Motivic degree zero Donaldson–Thomas invariants

Abstract

Given a smooth complex threefold X, we define the virtual motive $[\operatorname{Hilb}^{n}(X)]_{\operatorname {vir}}$ of the Hilbert scheme of n points on X. In the case when X is Calabi–Yau, $[\operatorname{Hilb}^{n}(X)]_{\operatorname{vir}}$ gives a motivic refinement of the n-point degree zero Donaldson–Thomas invariant of X. The key example is X=ℂ³, where the Hilbert scheme can be expressed as the critical locus of a regular function on a smooth variety, and its virtual motive is defined in terms of the Denef–Loeser motivic nearby fiber. A crucial technical result asserts that if a function is equivariant with respect to a suitable torus action, its motivic nearby fiber is simply given by the motivic class of a general fiber. This allows us to compute the generating function of the virtual motives $[\operatorname{Hilb}^{n} (\mathbb{C}^{3})]_{\operatorname{vir}}$ via a direct computation involving the motivic class of the commuting variety. We then give a formula for the generating function for arbitrary X as a motivic exponential, generalizing known results in lower dimensions. The weight polynomial specialization leads to a product formula in terms of deformed MacMahon functions, analogous to Göttsche’s formula for the Poincaré polynomials of the Hilbert schemes of points on surfaces.

Appendices

Appendix A: q-Deformations of the MacMahon function

Let $\mathcal{P}$ denote the set of all finite 3-dimensional partitions. For a partition $\alpha\in\mathcal{P}$, let w(α) denote the number of boxes in α. The combinatorial generating series

$$M(t) = \sum_{\alpha\in\mathcal{P}} t^{w(\alpha)} $$

was determined in closed form by MacMahon [26] to be

$$M(t)= \prod_{m=1}^\infty \bigl(1-t^m\bigr)^{-m}. $$

Motivated by work of Okounkov and Reshetikhin [31], in a recent paper [19], Iqbal–Kozçaz–Vafa discussed a family of q-deformations of this formula. Think of a 3-dimensional partition $\alpha\in\mathcal{P}$ as a subset of the positive octant lattice ℕ³, and break the symmetry by choosing one of the coordinate directions. Define w ₋(α),w ₀(α) and w ₊(α), respectively, as the number of boxes (lattice points) in α∩{x−y<0}, α∩{x−y=0} and α∩{x−y>0}. For a half-integer $\delta\in\frac {1}{2}\mathbb{Z}$, consider the generating series

$$M_\delta(t_1, t_2) = \sum _{\alpha\in\mathcal{P}} t_1^{w_-(\alpha) + (\frac{1}{2}+\delta )w_0(\alpha)}t_2^{w_+(\alpha) + (\frac{1}{2}-\delta )w_0(\alpha)}. $$

Clearly M _δ (t,t)=M(t) for all δ.

Theorem A.1

(Okounkov–Reshetikhin [31, Theorem 2])

The series M _δ (t ₁,t ₂) admits the product form

$$M_\delta(t_1, t_2)=\prod _{i,j=1}^\infty \bigl(1-t_1^{i-\frac {1}{2}+\delta}t_2^{j-\frac{1}{2}-\delta} \bigr)^{-1}. $$

In the main body of the paper, we use a different set of variables. Namely, we set

$$t_{1} = tq^{\frac{1}{2}},\qquad t_{2} = tq^{-\frac{1}{2}}. $$

Then the product formula becomes

$$M_\delta\bigl(t,q^{\frac{1}{2}}\bigr)=\prod _{m=1}^\infty\prod_{k=0}^{m-1} \bigl(1-t^mq^{k+\frac{1}{2}-\frac{m}{2}+\delta} \bigr)^{-1}. $$

The specialization to the MacMahon function is $M_{\delta}(t,q^{\frac{1}{2}}=1)=M(t)$ for all δ.

Appendix B: The motivic nearby fiber of an equivariant function

In this appendix we prove Proposition 2.12, which asserts that if a regular function f:X→ℂ on a smooth variety is equivariant with respect to a torus action satisfying certain assumptions, then Denef-Loeser’s motivic nearby fiber [ψ _f ] is simply equal to the motivic class of the geometric fiber [f ⁻¹(1)]. To make this appendix self-contained, we recall the definitions and restate the result below.

Let

$$f\colon X\to\mathbb{C} $$

be a regular function on a smooth quasi-projective variety X, and let X ₀=f ⁻¹(0) be the central fiber. Denef and Loeser define $[\psi_{f}]\in\mathcal{M}^{{\hat{\mu}}}_{\mathbb{C}}$, the motivic nearby cycle of f using arc spaces and the motivic zeta function [9, 25]. Using motivic integration, they give an explicit formula for [ψ _f ] in terms of any embedded resolution which we now recall.

Let h:Y→X be an embedded resolution of X ₀, namely Y is non-singular and

$$\widetilde{f}=h\circ f:Y\to\mathbb{C} $$

has central fiber Y ₀ which is a normal crossing divisor with non-singular components {E _j :j∈J}. For I⊂J, let

$$E_{I} = \bigcap_{i\in I} E_{i} $$

and let

$$E_{I}^{o} = E_{I} - \bigcup _{j\in I^{c}} (E_{j}\cap E_{I} ). $$

By convention, E _∅=Y and $E^{o}_{\emptyset}=Y-Y_{0}$.

Let N _i be the multiplicity of E _i in the divisor $\widetilde{f}^{-1} (0)$. Letting

$$m_{I} = gcd (N_{i})_{i\in I}, $$

there is a natural etale cyclic $\mu_{m_{I}}$-cover

$$\widetilde{E}^{o}_{I}\to E^{o}_{I}. $$

The formula of Denef and Loeser for the (absolute) motivic nearby cycles of f is given by

$$ [\psi_f]= \sum_{I \neq\emptyset} (1-\mathbb{L})^{|I|-1}\bigl[ \widetilde{E}^{o}_{I},\mu_{m_{I}}\bigr]\in \mathcal{M}_{\mathbb{C}}^{{\hat{\mu}}}. $$

(B.1)

Since all the $E_{I}^{o}$ appearing in the above sum have natural maps to X ₀, the above formula determines the relative motivic nearby cycle $[\psi_{f}]_{X_{0}}\in\mathcal{M}^{{\hat{\mu}}}_{X_{0}}$. The relative motivic vanishing cycle is supported on Z={df=0}, the degeneracy locus of f:

$$[\varphi_{f}]_{Z} = [\psi_{f}]_{X_{0}}-[X_{0}]_{X_{0}} \in\mathcal{M}_{Z}^{{\hat{\mu}}} \subset\mathcal{M}_{X_{0}}^{{\hat{\mu}}}. $$

Recall that an action of ℂ^∗ on a variety V is circle compact, if the fixed point set $V^{\mathbb{C}^{*}}$ is compact and moreover, for all v∈V, the limit lim _t→0 t⋅y exists.

The following is a restatement of Proposition 2.12 and Proposition 2.13. It is the main result of this appendix.

Theorem B.1

Let f:X→ℂ be a regular morphism on a smooth quasi-projective complex variety. Let Z={df=0} be the degeneracy locus of f and let $Z_{\operatorname{aff}}\subset X_{\operatorname{aff}}$ be the affinization of Z and X respectively. Assume that there exists an action of a connected complex torus T on X so that f is T-equivariant with respect to a primitive character χ:T→ℂ^∗, namely f(t⋅x)=χ(t)f(x) for all x∈X and t∈T. We further assume that there exists a one parameter subgroup ℂ^∗⊂T such that the induced action is circle compact. Then the motivic nearby cycle class [ψ _f ] is in $\mathcal{M}_{\mathbb{C}}\subset\mathcal{M}^{{\hat{\mu}}}_{\mathbb{C}}$ and is equal to [X ₁]=[f ⁻¹(1)]. Consequently the motivic vanishing cycle class [φ _f ] is given by

$$[\varphi_{f}] = \bigl[f^{-1} (1)\bigr] - \bigl[f^{-1} (0)\bigr]. $$

If we further assume that X ₀ is reduced then $[\varphi_{f}]_{Z_{\operatorname{aff}}}$, the motivic vanishing cycle, considered as a relative class on $Z_{\operatorname{aff}}$, lies in the subring $\mathcal{M}_{Z_{\operatorname{aff} }}\subset\mathcal{M}_{Z_{\operatorname{aff}}}^{{\hat{\mu}}}$.

By equivariant resolution of singularities [36, Corollary 7.6.3], we may assume that h:Y→X, the embedded resolution of X ₀, is T-equivariant. Namely Y is a non-singular T-variety and

$$\widetilde{f}=h\circ f:Y\to\mathbb{C} $$

is T-equivariant with central fiber E which is a normal crossing divisor with non-singular components E _j , j∈J. Let ℂ^∗⊂T be the one-parameter subgroup whose action on X is circle compact. Then the action of ℂ^∗ on Y is circle compact (since h is proper) and each E _j is invariant (but not necessarily fixed).

We will make use of the Białynicki-Birula decomposition for smooth varieties [3]. This result states that if V is a smooth projective variety with a ℂ^∗-action, then there is a locally closed stratification:

$$V=\bigcup_{F}Z_{F} $$

where the union is over the components of the fixed point locus and Z _F →F is a Zariski locally trivial affine bundle. The rank of the affine bundle Z _F →F is given by

$$n (F) = \operatorname{index}(N_{F/V}) $$

where the index of the normal bundle N _F/V is the number of positive weights of the fiberwise action of ℂ^∗. The morphisms Z _F →F are defined by x↦lim _t→0 t⋅x and consequently, the above stratification also exists for smooth varieties with a circle compact action. As a corollary of the Białynicki-Birula decomposition, we get the following relation in the ring of motivic weights.

Lemma B.2

Let V be a smooth quasi-projective variety with a circle compact ℂ^∗-action. For each component F of the fixed point locus, we define the index of F, denoted by n(F), to be the number of positive weights in the action of ℂ^∗ on N _F/V . Then in $\mathcal{M}_{\mathbb{C}}$ we have

$$[V] = \sum_{F} \mathbb{L}^{n (F)} [F], $$

where the sum is over the components of the fixed point locus and $\mathbb{L}=[\mathbb{A}^{1}_{\mathbb{C}}]$ is the Lefschetz motive.

We call this decomposition a BB decomposition.

We begin our proof of Theorem B.1 with a “no monodromy” result.

Lemma B.3

The following equation holds in $\mathcal{M}_{\mathbb{C}}^{{\hat{\mu}}}$:

$$\bigl[\widetilde{E}^{o}_{I},\mu_{m_{I}}\bigr] = \bigl[E_{I}^{o}\bigr]. $$

Under the further assumption that X ₀ is reduced, the above equation holds in $\mathcal{M}_{X_{\operatorname {aff}}}^{{\hat{\mu}}}$.

An immediate corollary of Lemma B.3 is that [ψ _f ] lies in the subring $\mathcal{M}_{\mathbb{C}}\subset\mathcal {M}_{\mathbb{C}}^{{\hat{\mu}}}$ and that if X ₀ is reduced, then $[\varphi_{f}]_{Z_{\operatorname {aff}}}$ lies in the subring $\mathcal{M}_{Z_{\operatorname{aff}}}\subset\mathcal {M}_{Z_{\operatorname{aff}}}^{{\hat{\mu}}}$.

To prove Lemma B.3, we recall the construction of the $\mu_{m_{I}}$-cover $\widetilde{E}_{I}^{o}\to E^{o}_{I}$ given in [25, §5]. Let N=lcm(N _i ) and let $\widetilde {Y}\to Y$ be the μ _N -cover obtained by base change over the Nth power map (⋅)^N:ℂ→ℂ followed by normalization. Define $\widetilde{E}^{o}_{I}$ to be any connected component of the preimage of $E^{o}_{I}$ in $\widetilde{Y}$. The component $\widetilde{E}^{o}_{I}$ is stabilized by $\mu_{m_{I}}\subset \mu_{N}$ whose action defines the cover $\widetilde{E}^{o}_{I}\to E^{o}_{I}$.

Observe that the composition

$$E_{i} \to Y \to X \to X_{\operatorname{aff}} $$

contracts E _i to a point unless E _i is a component of the proper transform of X ₀. Thus for these components (and their intersections), the proof given below (stated for absolute classes), applies to relative classes over $X_{\operatorname{aff}}$ as well. Under the assumption that X ₀ is reduced, those E _i which are components of the proper transform of X ₀ have multiplicity one and so $\widetilde{E}_{i}=E_{i}$ and there is nothing to prove.

We define $\widetilde{T}$ by the fibered product

Thus $\widetilde{T}$ is an extension of T by μ _N ⊂ℂ^∗ and it has character $\widetilde{\chi}$ satisfying $\widetilde{\chi }^{N}=\chi$. Moreover, $\widetilde{\chi}$ is the identity on the subgroup $\mu_{N}\subset\widetilde{T}$. The key fact here is that since χ is primitive, $\widetilde{T}$ is connected.

By construction, $\widetilde{T}$ acts on the base change of Y over the Nth power map and hence it acts on the normalization $\widetilde {Y}$. Thus we have obtained an action of a connected torus $\widetilde {T}$ on $\widetilde{Y}$ covering the T-action on Y. Since T acts on each $E^{o}_{I}$, $\widetilde{T}$ acts on each component of the preimage of $E^{o}_{I}$ in $\widetilde{Y}$. Thus we have an action of the connected torus $\widetilde{T}$ on $\widetilde{E}^{o}_{I}$ such that the $\mu_{m_{I}}$-action is induced by the subgroup

$$\mu_{m_{I}}\subset \mu_{N}\subset\widetilde{T}. $$

Lemma B.3 then follows from the following:

Lemma B.4

Let W be a smooth quasi-projective variety with the action of a connected torus $\widetilde{T}$. Then for any finite cyclic subgroup $\mu \subset\widetilde{T}$, the equation

$$[W,\mu] = [W] = [W/\mu] $$

holds in $\mathcal{M}^{{\hat{\mu}}}_{\mathbb{C}}$.

Proof

Let $\mathbb{C}^{*}\subset\widetilde{T}$ be the 1-parameter subgroup generated by $\mu\subset\widetilde{T}$. The ℂ^∗-action on W gives rise to a ℂ^∗-equivariant stratification of W into varieties W _i of the form (V−{0})×F where F is fixed and V is a ℂ^∗-representation. This assertion follows from applying the Białynicki-Birula decomposition to $\overline{W}$, any ℂ^∗-equivariant smooth compactification of W and stratifying further to trivialize all the bundles and to make the zero sections separate strata. The induced stratification of W then is of the desired form. Thus to prove Lemma B.4, it then suffices to prove it for the case of μ acting on V−{0} where V is a μ-representation. By the relation given in (2.4), we have [V,μ]=[V] and hence [V−{0},μ]=[V−{0}]. The equality [V−{0}]=[(V−{0})/μ] follows from [25, Lemma 5.1].

Applying Lemma B.4 to (B.1), we see that in order to prove [ψ _f ]=[X ₁], we must prove

$$[X_{1}] = \sum_{I \neq\emptyset} (1- \mathbb{L})^{|I|-1} \bigl[E^{o}_{I}\bigr]. $$

As explained in the beginning of Sect. 2.7, there is an isomorphism

$$X_{1}\times\mathbb{C}^{*} \cong X-X_{0}. $$

Consequently we get $(\mathbb{L}-1)[X_{1}] = [X]-[X_{0}]$ or equivalently

$$[Y] = [Y_{0}] + (\mathbb{L}-1) [X_{1}]. $$

Combining this with the previous equation we find that the equation we wish to prove, [ψ _f ]=[X ₁], is equivalent to

$$[Y] = [Y_{0}] - \sum_{I\neq\emptyset} (1- \mathbb{L})^{|I|} \bigl[E_{I}^{o}\bigr], $$

which can also be written (using the conventions about the empty set) as

$$ 0=\sum_{I} (1- \mathbb{L})^{|I|} \bigl[E_{I}^{o}\bigr]. $$

(B.2)

By the principle of inclusion/exclusion, we can write

thus we have

$$(-1)^{|I|}\bigl[E^{o}_{I}\bigr] = \sum _{A\supset I} (-1)^{|A|}[E_{A}]. $$

Thus the right hand side of (B.2) becomes

$$\sum_{I} (\mathbb{L}-1)^{|I|} \sum _{A\supset I} (-1)^{|A|} [E_{A}] = \sum_{A} (-1)^{|A|} [E_{A}] \sum_{I\subset A} (\mathbb{L}-1)^{|I|}. $$

Since

$$\sum_{I\subset A} (\mathbb{L}-1)^{|I|} =\sum _{n=0}^{|A|} \binom{|A|}{n} (\mathbb{L} -1)^{n} = \mathbb{L}^{|A|}, $$

we can reformulate the equation we need to prove as

$$ 0 = \sum_{A} (- \mathbb{L})^{|A|} [E_{A}]. $$

(B.3)

Note that each E _A is smooth and ℂ^∗ invariant and that the induced ℂ^∗-action on E _A is circle compact, so we have a BB decomposition for each.

Let F denote a component of the fixed point set $Y^{\mathbb{C}^{*}}$ and let θ _F,A denote a component of F∩E _A . Since E _A is smooth and ℂ^∗ invariant and the induced ℂ^∗-action is circle compact, E _A admits a BB decomposition

$$[E_{A}] = \sum_{F}\sum _{\theta_{F,A}} \mathbb{L}^{n (\theta_{F,A})} [\theta_{F,a}] $$

where

$$n (\theta_{F,A}) = \operatorname{index}(N_{\theta_{F,A}/E_{A}}) . $$

Therefore the sum in (B.3) (which we wish to prove is zero) is given by

$$\sum_{A} (-\mathbb{L})^{|A| } [E_{A}] = \sum_{A}\sum _{F}\sum_{\theta _{F,A}} (-1)^{|A|}\mathbb{L}^{|A|+n (\theta_{F,A})} [ \theta_{F,A}]. $$

We define a set I(F) by

$$I (F) = \{i: F\subset E_{i} \}. $$

Then clearly

$$\theta_{F,A} = \theta_{F,A'}\quad\text{if } A\cup I (F) =A'\cup I (F). $$

So writing A=B∪C where B⊂I(F) and C⊂I(F)^c, we can rewrite the above sum as

$$ \sum_{F}\sum _{C\subset I (F)^{c}}\sum_{\theta_{F,C}} (- \mathbb{L} )^{|C|} [\theta_{F,C}]\sum _{B\subset I (F)} (-1)^{|B|} \mathbb{L}^{|B|+n (\theta_{F,B\cup C})}. $$

(B.4)

We will show that the inner most sum is always zero which will prove Theorem B.1.

Let y∈θ _F,A where A=B∪C. We write the ℂ^∗ representation T _y Y in two ways:

where restriction to the point y is implicit in the above equation. Counting positive weights on each side, we get

$$n (F) = n (\theta_{F,A}) +\sum_{i\in A} m_{i} (\theta_{F,A}) $$

where

$$m_{i} (\theta_{F,A}) = \operatorname{index}(N_{E_{i}/Y}|_{\theta_{F,A}}). $$

Note that m _i (θ _F,A ) is 1 or 0 depending on if the weight of the ℂ^∗-action on E _i |y is positive or not. Note also that if i∈I(F) then

$$m_{i} (\theta_{F,A}) = m_{i,F} = \operatorname{index}(N_{E_{i}/Y}|_{F}). $$

Moreover, if i∈I(F)^c, then

$$N_{E_{i}/Y}|_{y} \subset TF|_{y} $$

and so m _i (θ _F,A )=0.

Thus writing A=B∪C with B⊂I(F) and C⊂I(F)^c, we get

$$n (\theta_{F,A}) = n (F) - \sum_{i\in B} m_{i,F}, $$

and so

$$\sum_{B\subset I (F)} (-1)^{|B|} \mathbb{L}^{|B|+n (\theta_{F,A})} = \mathbb{L}^{n (F)}\sum _{B\subset I (F)} (-1)^{|B|} \mathbb{L}^{\sum_{i\in B} (1-m_{i,F})}. $$

For k=0,1, we define

$$I_{k} (F) = \bigl\{i\in I (F): m_{i,F}=k \bigr\}. $$

Then

$$\sum_{B\subset I (F)} (-1)^{|B|} \mathbb{L}^{\sum_{i\in B} (1-m_{i,F})} = \sum_{B_{0}\subset I_{0} (F)} (- \mathbb{L})^{|B_{0}|} \sum_{B_{1}\subset I_{1} (F)} (-1)^{|B_{1}|}, $$

but

$$\sum_{B_{1}\subset I_{1} (F)} (-1)^{|B_{1}|} = 0 $$

unless I ₁(F)=∅. Since the above equation implies that the expression in (B.4) is zero, and that in turn verifies (B.2) and (B.3) which are equivalent to Theorem B.1, it only remains for us to prove that I ₁(F)≠∅ for all F.

Let y∈F. We need to show that for some i, the action of ℂ^∗ on $N_{E_{i}/Y}|_{y}$ has positive weight.

By the Luna slice theorem, there is an etale local neighborhood of y∈Y which is equivariantly isomorphic to T _y Y. Over the point y, we have a decomposition

$$TY = TF + N_{F/E_{I (F)}} +\sum_{i\in I (F)} N_{E_{i}/Y}. $$

Let (u ₁,…,u _s ,v ₁,…,v _p ,{w _i } _i∈I(F)) be linear coordinates on T _y Y compatible with the above splitting. The action of t∈ℂ^∗ on T _y Y is given by

$$t\cdot(u,v,w) = \bigl(u_{1},\dots,u_{s},t^{a_{1}}v_{1}, \dots ,t^{a_{p}}v_{p},\bigl\{t^{b_{i}}w_{i} \bigr\}_{i\in I (F)}\bigr). $$

In these coordinates, the function $\widetilde{f}$ is given by

$$\widetilde{f} (u,v,w) = g (u) \prod_{i\in I (F)} w_{i}^{N_{i}} $$

where g(u) is a unit. Since the ℂ^∗-action on Y is circle compact, $\widetilde{f}$ is equivariant with respect to an action on ℂ of positive weight l, that is

$$\widetilde{f} \bigl(t\cdot(u,v,w)\bigr) = t^{l} \widetilde{f} (u,v,w). $$

This implies that

$$l=\sum_{i\in I (F)} b_{i}N_{i}. $$

Then since N _i >0 and l>0 we have that b _i >0 for some i∈I(F) and so for this i, we have m _i,F =1 which was what we needed to prove. The proof of Theorem B.1 is now complete. □