Integration on Deligne–Mumford Stacks

We now define integration on stacks. Intuitively, a point on a stack has an automorphism group \(G\), which produces the effect of the point being repeated \(\lvert G\rvert\) times, so a correction by \(1/\lvert G\rvert\) is necessary in integration. On a proper finite-type Deligne–Mumford stack, which we consider in this post, the geometric stabilizers are finite, and thus these weights are well-defined.

Meanwhile, on a pure-dimensional algebraic stack, the fundamental cycle itself is defined regardless of properness, but the reason we integrate is to push it forward along the structure morphism to a point to obtain a numerical degree. In addition, on a smooth stack, Chow cohomology and Chow homology correspond to each other via the cap product with the fundamental class, making it convenient to handle integration using cohomological notation as well. Therefore, all stacks appearing in this post refer to smooth proper connected Deligne–Mumford stacks over \(\mathbb{C}\).

Chow Groups of Stacks

In [Algebraic Varieties] §Chow Groups, ⁋Definition 5, we defined the Chow group of a variety by quotienting algebraic cycles by rational equivalence. On a Deligne–Mumford stack, cycles and rational equivalence can be defined in the same way. For \(k\)-dimensional integral closed substacks \(\mathcal{V}\subseteq\mathcal{X}\), let the vector space consisting of their formal \(\mathbb{Q}\)-linear combinations be

\[Z_k(\mathcal{X})_\mathbb{Q}\]

If we let the subspace generated by the principal divisors given by \((k+1)\)-dimensional integral closed substacks \(\mathcal{W}\subseteq\mathcal{X}\) and rational functions \(f\in K(\mathcal{W})^\times\) be \(R_k(\mathcal{X})_\mathbb{Q}\), we define the rational Chow group as

\[A_k(\mathcal{X})_\mathbb{Q}:=Z_k(\mathcal{X})_\mathbb{Q}/R_k(\mathcal{X})_\mathbb{Q}\]

Now, integration on stacks and the factor \(1/\lvert G\rvert\) arising in the process are naturally deduced from the following definition extending the proper pushforward of [Algebraic Varieties] §Chow Groups, ⁋Proposition 6 to stacks.

Definition 1 Suppose two Deligne–Mumford stacks \(\mathcal{X}, \mathcal{Y}\) and a proper morphism \(f: \mathcal{X}\rightarrow \mathcal{Y}\) between them are given. On \(\mathcal{X}\), given a \(k\)-dimensional integral closed substack \(\mathcal{V}\), and under \(f\), the image of \(\mathcal{V}\) with its closure \(\mathcal{W}=\cl f(\mathcal{V})\), we define as follows.

1. If \(\dim\mathcal{W}<\dim\mathcal{V}\), we define \(f_\ast[\mathcal{V}]=0\).

2. In the case where \(\dim\mathcal{W}=\dim\mathcal{V}\), for \(\mathcal{V},\mathcal{W}\) having coarse moduli spaces \(V,W\) and generic stabilizer orders \(e_\mathcal{V},e_\mathcal{W}\), we define

   \[f_\ast[\mathcal{V}]:=[K(V):K(W)]\frac{e_\mathcal{W}}{e_\mathcal{V}}[\mathcal{W}]\]

Extending this linearly preserves rational equivalence, thereby defining the pushforward \(f_\ast:A_k(\mathcal{X})_\mathbb{Q}\longrightarrow A_k(\mathcal{Y})_\mathbb{Q}\).

Recall that in [Algebraic Varieties] §Chow Groups, ⁋Proposition 6, the pushforward for an integral subvariety \(V\rightarrow W\) of the same dimension was simply given by

\[[V]\longmapsto [K(V):K(W)][W]\]

The above definition is precisely the extension of this formula to Deligne–Mumford stacks, the difference being the addition of the ratio of generic stabilizers \(e_{\mathcal W}/e_{\mathcal V}\). Fundamentally, as observed right after §Proper Stacks, ⁋Proposition 4 (Local quotient chart), the degree seen scheme-theoretically on the coarse moduli and the automorphism data inherently possessed by the stack are seen separately; for this reason, the product

\[[K(V):K(W)]\frac{e_{\mathcal W}}{e_{\mathcal V}}\]

can be thought of as the stack-theoretic generic degree of \(f\). For this reason, rational coefficients appear naturally on stacks. A property of the pushforward defined in this way is the following functoriality.

Proposition 2 (Functoriality) For proper morphisms \(f:\mathcal{X}\rightarrow\mathcal{Y}\) and \(g:\mathcal{Y}\rightarrow\mathcal{Z}\) between Deligne–Mumford stacks,

\[(g\circ f)_\ast=g_\ast\circ f_\ast\]

holds.

On an ordinary scheme, integration is the process of taking the cap product with the fundamental class to adjust the degree to \(0\), and then pushing it forward via the structure morphism to obtain the numerical value. Now in the case of a stack, the zero-dimensional (rational coefficient) Chow group of the target is

\[A_0(\Spec\mathbb{C})_\mathbb{Q}=\mathbb{Q}[\Spec\mathbb{C}]\cong\mathbb{Q}\]

and the proper pushforward directly defines the degree map

\[\deg:=p_\ast:A_0(\mathcal{X})_\mathbb{Q}\longrightarrow\mathbb{Q}\]

Then the following holds.

Lemma 3 Let \(\mathcal{Z}\subseteq\mathcal{X}\) be a \(0\)-dimensional integral closed substack, and let \(x\) be its unique geometric point. Then

\[\deg[\mathcal{Z}]=\frac{1}{\lvert\Aut(x)\rvert}\]

holds.

Proof

Since \(\mathcal{Z}\) is a \(0\)-dimensional integral proper stack, its coarse moduli space is a single point \(\Spec\mathbb{C}\), and \(\mathcal{Z}\cong[\Spec\mathbb{C}/\Aut(x)]\). Therefore, applying Definition 1 to the structure morphism \(p:\mathcal{Z}\rightarrow\Spec\mathbb{C}\), the coarse function field degree is \(1\), the generic stabilizer order of the source is \(\lvert\Aut(x)\rvert\), and the stabilizer order of the target is \(1\). Therefore,

\[p_\ast[\mathcal{Z}]=\frac{1}{\lvert\Aut(x)\rvert}[\Spec\mathbb{C}]\]

holds.

Integration

In general, it is known that on a pure \(d\)-dimensional stack \(\mathcal{X}\), there exists a fundamental cycle \([\mathcal{X}]\in A_d(\mathcal{X})_\mathbb{Q}\). Additionally, if \(\mathcal{X}\) is smooth, the cap product with the fundamental class yields an identification

\[A^k(\mathcal{X})_\mathbb{Q}\xrightarrow{\sim} A_{d-k}(\mathcal{X})_\mathbb{Q},\qquad\alpha\longmapsto\alpha\cap[\mathcal{X}]\]

between Chow cohomology and Chow homology. Therefore, a top codimension class naturally determines a zero-cycle, and we can define the integral of this cohomology class as its image under the degree map.

Definition 4 Consider a smooth proper Deligne–Mumford stack \(\mathcal{X}\) of pure dimension \(d\). We define the integral of a top class \(\alpha\in A^d(\mathcal{X})_\mathbb{Q}\) by

\[\int_\mathcal{X}\alpha:=\deg(\alpha\cap[\mathcal{X}])=p_\ast(\alpha\cap[\mathcal{X}])\in\mathbb{Q}\]

In Lemma 3 above, we showed that into this degree map \(\deg\), the factor \(1/\lvert G\rvert\) coming from automorphisms intervenes. First, let us verify this in a concrete example.

Example 5 Suppose a smooth proper variety \(U\) is equipped with an action of a finite group \(G\), and consider the quotient stack \(\mathcal{X}=[U/G]\). Then we can write the integral on it using the atlas \(p:U\rightarrow \mathcal{X}\). That is, if \(d=\dim U\), then for any \(\alpha\in A^d(\mathcal{X})_\mathbb{Q}\),

\[\int_{\mathcal{X}}\alpha=\frac{1}{\lvert G\rvert}\int_U p^\ast\alpha\]

holds. To verify this directly, let the zero-cycle corresponding to \(\alpha\) be

\[z=\alpha\cap[\mathcal{X}]=\sum_i m_i[\mathcal{Z}_i]\]

Then for each \(0\)-dimensional integral closed substack \(\mathcal{Z}_i\), letting \(x_i\) be its geometric point and \(G_i=\Aut(x_i)\) its stabilizer, we have \(\deg[\mathcal{Z}_i]=1/\lvert G_i\rvert\) by Lemma 3.

Meanwhile, if we represent \(x_i\) by a point in \(U\), denoted \(u_i\), the pullback under \(p\) of \(\mathcal{Z}_i\) corresponds to the orbit of \(u_i\) under \(G\). Then by the orbit-stabilizer formula, this orbit consists of \(\lvert G\rvert/\lvert G_i\rvert\) distinct points, and since each point is a point of the scheme \(U\), each has degree \(1\). Therefore,

\[\deg\bigl(p^\ast[\mathcal{Z}_i]\bigr)=\frac{\lvert G\rvert}{\lvert G_i\rvert}=\lvert G\rvert\deg[\mathcal{Z}_i]\]

holds, and applying this to each term and summing yields

\[\deg(p^\ast z)=\lvert G\rvert\deg z\]

Since \(p\) is flat, we have \(p^\ast z=p^\ast\alpha\cap[U]\), and therefore

\[\int_U p^\ast\alpha=\lvert G\rvert\int_{\mathcal{X}}\alpha\]

holds.

Similarly, the integral on the coarse moduli space and that on the entire stack can also be compared as follows.

Example 6 Let \(\pi:\mathcal{X}\rightarrow X\) be the coarse moduli morphism, and let the generic stabilizer of \(\mathcal{X}\) have order \(e\). Then

\[\pi_\ast[\mathcal{X}]=\frac{1}{e}[X]\]

holds. Indeed, the morphism induced between coarse spaces by the coarse moduli morphism \(\pi\) is \(\mathrm{id}_X\), so the degree of the function field extension is \(1\). Meanwhile, the order of the generic stabilizer of the source \(\mathcal{X}\) is \(e\), and since the target \(X\) is an algebraic space (or scheme), its stabilizer is trivial. Therefore, by Definition 1, we obtain

\[\pi_\ast[\mathcal{X}]=1\cdot\frac{1}{e}[X]=\frac{1}{e}[X]\]

In other words, the generic stabilizer, which is invisible in the coarse moduli space, appears with a weight of exactly \(1/e\) in the pushforward of the fundamental cycle.

Integration along fibers

Meanwhile, as explained above, since integration is the proper pushforward along the structure morphism, we can extend this to an arbitrary proper morphism. To this end, consider a proper morphism \(f:\mathcal{X}\rightarrow\mathcal{Y}\) between smooth proper Deligne-Mumford stacks. Let \(\dim\mathcal{X}=m\) and \(\dim\mathcal{Y}=n\), and let the relative dimension be \(r=m-n\). Using smoothness to pass from homology classes to cohomology classes, the proper pushforward above can be written as

\[f_\ast:A^k(\mathcal{X})_\mathbb{Q}\longrightarrow A^{k-r}(\mathcal{Y})_\mathbb{Q}\]

That is, for a cohomology class \(\alpha\), we define \(f_\ast \alpha\) by the formula

\[(f_\ast\alpha)\cap[\mathcal{Y}]:=f_\ast(\alpha\cap[\mathcal{X}])\]

Then the following projection formula holds. ([Algebraic Varieties] §Intersection Product, ⁋Proposition 14 (Projection Formula))

Proposition 7 (Projection formula) In the situation above, for any \(\alpha\in A^\ast(\mathcal{X})_\mathbb{Q}\) and \(\beta\in A^\ast(\mathcal{Y})_\mathbb{Q}\),

\[f_\ast(\alpha\cdot f^\ast\beta)=f_\ast\alpha\cdot\beta\]

holds. In particular, when both sides are of top degree,

\[\int_\mathcal{X}\alpha\cdot f^\ast\beta=\int_\mathcal{Y}f_\ast\alpha\cdot\beta\]

holds.

Intuitively, this is integration along fibers, but in order to interpret it as an actual integration over fibers, we need an operation that restricts a given class to a point of the base, namely the Gysin map. Suppose that a codimension-\(c\) closed regular embedding \(i:Z\hookrightarrow Y\) and a morphism \(g:V\rightarrow Y\) are given. In the Cartesian square

\[W=V\times_Y Z\]

let \(i':W\hookrightarrow V\) be the induced closed embedding; then it is known that the refined Gysin pullback

\[i^!:A_k(V)_\mathbb{Q}\longrightarrow A_{k-c}(W)_\mathbb{Q}\]

is defined. In addition to the usual Gysin pullback, this is compatible with proper pushforward and base change. Specifically, suppose given a codimension-\(c\) closed regular embedding \(i:Z\hookrightarrow Y\) and a proper morphism \(g:V\rightarrow Y\), and let

\[W=V\times_Y Z\]

If \(i':W\hookrightarrow V\) and \(g':W\rightarrow Z\) are the morphisms induced by the fiber product, then

\[i^!g_\ast=g'_\ast{i'}^!\]

holds.

Applying this to a proper family \(f:\mathcal{C}\rightarrow\mathcal{M}\) and a point \(y\rightarrow\mathcal{M}\) of the base, restricting the class pushed forward by \(f_\ast\) to \(y\) is equivalent to first restricting to the fiber \(\mathcal{C}_y\) and then pushing forward along \(f_y:\mathcal{C}_y\rightarrow y\). Therefore, we obtain the following fiber integration formula comparing the proper pushforward with the actual integration over the fibers.

Proposition 8 (Fiber integration) Let \(f:\mathcal{C}\rightarrow\mathcal{M}\) be a proper flat representable morphism of relative dimension \(1\), and let \(\mathcal{C},\mathcal{M}\) be smooth proper Deligne-Mumford stacks. Assume also that \(\mathcal{M}\) is connected.

Then for \(\alpha\in A^1(\mathcal{C})_\mathbb{Q}\), we have

\[f_\ast\alpha\in A^0(\mathcal{M})_\mathbb{Q}\]

and its value at a geometric point \(y:\Spec\mathbb{C}\rightarrow\mathcal{M}\) is given by the degree on the fiber

\[\left.f_\ast\alpha\right\vert_y=\deg\bigl(\alpha\vert_{\mathcal{C}_y}\cap[\mathcal{C}_y]\bigr)\]

In particular,

\[f_\ast1=0\]

Also, when \(X\) is a smooth projective variety and \(h:\mathcal{C}\rightarrow X\) represents the same curve class \(\beta\in A_1(X)_\mathbb{Q}\) on every geometric fiber, if \(D\in A^1(X)_\mathbb{Q}\), then

\[f_\ast h^\ast D=(D\cdot\beta)1\]

Proof

Applying refined Gysin base change étale-locally, we obtain

\[\left.f_\ast\alpha\right\vert_y=(f_y)_\ast\bigl(\alpha\vert_{\mathcal{C}_y}\cap[\mathcal{C}_y]\bigr)\]

Here, since \(f\) is flat, the refined Gysin pullback to the fiber gives the fundamental class \([\mathcal{C}_y]\). Since the right-hand side is the pushforward of a zero-cycle by \(f_y:\mathcal{C}_y\rightarrow\Spec\mathbb{C}\), we have

\[\left.f_\ast\alpha\right\vert_y=\deg\bigl(\alpha\vert_{\mathcal{C}_y}\cap[\mathcal{C}_y]\bigr)\]

On the other hand, since the relative dimension is \(1\), we have \(f_\ast1\in A^{-1}(\mathcal{M})_\mathbb{Q}=0\). Moreover, since the degree of \(h^\ast D\) on each fiber is \(D\cdot\beta\), we obtain

\[f_\ast h^\ast D=(D\cdot\beta)1\]

Since \(f\) is representable, the geometric fiber \(\mathcal{C}_y\) is an algebraic space. In particular, in the case arising in the moduli of curves, it is a proper nodal curve. Therefore, no new separate denominator coming from stack stabilizers appears in the fiber degree above. In the case of a reducible fiber, the fundamental cycle \([\mathcal{C}_y]\) counts each irreducible component with its scheme-theoretic multiplicity.

Product Spaces and Diagonal Intersections

We now conclude this post by introducing several useful formulas. Intersections given independently on two stacks can be treated together on their product. For smooth proper Deligne-Mumford stacks \(\mathcal{X},\mathcal{Y}\) and top-degree classes \(\alpha\in A^{\dim\mathcal{X}}(\mathcal{X})_\mathbb{Q}\), \(\beta\in A^{\dim\mathcal{Y}}(\mathcal{Y})_\mathbb{Q}\), from the external product and the multiplicativity of the degree, we obtain

\[\int_{\mathcal{X}\times\mathcal{Y}}\pr_\mathcal{X}^\ast\alpha\cdot\pr_\mathcal{Y}^\ast\beta=\left(\int_\mathcal{X}\alpha\right)\left(\int_\mathcal{Y}\beta\right)\]

That is, the integral for two mutually independent conditions separates on the product into the product of each integral.

While we use products when simply considering two stacks together as above, to compare two morphisms over the same base we must use the fiber product. To this end, suppose we are given morphisms

\[u:\mathcal{M}\rightarrow X,\qquad v:\mathcal{N}\rightarrow X\]

Then the fiber product

\[\mathcal{Z}=\mathcal{M}\times_X\mathcal{N}\]

represents the locus where the images of the two morphisms agree in \(X\), and this fiber product can be expressed as an intersection inside the ordinary product using the diagonal. For this, considering

\[(u,v):\mathcal{M}\times\mathcal{N}\longrightarrow X\times X\]

we have

\[\mathcal{M}\times_X\mathcal{N}=(\mathcal{M}\times\mathcal{N})\times_{X\times X}X\]

where \(X\rightarrow X\times X\) is the diagonal \(\Delta_X\). That is, intuitively, forming the fiber product can be viewed as intersecting \(\mathcal{M}\times\mathcal{N}\) by base change with the diagonal \(\Delta_X\), and we have already defined the refined Gysin pullback earlier for use in precisely this situation. That is, since the diagonal \(\Delta_X\) is a closed regular embedding of codimension \(d\), we can apply the refined Gysin pullback defined earlier, through which we can define

\[[\mathcal{Z}]_\Delta:=\Delta_X^!\bigl([\mathcal{M}]\times[\mathcal{N}]\bigr)\in A_{\dim\mathcal{M}+\dim\mathcal{N}-d}(\mathcal{Z})_\mathbb{Q}\]

We call this the refined intersection class of \(\mathcal{Z}\).

Proposition 9 (Diagonal intersection formula) In the situation above, let \(j:\mathcal{Z}\rightarrow\mathcal{M}\times\mathcal{N}\) be the natural closed immersion. Then

\[j_\ast[\mathcal{Z}]_\Delta=(u,v)^\ast[\Delta_X]\cap\bigl([\mathcal{M}]\times[\mathcal{N}]\bigr)\]

holds. Therefore, for \(\eta\in A^\ast(\mathcal{M}\times\mathcal{N})_\mathbb{Q}\) of appropriate codimension,

\[\deg\bigl(j^\ast\eta\cap[\mathcal{Z}]_\Delta\bigr)=\int_{\mathcal{M}\times\mathcal{N}}\eta\cdot(u,v)^\ast[\Delta_X]\]

holds.

As for the proof of this, the first identity holds by the compatibility of the refined Gysin pullback and proper pushforward, and the second identity follows immediately by applying the projection formula. That is, the intersection on the fiber product \(\mathcal{Z}\) can be converted into a computation of multiplying by an additional diagonal class on the product \(\mathcal{M}\times\mathcal{N}\).

Above, by forming the fiber product along the diagonal \(\Delta_X\hookrightarrow X\times X\), we expressed the condition that the values of the two morphisms agree as an intersection. In a similar way, we can apply this method to the condition of passing through a subspace of \(X\).

Suppose that a smooth Cartier divisor \(i_D:D\hookrightarrow X\) and a morphism \(u:\mathcal{M}\rightarrow X\) are given. The fiber product

\[\mathcal{Z}_D:=\mathcal{M}\times_XD\]

represents the locus of points in \(\mathcal{M}\) whose image lies on \(D\). That is, \(\mathcal{Z}_D\) is the locus obtained by requiring the geometric condition

\[u(m)\in D\]

to hold.

Now, to apply the method above, let \(j_D:\mathcal{Z}_D\rightarrow\mathcal{M}\) be the natural closed immersion. Since \(D\hookrightarrow X\) is a regular embedding of codimension \(1\), by the refined Gysin pullback the refined intersection class

\[[\mathcal{Z}_D]_D:=i_D^![\mathcal{M}]\]

is defined. Pushing this forward to \(\mathcal{M}\), we obtain

\[j_{D\ast}[\mathcal{Z}_D]_D=u^\ast[D]\cap[\mathcal{M}]\]

and since the class of a Cartier divisor is \([D]=c_1(\mathcal{O}_X(D))\), this can be written as

\[j_{D\ast}[\mathcal{Z}_D]_D=u^\ast c_1(\mathcal{O}_X(D))\cap[\mathcal{M}]\]

as well.

Therefore, for an appropriate class \(\alpha\in A^\ast(\mathcal{M})_\mathbb{Q}\),

\[\deg\bigl(j_D^\ast\alpha\cap[\mathcal{Z}_D]_D\bigr)=\int_\mathcal{M}\alpha\cdot u^\ast[D]\]

holds. That is, instead of dealing with the geometric condition \(u(m)\in D\) directly on the fiber product, we can compute it by converting it into an intersection with the divisor class \(u^\ast[D]\) on \(\mathcal{M}\).

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References

[Vis] A. Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989), 613–670.

[Kre] A. Kresch, Cycle construction for Artin stacks, Invent. Math. 138 (1999), 495–536.