpha )]$ , we a get locally closed subset

where $i\sim _B j$ denotes that i and j are in the same set of the partition B. We have a partial order on $P[b(\alpha )]$ , for which $B \leq A$ if B is a refinement of A. $P[b(\alpha )]$ carries a natural $S_{b(\alpha )}$ -action with $\sigma (X_B)=X_{\sigma (B)}$ . Define

For use below, for A and B in $P[b(\alpha )]$ , we write $A\cap B\in P[b(\alpha )]$ for the partition obtained by taking all intersections of sets in A and B. For every $A\in P[b(\alpha )]$ , we can define an equivalence relation on $P[b(\alpha )]$ by $B_1\sim _A B_2$ if and only if $B_1\cap A$ is the same partition as $B_2\cap A$ .

Now consider $P[\alpha ]$ , which carries a natural $S_{b(\alpha )}$ -action. This also comes with a partial order, for which $\Xi _1\leq \Xi _2$ if $\Xi _1$ is subordinate to $\Xi _2$ . For every $B\in P[b(\alpha )]$ we can define an equivalence relation on $P[\alpha ]$ by $\Xi _1 \sim _B \Xi _2$ if and only if the partitions $B(\Xi _1)\cap B$ and $B(\Xi _2)\cap B$ agree as partitions of $[b(\alpha )]$ . The $S_{b(\alpha )}$ -action respects the partial ordering and the equivalence relations. Moreover, we see immediately that for $B{\color {black} {\leq }} A$ in $P[b(\alpha )]$ , $\Xi _1 \sim _A \Xi _2$ implies $\Xi _1 \sim _B \Xi _2$ , and that for $\Xi _1\leq \Xi _2\leq \Xi _3$ , $\Xi _1 \sim _B \Xi _3$ holds if and only if both $\Xi _1 \sim _B \Xi _2$ and $\Xi _2 \sim _B \Xi _3$ hold.

Given the above data, we define an $\alpha $ -level system to be a collection of $S_{b(\alpha )}$ -equivariant complexes of coherent sheaves $\left\{\mathcal {F}_\Xi \right\}_{\Xi \in P[\alpha ]}$ on $X^{b(\alpha )}$ , together with, for each $\Xi _1\leq \Xi _2$ in $P[\alpha ]$ and $B\in P[b(\alpha )]$ such that $\Xi _1 \sim _B \Xi _2$ , a morphism

$$ \begin{align*} \phi_{\Xi_1 \Xi_2 B}: \left.{\mathcal{F}_{\Xi_1}}\right\vert_{U_B} \to \left.{\mathcal{F}_{\Xi_2}}\right\vert_{U_B} \end{align*} $$

compatible with the $S_{b(\alpha )}$ -equivariant structure, such that

1. (a) For each $\Xi _1\leq \Xi _2$ in $P[\alpha ]$ and $A,B\in P[b(\alpha )]$ , the morphisms $\phi _{\Xi _1 \Xi _2 A}$ and $\phi _{\Xi _1 \Xi _2 B}$ agree on $U_A\cap U_B$ whenever they exist. As a consequence, the morphisms glue to

   $$ \begin{align*} \phi_{\Xi_1 \Xi_2}: \left.{\mathcal{F}_{\Xi_1}}\right\vert_{U_{\Xi_1 \Xi_2}} \to \left.{\mathcal{F}_{\Xi_2}}\right\vert_{U_{\Xi_1 \Xi_2}}, \end{align*} $$
   where .

2. (b) For any $\Xi _1\leq \Xi _2\leq \Xi _3$ we have over $U_{\Xi _1 \Xi _2}\cap U_{\Xi _2 \Xi _3}=U_{\Xi _1 \Xi _3}$

   $$ \begin{align*} \phi_{\Xi_2 \Xi_3}\circ \phi_{\Xi_1 \Xi_2} = \phi_{\Xi_1 \Xi_3}. \end{align*} $$

3. (c) $\phi _{\Xi \Xi }$ is the identity for any $\Xi $ in $P[\alpha ]$ .

We additionally assume that given $\Xi _2\in P[\alpha ]$ with a refinement B of $B(\Xi _2)$ , for every A with $B\sim _A B(\Xi _2)$

(20) $$ \begin{align} \bigoplus_{\Xi_1\leq \Xi_2, B(\Xi_1)=B}\phi_{\Xi_1 \Xi_2 A}: \left.{\left(\bigoplus_{\Xi_1\leq \Xi_2, B(\Xi_1)=B}\mathcal{F}_{\Xi_1}\right)}\right\vert_{U_A} \to \left. \vphantom{\left(\bigoplus_{\Xi_1\leq \Xi_2, B(\Xi_1)=B}\mathcal{F}_{\Xi_1}\right)} {\mathcal{F}_{\Xi_2}}\right\vert_{U_A} \end{align} $$

is an isomorphism.

Morphisms between $\alpha $ -level systems $\left(\mathcal {F},\phi \right)$ and $\left(\mathcal {F}',\phi '\right)$ are given by morphisms $\mathcal {F}_\Xi \to \mathcal {F}^{\prime }_\Xi $ that commute with the $\phi $ and $\phi '$ in the obvious way on the open subschemes, where they are defined. Denote the abelian category of $\alpha $ -level systems by $\mathbf {Syst}^\alpha $ .