2.1 Proof of Theorem 3

Recall that the tautological rings $(R^\star (\overline {\mathcal {M}}_{g,n}))_{g,n}$ are defined as the smallest system of $\mathbb {Q}$-subalgebras with unit of the Chow rings $(\mathsf {CH}^\star (\overline {\mathcal {M}}_{g,n}))_{g,n}$ closed under push-forwards by gluing and forgetful maps (see [Reference Faber and PandharipandeFP00b, Reference PandharipandePan18] for more details). The tautological subring $\mathsf {RH}^\star (\overline {\mathcal {M}}_{g,n})$ is defined as the image of the cycle map

\[ \mathsf{R}^\star(\overline{\mathcal{M}}_{g,n}) \twoheadrightarrow \mathsf{RH}^\star(\overline{\mathcal{M}}_{g,n}) \subset \mathsf{H}^{2\star}(\overline{\mathcal{M}}_{g,n}). \]

We will use the complex degree grading for $\mathsf {RH}^\star$ and the real degree grading (as usual) for $\mathsf {H}^\star$. Let

\[ \mathsf{divRH}^\star(\overline{\mathcal{M}}_{g,n}) \subset \mathsf{RH}^{\star}(\overline{\mathcal{M}}_{g,n})\quad {\text{and}}\quad \mathsf{divH}^\star(\overline{\mathcal{M}}_{g,n}) \subset \mathsf{H}^{2\star}(\overline{\mathcal{M}}_{g,n}) \]

be the subrings generated respectively by $\mathsf {RH}^1(\overline {\mathcal {M}}_{g,n})$ and $\mathsf {H}^2(\overline {\mathcal {M}}_{g,n})$. Since

\[ \mathsf{RH}^1(\overline{\mathcal{M}}_{g,n}) = \mathsf{H}^2(\overline{\mathcal{M}}_{g,n}), \]

by [Reference Arbarello and CornalbaAC98, Theorem 2.2] we have

(4)\begin{equation} \mathsf{divRH}^\star(\overline{\mathcal{M}}_{g,n}) = \mathsf{divH}^{2\star}(\overline{\mathcal{M}}_{g,n}). \end{equation}

We will use the complex degree grading for both $\mathsf {divRH}^\star$ and $\mathsf {divH}^\star$. Since

\[ \mathsf{CH}^1(\overline{\mathcal{M}}_{g,n}) \stackrel{\sim}= \mathsf{H}^2(\overline{\mathcal{M}}_{g,n}) \]

via the cycle class map, we obtain a surjection

\[ \mathsf{divCH}^\star(\overline{\mathcal{M}}_{g,n})\twoheadrightarrow \mathsf{divH}^{\star}(\overline{\mathcal{M}}_{g,n})\subset \mathsf{H}^{2\star}(\overline{\mathcal{M}}_{g,n}). \]

The following stronger result implies Theorem 3.

Theorem 3/Cohomology. For all $g\geq 3$, we have $\lambda _g\notin \mathsf {divH}^\star (\overline {\mathcal {M}}_g)$.

Proof. For $g=3$, we have complete control of the tautological rings in Chow and cohomology since the intersection pairing to $\mathsf {R}_0(\overline {\mathcal {M}}_g) \stackrel {\sim }= \mathbb {Q}$ is non-degenerate for tautological classes (see [Reference FaberFab90]). In particular,

\[ \mathsf{R}^\star(\overline{\mathcal{M}}_3) \stackrel{\sim}= \mathsf{RH}^\star(\overline{\mathcal{M}}_3). \]

In degree 3,

\[ \mathsf{divRH}^3(\overline{\mathcal{M}}_3) \subset \mathsf{RH}^3(\overline{\mathcal{M}}_3) \]

is a nine-dimensional subspace of a 10-dimensional space. Explicit calculations with the Sage program admcycles [Reference Delecroix, Schmitt and van ZelmDSvZ21] show that $\lambda _3 \notin \mathsf {divRH}^3(\overline {\mathcal {M}}_3)$. We conclude that $\lambda _3 \notin \mathsf {divH}^\star (\overline {\mathcal {M}}_3)$ by (4).

Adding one marked point, we can consider the case of $\overline {\mathcal {M}}_{3,1}$. Again it is known that all (even) cohomology classes on $\overline {\mathcal {M}}_{3,1}$ are tautological (see [Reference Schmitt and van ZelmSvZ20, § 5.1]). Thus, again by Poincaré duality, the intersection pairing on $\mathsf {RH}^*(\overline {\mathcal {M}}_{3,1})$ is perfect and hence we can completely identify these groups in terms of generators and relations. One finds that

\[ \mathsf{divRH}^3(\overline{\mathcal{M}}_{3,1}) \subset \mathsf{RH}^3(\overline{\mathcal{M}}_{3,1}) \]

is a 28-dimensional subspace of a 29-dimensional space. But remarkably, a calculation by admcycles shows

\[ \lambda_3 \in \mathsf{divRH}^3(\overline{\mathcal{M}}_{3,1}). \]

The containment appears miraculous. Is there a geometric explanation?

The tautological ring $\mathsf {RH}^*(\overline {\mathcal {M}}_{4,1})$ is also completely under control in codimension 4:

\[ \mathsf{divRH}^4(\overline{\mathcal{M}}_{4,1}) \subset \mathsf{RH}^4(\overline{\mathcal{M}}_{4,1}) \]

is a 103-dimensional subspace of a 191-dimensional space. An admcycles calculation shows that

(5)\begin{equation} \lambda_4 \notin \mathsf{divRH}^4(\overline{\mathcal{M}}_{4,1}). \end{equation}

Result (5) implies $\lambda _4 \notin \mathsf {divRH}^4(\overline {\mathcal {M}}_{4})$ by a pull-back argument and

\[ \lambda_4 \notin \mathsf{divH}^\star(\overline{\mathcal{M}}_{4}) \]

since divisor classes are tautological.

For $g\geq 5$, a boundary restriction argument is pursued. Suppose, for contradiction, that

(6)\begin{equation} \lambda_g \in \mathsf{divH}^g(\overline{\mathcal{M}}_g). \end{equation}

Then, by pull-back, we have

(7)\begin{equation} \lambda_g \in \mathsf{divH}^g(\overline{\mathcal{M}}_{g,1}). \end{equation}

Consider the standard boundary inclusion

\[ \delta: \overline{\mathcal{M}}_{g-1,1} \times \overline{\mathcal{M}}_{1,2} \rightarrow\overline{\mathcal{M}}_{g,1}. \]

As usual, we have

(8)\begin{equation} \delta^*(\lambda_g) = \lambda_{g-1} \otimes \lambda_1. \end{equation}

Then (7) implies

(9)\begin{equation} \lambda_{g-1} \otimes \lambda_1 \in \mathsf{divH}^g( \overline{\mathcal{M}}_{g-1,1} \times \overline{\mathcal{M}}_{1,2}). \end{equation}

Since $\mathsf {H}^1(\overline {\mathcal {M}}_{g-1,1})$ and $\mathsf {H}^1(\overline {\mathcal {M}}_{1,2})$ both vanish,

\[ \mathsf{divH}^\star( \overline{\mathcal{M}}_{g-1,1} \times \overline{\mathcal{M}}_{1,2} ) = \mathsf{divH}^\star( \overline{\mathcal{M}}_{g-1,1}) \otimes \mathsf{divH}^*(\overline{\mathcal{M}}_{1,2}). \]

We can therefore write $\mathsf {divH}^g( \overline {\mathcal {M}}_{g-1,1} \times \overline {\mathcal {M}}_{1,2} )$ as

(10)\begin{align} \mathsf{divH}^{g}(\overline{\mathcal{M}}_{g-1,1})&\otimes \mathsf{divH}^{0}(\overline{\mathcal{M}}_{1,2}) \nonumber\\ \oplus\quad \mathsf{divH}^{g-1}(\overline{\mathcal{M}}_{g-1,1}) &\otimes \mathsf{divH}^{1}(\overline{\mathcal{M}}_{1,2}) \nonumber\\ \oplus\quad \mathsf{divH}^{g-2}(\overline{\mathcal{M}}_{g-1,1}) &\otimes \mathsf{divH}^{2}(\overline{\mathcal{M}}_{1,2}). \end{align}

Since by (8) the degree of $\delta ^*(\lambda _g)$ splits as $(g-1)+1$ on the two factors, we conclude that

\begin{align*} \lambda_{g-1}\otimes \lambda_1 &\in \mathsf{divH}^{g-1} (\overline{\mathcal{M}}_{g-1,1}) \otimes \mathsf{divH}^{1} (\overline{\mathcal{M}}_{1,2})\\ \implies \lambda_{g-1} &\in \mathsf{divH}^{g-1}(\overline{\mathcal{M}}_{g-1,1}), \end{align*}

using that $\lambda _1 \neq 0 \in \mathsf {divH}^{1}(\overline {\mathcal {M}}_{1,2})$. By descending induction, we contradict (5). Therefore (7) and hence also (6) must be false.

2.2 With marked points

The proof of Theorem 3 in cohomology shows that

(11)\begin{equation} \lambda_{g} \notin \mathsf{divH}^{g}(\overline{\mathcal{M}}_{g,1}) \end{equation}

for $g\geq 4$. By using (11) as a starting point, we can study

\[ \lambda_{g} \in \mathsf{divH}^{g}(\overline{\mathcal{M}}_{g,n}) \]

for $g\geq 4$ and $n\geq 2$ using the boundary restrictions

\[ \widehat{\delta}: \overline{\mathcal{M}}_{g,n-1} \times \overline{\mathcal{M}}_{0,3} \rightarrow \overline{\mathcal{M}}_{g,n}. \]

The argument used in the proof then easily yields the following statement with markings.

Theorem 3/Markings. For all $g\geq 4$ and $n\geq 0$, we have

\[ \lambda_g\notin \mathsf{divH}^\star(\overline{\mathcal{M}}_{g,n}). \]

2.3 Proof of Theorem 4

Define the subalgebra of tautological classes

\[ \mathsf{RH}^\star_{\leq k}(\overline{\mathcal{M}}_{g,n}) \subset \mathsf{RH}^\star(\overline{\mathcal{M}}_{g,n}) \]

generated by classes of complex degrees less than or equal to $k$. Since all divisors are tautological,

\[ \mathsf{divRH}^\star(\overline{\mathcal{M}}_{g,n}) = \mathsf{RH}^\star_{\leq 1}(\overline{\mathcal{M}}_{g,n}). \]

The arguments in §§ 2.1 and 2.2 naturally generalize to address the following question: when does

\[ \lambda_{g-r} \in \mathsf{RH}^{g-r}_{\leq k}(\overline{\mathcal{M}}_{g,n}) \]

hold?

A crucial case of the question (from the point of view of boundary restriction arguments) is for $n=1$. Let $\mathsf {Q}_g(r,k)$ be the statement

\[ \lambda_{g-r} \notin \mathsf{RH}^{g-r}_{\leq k}(\overline{\mathcal{M}}_{g,1}), \]

which may be true or false.

For example, $\mathsf {Q}_g(r,g-r)$ is false essentially by definition. In fact,

\[ \mathsf{Q}_g(s,g-r)\ {\text{is false for all $s\geq r$}} \]

for the same reason. In fact, depending on the parity of $g-r$, it is also false for $s$ slightly below $r$:

\[ \mathsf{Q}_g(r-1,g-r)\ {\text{is false whenever $g-r$ is odd}}. \]

To see this, note that the even Chern character $\mathrm {ch}_{g-(r-1)}(\mathbb {E}_g)$ vanishes by [Reference MumfordMum83, Corollary (5.3)]. Expressing it in terms of Chern classes $\lambda _i = c_i(\mathbb {E}_g)$ using Newton's identities, we have

\[ 0 = \mathrm{ch}_{g-(r-1)}(\mathbb{E}_g) = \frac{(-1)^{g-r+1}}{(g-r+1)!} \lambda_{g-r+1} + \big(\text{polynomial in }\lambda_1, \ldots, \lambda_{g-r} \big). \]

This proves that $\lambda _{g-r+1}$ can be written in terms of tautological classes of degrees $1, \ldots, g-r$, showing $\mathsf {Q}_g(r-1,g-r)$ to be false.

The boundary arguments used in §§ 2.1 and 2.2 yield the following two results.

Proposition 9 If $\mathsf {Q}_g(r,k)$ is true, then

\[ \lambda_{g-r} \notin \mathsf{RH}^{g-r}_{\leq k}(\overline{\mathcal{M}}_{g,n}) \]

for all $n\geq 0$.

Since the $k=1$ case has already been analyzed, we now consider $k= 2$. The first relevant admcycles calculation is

\[ \lambda_3 \notin \mathsf{RH}^{3}_{\leq 2}(\overline{\mathcal{M}}_{4,1}), \]

so $\mathsf {Q}_4(1,2)$ is true. The corresponding subspace here is of dimension 91 inside a 93-dimensional space. As a consequence of Propositions 8 and 9, we obtain the following result.

Proposition 10 For all $g\geq 4$ and $n\geq 0$, we have

\[ \lambda_{g-1} \notin \mathsf{RH}^{g-1}_{\leq 2}(\overline{\mathcal{M}}_{g,n}). \]

A much more complicated admcycles calculation shows that

\[ \lambda_5 \notin \mathsf{RH}^{5}_{\leq 2}(\overline{\mathcal{M}}_{5,1}), \]

so $\mathsf {Q}_5(0,2)$ is true. The corresponding subspace here is of dimension 1314 inside a 1371-dimensional space. As a consequence of Propositions 8 and 9, we find that

(12)\begin{equation} \lambda_{g} \notin \mathsf{RH}^{g}_{\leq 2}(\overline{\mathcal{M}}_{g,n}) \end{equation}

for all $g\geq 5$ and $n\geq 0$. For $g\geq 7$, the equality

\[ \mathsf{RH}^2(\overline{\mathcal{M}}_g) = \mathsf{H}^4(\overline{\mathcal{M}}_g) \]

is shown by combining results of Edidin [Reference EdidinEdi92] and Boldsen [Reference BoldsenBol12]. We provide a summary of the argument in Appendix A. For $g\geq 7$, the cycle map

\[ \mathsf{CH}_{\leq 2}^\star(\overline{\mathcal{M}}_g) \rightarrow \mathsf{H}^{2\star}(\overline{\mathcal{M}}_g) \]

therefore factors through $\mathsf {RH}_{\leq 2}^\star (\overline {\mathcal {M}}_g)$. Then the non-containment (12) completes the proof of Theorem 4.

2.4 Cases of Pixton's conjecture (Proposition 5)

For the proofs of Theorem 3 and 4, dimensions and bases of the following graded parts of tautological rings are required:

\begin{align*} & \mathsf{RH}^4(\overline{\mathcal{M}}_{4,1}),\quad \text{dim}_{\mathbb{Q}} = 191, \\ & \mathsf{RH}^5(\overline{\mathcal{M}}_{5,1}),\quad \text{dim}_{\mathbb{Q}} = 1314. \end{align*}

These cases can be analyzed (via admcycles) since the dual pairings are found to have kernels exactly spanned by Pixton's relations. A discussion of the admcycles calculation is presented in Appendix B.

Pixton has conjectured that his relations always provide all tautological relations. Dual pairings are known to be insufficient to prove Pixton's conjecture in all cases; see [Reference PandharipandePan18, Reference Pandharipande, Pixton and ZvonkinePPZ15] for a more complete discussion.

3.1 Definitions

Let $(X,D)$ be a non-singular varietyFootnote ¹⁰ $X$ with a normal crossings divisor

\[ D = D_1\cup \cdots \cup D_\ell\subset X \]

with $\ell$ irreducible components. The divisor $D\subset X$ is called the logarithmic boundary. An open stratum

\[ S\subset X \]

is an irreducible quasi-projective subvariety satisfying two properties.

1. (i) $S$ is étale locally the transverse intersections of the branches of the $D_i$ which meet $S$.

2. (ii) $S$ is maximal with respect to (i).

The set $U=X{\setminus} D$ is an open stratum. Every open stratum is non-singular. A closed stratum is the closure of an open stratum.

If all $D_i$ are non-singular and all intersections

\[ D_{i_1} \cap \cdots \cap D_{i_k} \]

are irreducible and non-empty, then there are exactly $2^\ell$ open strata.

Our main interest will be in the case $(\overline {\mathcal {M}}_{g,n}, \partial \overline {\mathcal {M}}_{g,n})$ where the normal crossings divisors have self-intersections. The open strata defined above for $(\overline {\mathcal {M}}_{g,n}, \partial \overline {\mathcal {M}}_{g,n})$ are the same as the usual open strata of the moduli space of stable curves.

An open stratum $S\subset X$ is simple if the closure

\[ \overline{S} \subset X \]

is non-singular. A simple blow-up of $(X,D)$ is a blow-up of $X$ along the closure $\overline {S}\subset X$ of a simple stratum. Let

(13)\begin{equation} \widetilde{X} \rightarrow X \end{equation}

be a simple blow-up along $\overline {S}$. Let

\[ \widetilde{D}= \widetilde{D}_1 \cup \cdots \widetilde{D}_\ell \cup E \subset \widetilde{X} \]

be the union of the strict transforms $\widetilde {D}_i$ of $D_i$ along with the exceptional divisor $E$ of the blow-up (13). Then $(\widetilde {X},\widetilde {D})$ is also a non-singular variety with a normal crossings divisor. An iterated blow-up

\[ (\widehat{X},\widehat{D}) \rightarrow (X,D) \]

is a finite sequence of simple blow-ups of varieties with normal crossings divisors.Footnote ¹¹

The log Chow group of $(X,D)$ is defined as a colimit over all iterated blow-ups,

\[ \mathsf{logCH}^*(X,D) = \varinjlim_{Y \in \mathsf{logB}(X,D)} \mathsf{CH}^*(Y). \]

Here, $\mathsf {logB}(X,D)$ is the category of iterated blow-ups of $(X,D)$: objects in $\mathsf {logB}(X,D)$ are iterated blow-ups of $(X,D)$ and morphisms in $\mathsf {logB}(X,D)$ are iterated blow-ups.

Since $(X,D)$ is the trivial iterated blow-up of itself, there is canonical algebra homomorphism

\[ \mathsf{CH}^\star(X) \rightarrow \mathsf{logCH}^\star(X,D) \]

which is injective (since an inverse map of $\mathbb {Q}$-vectors spaces is obtained by proper push-forward). We therefore view $\mathsf {CH}^\star (X)$ as a subalgebra of $\mathsf {logCH}^\star (X,D)$. Every Chow class on $X$ canonically determines a log Chow class for $(X,D)$.

3.2 Calculation in genus 2

We will prove Proposition 7: there does not exist a class $\mathsf {T}\in \mathsf {logCH}^1(\overline {\mathcal {M}}_2)$ satisfying

\[ \mathsf{T}|_{\mathcal{M}^{\mathsf{ct}}_2}=0\quad \textit{and}\quad \lambda_2 = \frac{\mathsf{T}^2}{2!} \in \mathsf{logCH}^2(\overline{\mathcal{M}}_2). \]

Proof. Denote by $\pi _* : \mathsf {logCH}^\star (\overline {\mathcal {M}}_{2}) \to \mathsf {CH}^\star (\overline {\mathcal {M}}_{2})$ the push-forward from log Chow to ordinary Chow. We will prove a stronger claim: there does not exist a class $\mathsf {T}\in \mathsf {logCH}^1(\overline {\mathcal {M}}_{2})$ satisfying

(14)\begin{equation} \mathsf{T}|_{\mathcal{M}^{\mathsf{ct}}_2}=0\quad \textit{and}\quad \pi_* \bigg(\lambda_2 - \frac{\mathsf{T}^2}{2!}\bigg) = 0 \in \mathsf{CH}^2(\overline{\mathcal{M}}_{2}). \end{equation}

Denote by $U_2 \subseteq \overline {\mathcal {M}}_{2}$ the open subset obtained by removing all closed strata of codimension at least $3$. By the excision exact sequence of Chow groups, we have

\[ \mathsf{CH}^2(U_2) \cong \mathsf{CH}^2(\overline{\mathcal{M}}_{2}) \]

and thus we can verify the stronger claim by working over $U_2$.

The open set $U_2$ has open strata of codimension 1 and 2. Since blow-ups along codimension 1 strata do not change $U_2$, the only simple blow-ups

\[ U_2'\rightarrow U_2 \]

are along codimension 2 open strata (all of which are special in $U_2$). Since the codimension 2 open strata of $U_2$ do not intersect (or self-intersect), we obtain a $\mathbb {P}^1$-bundle as an exceptional divisor which contains $0$- and $\infty$-sectionsFootnote ¹² which are codimension 2 strata of $U'_2$. The iterated blow-ups

\[ \widehat{U}_2 \rightarrow U_2 \]

are then simply towers of blow-ups of these codimension 2 toric strata in successive exceptional divisors.

Assume $\mathsf {T} \in \mathsf {logCH}^1(U_{2})$ satisfies the conditions (14). Since $\mathsf {T}$ restricts to zero over the compact type locus, $\mathsf {T}$ can be represented as

\[ \mathsf{T} \in \mathsf{CH}^1(\widehat{U}_{2}) \]

on an iterated blow-up

\[ \widehat{U}_{2}\rightarrow U_{2} \]

with all blow-up centers living over strata in the complement of the compact type locus.

There are a single codimension 1 stratum $\Delta _0\subset U_2$ and two codimension 2 strata $B,C \subset U_2$ contained in the complement of the compact type locus (see Figure 1).

Figure 1. The stable graphs associated to the codimension $2$ boundary strata $B$ and $C$ contained in $U_2$.

Denote by $E_B^1, \ldots, E_B^\ell$ and $E_C^1, \ldots, E_C^m$ the exceptional divisors of blow-ups with centers lying over $B$ and $C$. Then $\mathsf {T}$ has a representationFootnote ¹³

\[ \mathsf{T} = a \cdot [\Delta_0] + \sum_{i=1}^\ell b_i [E_B^i] + \sum_{j=1}^m c_j [E_C^j]. \]

After taking the square and pushing forward, we claim that

(15)\begin{equation} \pi_* (\mathsf{T}^2) = x \cdot [\Delta_0]^2 + y \cdot [B] + z \cdot [C], \end{equation}

with $x, y, z \in \mathbb {Q}$ satisfying

\[ x = a^2 \geq 0\quad {\text{and}}\quad z \leq 0. \]

The claim follows from the following observations.

• • In $\mathsf {T}^2$, all mixed terms $[\Delta _0] \cdot [E_B^i]$ and $[\Delta _0] \cdot [E_C^j]$ vanish after push-forward to $U_{2}$, since

  \[ \pi_*([\Delta_0] \cdot [E_B^i]) = [\Delta_0] \cdot \pi_* [E_B^i] =[\Delta_0] \cdot 0 = 0. \]

• • Similarly, since $B \cap C = \emptyset$ in $U_2$ (as we have removed the codimension $3$ stratum of $\overline {\mathcal {M}}_2$), we have $[E_B^i] \cdot [E_C^j] = 0$.

• • Denote by $\textbf {M} \in \mathsf {Mat}_{\mathbb {Q}, m \times m}$ the matrix defined by

  \[ \pi_*([E_C^{j_1}] \cdot [E_C^{j_2}]) = \textbf{M}_{j_1, j_2} [C]. \]
  A basic fact is that $\textbf {M}$ is negative definite (see [Reference MumfordMum61, § 1]). Therefore, for $\textbf {b} = (b_i)_{i=1}^\ell$, we have
  \[ \pi_* \bigg(\sum_{j=1}^m b_j [E_C^j]\bigg)^2 = \underbrace{( \textbf{b}^\top \textbf{M} \textbf{b} )}_{= z \leq 0} [C]. \]

• • The push-forward

  \[ \pi_* \bigg(\sum_{i=1}^\ell b_i [E_B^i]\bigg)^2 \]
  is supported on $B$ and thus is a multiple $y \cdot [B]$ of the fundamental class of $B$.

After substituting (15) in the second condition of (14), we conclude the existence of $x, y, z \in \mathbb {Q}$ with $x \geq 0$ and $z \leq 0$ satisfying

(16)\begin{equation} x \cdot [\Delta_0]^2 + y \cdot [B] + z \cdot [C] = 2 \lambda_2 \in \mathsf{CH}^2(U_2). \end{equation}

Using admcycles (see Appendix B.3), we can explicitly identify all classes in (16) in

\[ \mathsf{CH}^2(U_2) \cong \mathbb{Q}^2. \]

The corresponding affine linear equation has the solution space

\[ x = z - \tfrac{1}{120},\quad y = -\tfrac{5}{24}\cdot z + \tfrac{11}{2880}. \]

But for $z \leq 0$, we have

\[ z - \tfrac{1}{120} < 0, \]

which contradicts the assumption that $x \geq 0$. Therefore, there cannot exist a class

\[ \mathsf{T} \in\mathsf{logCH}^1(U_{2}) \]

satisfying conditions (14).

4.1 Overview

The definitions of § 3 are natural from the perspective of logarithmic geometry. The choice of the divisor $D$ on $X$ can be seen as the choice of a log structure on $X$. We briefly recall the relevant definitions and constructions of logarithmic geometry.

4.2 Definitions

A log structure on a scheme $X$ is a sheaf of monoids $M_X$ on the étale site of $X$ together with a homomorphismFootnote ¹⁴

\[ {\rm exp}:M_X \rightarrow \mathcal{O}_X \]

which induces an isomorphism ${\rm exp}^{-1}(\mathcal {O}_X^*) \cong \mathcal {O}_X^*$ on units.

• • Morphisms of log schemes $(X,M_X) \rightarrow (Y,M_Y)$ are morphisms of schemes

  \[ f:X \to Y \]
  together with homomorphisms of sheaves of monoids $f^{-1}M_Y \to M_X$ which are compatible with the structure map $f^{-1}\mathcal {O}_Y \to \mathcal {O}_X$ in the obvious sense.

• • Log structures can be pulled back. Given a morphism of schemes

  \[ f: X \to Y \]
  and a log structure $M_Y$ on $Y$, there is an induced log structure $f^*M_Y$ on $X$, generated by $f^{-1}M_Y$ and the units $\mathcal {O}_X^*$.

The basics of log schemes can be found in Kato's original article on the subject [Reference KatoKat89].

The category of log schemes is, in practice, too large for geometric study. It is therefore common to work in smaller categories by requiring additional properties to hold. For our purposes, we will work only with in the category of fine and saturated log schemes, usually termed f.s. log schemes. The prototype of such a log scheme is

\[ A_P = {\rm Spec} (k[P]), \]

the spectrum of the algebra generated by a fine and saturated monoid $P$: a finitely generated monoid $P$ which injects into its Grothendieck group $P^{\rm gp}$ and which is saturated there,

\[ nx \in P\quad \text{for}\quad n\in \mathbb{N},\quad x \in P^{\rm gp}\implies\ x \in P. \]

The sheaf $M_{A_P}$ here is the subsheaf of $\mathcal {O}_{A_P}$ generated by $P$ and the units of $\mathcal {O}_{A_P}$.

All of the log schemes which arise for us will be comparable to $A_P$ on the level of log structures. More precisely, we require our log schemes $X$ to admit the following local charts: for each $x \in X$, there must be an étale neighborhood

\[ i:U \to X, \]

an f.s. monoid $P$, and a map $g: U \to A_P$ such that

\[ i^*M_X = g^*M_{A_P}. \]

Since we are always working with f.s. log schemes, the chart $P$ at $x$ can in fact always be chosen to be isomorphic to the characteristic monoidFootnote ¹⁵

\[ \overline{M}_{X,\overline{x}}= M_{X,\overline{x}}/\mathcal{O}_{X,\overline{x}}^* \]

at $x$.

4.3 Normal crossings pairs

Let us now return to the situation of interest for this paper: a pair $(X,D)$ of a non-singular scheme (or Deligne–Mumford stack) with a normal crossings divisor $D\subset X$. The pair $(X,D)$ determines a sheaf $M_X$ on the étale site of $X$ by setting

\[ M_X(p:U \to X) = \{f \in \mathcal{O}_U: f \;{\rm is\, a\, unit\, on\, } p^{-1}(X-D)\} \]

for each étale map $p:U \rightarrow X$. The sheaf of units $\mathcal {O}_X^*$ is a subsheaf of $M_X$. We write

\[ \overline{M}_X = M_X/\mathcal{O}_X^* \]

for the characteristic monoid of $X$. Normal crossings pairs $(X,D)$, with the log structure described above, are precisely the log schemes which are log smooth over the base field ${\rm Spec } k$ with trivial log structure.

When the irreducible components of $D$ do not have self-intersections, the log structure $M_X$ of $(X,D)$ can be defined on the Zariski topology of $X$. The result is a technically simpler theory. The pair $(X,D)$ is then called a toroidal embedding (without self-intersection) in [Reference Kempf, Knudsen, Mumford and Saint-DonatKKMS73]. However, for a general pair $(X,D)$, $M_X$ can only be defined on the étale site of $X$. The general étale case differs from the Zariski case in two key aspects: the irreducible components of $D$ can self-intersect, and the characteristic monoid $\overline {M}_X$, while locally constant on a stratum, can globally acquire monodromy.

The characteristic monoid $\overline {M}_X$ is a constructible sheaf on $X$. The connected components of the loci on which $\overline {M}_X$ is locally constant define a stratification of $X$, which is precisely the stratification of § 3.1. Indeed, for a geometric point $x \in X$,

\[ \overline{M}_{X,\overline{x}} = \mathbb{N}^r \]

where $r$ is the number of branches (in the étale topology) of $D$ that contain $x$.

A combinatorial space can be built from the information contained in $\overline {M}_{X}$. There are two basic approaches. The first, which is more geometric and more evidently combinatorial, is to build the cone complex $C(X,D)$ of $(X,D)$. We briefly outline the construction (details can be found in [Reference Cavalieri, Chan, Ulirsch and WiseCCUW20, Reference Abramovich, Chen, Marcus, Ulirsch and WiseACMUW16]).

We begin with the case where $M_X$ is defined Zariski locally on $X$ (when the irreducible components of $D$ do not have self-intersections). Then $C(X,D)$ is a rational polyhedral cone complex (see [Reference Kempf, Knudsen, Mumford and Saint-DonatKKMS73]).

• • For each point $x \in X$, the characteristic monoid $\overline {M}_{X,\overline {x}}$ determines a rational polyhedral cone

  \[ \sigma_{X,x} = {\rm Hom}_{{\rm Monoids}}(\overline{M}_{X,\overline{x}}, \mathbb{R}_{\ge 0}) \]
  together with an integral structure
  \[ N_{X,x} = {\rm Hom}(\overline{M}_{X,\overline{x}}^{\rm gp},\mathbb{Z}). \]

• • When $x$ belongs to a stratum $S \subset X$ and $y$ belongs to the closure $\overline {S}\subset X$, there are canonical inclusions

  \[ \sigma_{X,x} \subset \sigma_{X,y},\quad N_{X,x} \subset N_{X,y}. \]

• • We glue the cones $\sigma _{X,x}$ together with their integral structures to form the complex

  \[ C(X,D) = \varinjlim_{x \in X} (\sigma_{X,x}, \sigma_{X,x} \cap N_{X,x}). \]

• • More effectively, instead of working with all points $x \in X$, we can take the finite set $\{x_S \}$ of the generic points of the strata of $(X,D)$. Then

  \[ C(X,D) = \varinjlim_{x_S}(\sigma_{X,x_S}, \sigma_{X,x_S} \cap N_{X,x_S}). \]
  In other words, $C(X,D)$ is the dual intersection complex of $(X,D)$.

When $M_X$ is defined only on the étale site, we build the cone complex $C(X,D)$ by descent.

• • We find an étale (but not necessarily proper), strict ($f^*M_X = M_Y)$ cover $f: Y \to X$ which is as fine as possible (called atomic or small in the literature): the log structure on $Y$ is defined on the Zariski site of $Y$, and each connected component of $Y$ has a unique closed stratum. Taking a further such cover $V$ of the fiber product $Y \times _X Y$ if necessary, we find a groupoid presentation

  \[ V \rightrightarrows Y \rightarrow X. \]

• • We define

  \[ C(X,D) = \varinjlim [C(V) \rightrightarrows C(Y)] \]
  in the category of stacks (with respect to the topology generated by face inclusions) over cone complexes. The construction is carried out in detail in [Reference Cavalieri, Chan, Ulirsch and WiseCCUW20], where it is also shown that it is independent of the choice of groupoid presentation.

Moreover, $C(X,D)$ is a complex of cones, but no longer a rational polyhedral cone complex. For each point $x \in X$, there is a canonical map

\[ \sigma_{X,x} \rightarrow C(X,D), \]

but the map may no longer be injective. As the étale local branches of the divisor $D$ may be connected globally on $X$, the faces of the cones $\sigma _{X,x}$ may be glued to each other in $C(X,D)$, and they may naturally acquire automorphisms coming from the monodromy of the branches of $D$.

4.4 Artin fans

An equivalent combinatorial space is the Artin fan $\mathcal {A}_X$ of $(X,D)$. The Artin fan is defined by gluing, instead of the dual cones $\sigma _{X,x}$ of $\overline {M}_{X,\overline {x}}$, the quotient stacks

\[ \mathcal{A}_{\overline{M}_{X,\overline{x}}}= [{\rm Spec} (k[\overline{M}_{X,\overline{x}}])/{\rm Spec} (k[\overline{M}_{X,\overline{x}}^{\rm gp}])]. \]

The gluing is exactly the same as for $C(X,D)$ as explained above. When $M_X$ is defined on the Zariski site of $X$,

\[ \mathcal{A}_X = \varinjlim_{x \in X} \mathcal{A}_{\overline{M}_{X,\overline{x}}} = \varinjlim_{x_S} \mathcal{A}_{\overline{M}_{X,\overline{x_S}}}, \]

and when $M_X$ is defined only on the étale site of $X$,

\[ \mathcal{A}_X = \varinjlim [\mathcal{A}_{V} \rightrightarrows \mathcal{A}_Y], \]

for an atomic presentation $\varinjlim [V \rightrightarrows Y] = X$ as before.

The Artin fan $\mathcal {A}_X$ captures exactly the same combinatorial information as the cone complex $C(X,D)$, but is geometrically less intuitive. Nevertheless, the Artin fan has the advantage of coming with a smooth morphism of stacks

\[ \alpha:X \rightarrow \mathcal{A}_X. \]

4.5 Logarithmic modifications

The cone complex $C(X,D)$ encodes an important operation: logarithmic modification of $X$. Logarithmic modifications correspond to subdivisions of $C(X,D)$. A subdivision of $C(X,D)$ is, by definition, a compatible subdivision of all the cones $\sigma _{X,x}$ compatible with the gluing relations. Each cone in the subdivision $\sigma _{X,x}' \rightarrow \sigma _{X,x}$ determines dually a map $\overline {M}_{X,\overline {x}} \rightarrow \overline {M}_{X,\overline {x}}'$, and so a map

\[ [{\rm Spec} (k[\overline{M}'_{X,\overline{x}}])/{\rm Spec} (k[\overline{M}_{X,\overline{x}}^{\rm gp}])] \rightarrow [{\rm Spec} (k[\overline{M}_{X,\overline{x}}])/{\rm Spec} (k[\overline{M}_{X,\overline{x}}^{\rm gp}])]. \]

The compatibility of the subdivisions with respect to the gluing relations in $C(X,D)$ implies that these maps glue to a proper and birational representable map

\[ \mathcal{A}_X' \to \mathcal{A}_X. \]

Then we define

\[ X' =X \times_{\mathcal{A}_X} \mathcal{A}_X'\rightarrow X \]

which is proper, birational, and representable over $X$. Moreover, $X'$ has an induced log structure, and there is a map

\[ \mathcal{A}_{X}' \to \mathcal{A}_{X'} \]

which is proper, Deligne–Mumford type, étale, and bijective.

The map $\mathcal {A}_{X}' \to \mathcal {A}_{X'}$ – called the relative Artin fan of $X'$ over $X$ in the literature – is not necessarily representable, as the various monodromy groups of the strata of $\mathcal {A}_X$ may act non-faithfully on the strata of $\mathcal {A}_{X}'$, whereas the monodromy groups of the strata of $X'$ act faithfully on $\mathcal {A}_{X'}$ by definition. In this way the strata of $\mathcal {A}_{X}'$ become trivial gerbes over the strata of $\mathcal {A}_{X'}$. In a sense, $\mathcal {A}_{X'}$ can be considered as a relative coarse moduli space for $\mathcal {A}_{X}'$.Footnote ¹⁶

Geometrically, subdivisions come in three levels of generality as follows.

• • General subdivisions simply produce proper birational maps $X' \to X$, which are isomorphisms over $X - D$. Such maps are called logarithmic modifications

• • Log blow-ups are a special kind of subdivision. They are the subdivisions of $C(X,D)$ into the domains of linearity of a piecewise linear function on $C(X,D)$, and they correspond to a sheaf of monomial ideals,

  \[ I \subset M_X. \]
  The map $X' \to X$ is then projective and is the normalization of the blow-up of $X$ along the sheaf of ideals ${\rm exp}(I) \subset \mathcal {O}_X$.

• • Star subdivisions along simple strata $S$ correspond to the most basic logarithmic modifications. The strata of $X$ are, by construction, in bijection with the cones of $C(X,D)$. We obtain a subdivision by subdividing $\sigma _{X,x_S}$ along its barycenter (see [Reference Cox, Little and SchenckCLS11, Definition 3.3.13]). A simple blow-up along $\overline {S}$ corresponds precisely to the star subdivision of the cone $\sigma _{X,x_S}$. Further applications of the star subdivision operation are discussed in § 5.3.

Although star subdivisions are the simplest and most basic subdivisions, we need not consider more general subdivisions for our purposes. We are only concerned with statements that are valid over some arbitrarily fine subdivision, and the star subdivisions along simple strata are cofinal in this setting: for each subdivision

\[ C(X,D)' \to C(X,D), \]

there is a further subdivision $C(X,D)'' \to C(X,D)'$ such that the composition $C(X,D)'' \to C(X,D)$ is the composition of star subdivisions along simple strata (see [Reference OdaOda88, Chapter 1.7]). So the reader can restrict attention to simple blow-ups without any loss of generality.

We define a category $\mathsf {logM}(X,D)$ whose objects are log modifications

\[ X' \to X \]

obtained via subdivisions of $C(X,D)$. There is a unique morphism $X'' \to X'$ if and only if $X''$ is a log modification of $X'$. Following [Reference BarrottBar18], we then define

\[ \mathsf{logCH}^\star(X,D) = \varinjlim_{X' \in \mathsf{logM}(X)}\mathsf{CH}^\star(X'). \]

As simple blow-ups are cofinal among log modifications, we have, equivalently,

\[ \mathsf{logCH}^\star(X,D) = \varinjlim_{X'\in \mathsf{logB}(X,D)}\mathsf{CH}^\star(X') \]

as defined in § 3.1.

5.1 Definitions

Let $(X,D)$ be a non-singular variety $X$ with a normal crossings divisor

\[ D = D_1\cup \cdots \cup D_\ell\subset X \]

with $\ell$ irreducible components. Let

\[ \mathsf{divlogCH}^\star(X,D) \subset \mathsf{logCH}^\star(X,D) \]

be the subalgebra generated by the classes of all the components of the associated normal crossings divisors of all iterated blow-ups of $X$.

Let $S\subset X$ be an open stratum of codimension $s$, let $\overline {S} \subset X$ be the closure, and let

\[ \epsilon: \widetilde{S} \rightarrow X \]

be the normalization of $\overline {S}$ equipped with a canonical map $\epsilon$ to $X$. The normalization $\widetilde {S}$ is non-singular and separates the branches of the self-intersections of $\overline {S}$. The map $\epsilon$ is an immersion locally on the source and therefore has a well-defined normal bundle

\[ \mathsf{N}_\epsilon = \epsilon^* T_X / T_{\widetilde{S}} \]

of rank $s$.

An open stratum $S\subset X$ of codimension $s$ is étale locally cut out by $s$ branches of the full divisor $D$. These $s$ branches are partitioned by monodromy orbits over $S$. Each monodromy orbit determines a summand of $\mathsf {N}_\epsilon$. We obtain a canonical splitting of $\mathsf {N}_\epsilon$ corresponding to monodromy orbits

\[ \mathsf{N}_\epsilon = \oplus_{\gamma\in \mathsf{Orb}(S)} \mathsf{N}^\gamma_\epsilon,\quad \mathsf{rank}(\mathsf{N}^\gamma_\epsilon)= |\gamma|, \]

where $\mathsf {Orb}(S)$ is the set of monodromy orbits of the branches of $D$ cutting out $S$, and $|\gamma |$ is the number of branches in the orbit $\gamma$. For polynomials $P_\gamma$ in the Chern classes of $\mathsf {N}^\gamma _\epsilon$, we define

(17)\begin{equation} [S, \{P_\gamma\}_{\gamma\in \mathsf{Orb}(S)}] = \epsilon_*\bigg(\prod_{\gamma\in \mathsf{Orb}(S)} P_\gamma( \mathsf{N}^\gamma_\epsilon)\bigg) \in \mathsf{CH}^\star(X). \end{equation}

We define normally decorated classes by the following more general construction. Let $G$ be the monodromy group of the $s$ branches of $D$ which cut out $S$. Over $\widetilde {S}$, there is a principal $G$-bundle

\[ \mu: \widetilde{P} \rightarrow \widetilde{S} \]

over which the $s$ branches determine $s$ line bundles

(18)\begin{equation} N_1,\ldots, N_s. \end{equation}

The $G$-action on $\widetilde {P}$ permutes the line bundles (18) via the original monodromy representation. Let $P_G$ be any $G$-invariant polynomial in the Chern classes $c_1(N_i)$. Since $P_G(c_1(N_1), \ldots,c_1(N_s))$ is $G$-invariant,

\[ P_G(c_1(N_1), \ldots,c_1(N_s)) \in \mathsf{CH}^\star(\widetilde{S}). \]

We define a normally decorated strata class by

\[ [S, P_G] =\epsilon_*(P_G(c_1(N_1), \ldots,c_1(N_s)))\in \mathsf{CH}^\star(X). \]

Construction (17) is a special case of a normally decorated strata class.

A fundamental result about the log Chow ring of $(X,D)$ is the following inclusion.

We give two proofs of Theorem 11. In § 5.2 we give a very concrete iterated blow-up of $X$ and an explicit computation expressing the normally decorated class as a sum of products of divisors. On the other hand, in Corollary 16 we give a more conceptual explanation based on the study of the Chow group of the Artin fan of the pair $(X,D)$.

5.2 Proof of Theorem 11

Theorem 11 is almost trivial if every irreducible component $D_i$ of $D$ is non-singular. The complexity of the argument occurs only if there are irreducible components with self-intersections.

Proof. Let $S\subset X$ be an open stratum of codimension $s$. The first case to consider is when $S$ is simple. Then the closure

\[ \overline{S} \subset X \]

in non-singular and no normalization is needed,

\[ \epsilon: \overline{S} \rightarrow X. \]

Let $G$ be the monodromy of the $s$ branches of $D$ which cut out $S$. We must prove

\[ [S, P_G] =\epsilon_*(P_G(c_1(N_1), \ldots,c_1(N_s)))\in \mathsf{divlogCH}^\star(X) \]

for every $G$-invariant polynomial $P_G$.

We argue by induction on the degree of $P_G$. The base case is when $P_G$ is of degree 0. We can take $P_G=1$, and we must prove

(19)\begin{equation} [S, 1] =\epsilon_*[S]\in \mathsf{divlogCH}^\star(X,D). \end{equation}

Our argument requires a blow-up construction which we term an explosion.

The explosion of $(X,D)$ along a simple stratum $S$,

(20)\begin{equation} e:\mathsf{E}_S(X,D) \rightarrow X, \end{equation}

is defined by a sequence of blow-ups of $X$. To describe the blow-ups locallyFootnote ¹⁷ near a point $p\in S$, let

\[ B_1,\ldots,B_s \]

be the branches of $D$ cutting out $S$ near $p$.

• • At the zeroth stage, we blow up $S$, the intersection of all $s$ branches $B_1,\ldots,B_s$.

Consider next the strict transform of the intersection of $s-1$ branches. For each choice of $s-1$ branches, the strict transform of the intersection is non-singular of codimension $s-1$ over an open set of $p\in X$. Moreover, the strict transforms of the intersections of different sets of $s-1$ branches are disjoint over an open set of $p\in X$.

• • At the first stage, we blow up all $s$ of these strict transforms of intersections of $s-1$ branches.

Then the strict transforms of the intersections of $s-2$ branches among $B_1,\ldots, B_s$ are non-singular of codimension $s-2$ and disjoint over an open set of $p\in X$.

• • At the second stage, we blow up all $\binom {s}{2}$ of these strict transforms of intersections of $s-2$ branches.

We proceed in the above pattern until we have completed $s-1$ stages.

• • At the $j$th stage, we blow up all $\binom {s}{j}$ strict transforms of intersections of $s-j$ branches.

The explosion (20) is the result after stage $j=s-1$.Footnote ¹⁸ Since the above blow-ups are defined symmetrically with respect to the branches $B_i$, the definition is well defined globally on $X$.

Near $S$, all the prescribed blow-ups are of simple loci, but non-simplicity may occur away from $S$. In order for the explosion to be a sequence of simple blow-ups, some extra blow-ups may be required far from $S$. Since we will only be interested in the geometry near $S$, the blow-ups related to non-simplicity away from $S$ are not important for our argument (and are not included in our notation).

A local study shows the following properties of the explosion

\[ e:\mathsf{E}_S(X,D) \rightarrow X, \]

near $S$.

1. (i) The inverse image $e^{-1}(S) \subset \mathsf {E}_S(X,D)$ is a non-singular irreducible subvariety which we denote by $\mathsf {E}_S(S)$ and call the exceptional divisor of the explosion. We denote the inclusion by

   \[ \iota: \mathsf{E}_S(S) \rightarrow \mathsf{E}_S(X,D). \]

2. (ii) Let $\mathsf {N}_S$ be the rank $s$ normal bundle of $S$ in $X$. The fibers of the projective normal bundle

   (21)\begin{equation} \mathsf{P}(\mathsf{N}_S) \rightarrow S \end{equation}
   have a canonical (unordered) set of $s$ coordinate hyperplanes determined by the $s$ local branches of $D$ cutting out $S$. In the fibers of (21), these relative hyperplanes determine $s$ coordinate points, $\binom {s}{2}$ coordinate lines, $\binom {s}{3}$ coordinate planes, and so on.

3. (iii) The restriction of the explosion morphism to the exceptional divisor

   \[ e_S: \mathsf{E}_S(S) \rightarrow S \]
   is obtained from $\mathsf {P}(\mathsf {N}_S) \rightarrow S$ by first blowing up the coordinate points, and then blowing up the strict transforms of the coordinate lines, and so on. For
   \[ 1\leq j \leq s-1, \]
   the $j$th stage of the construction of the explosion restricts to the blow-up of the strict transform of the $(j-1)$-dimensional coordinate linear spaces of the fibers of (21).

4. (iv) On $\mathsf {E}_S(S)$, we have a distinguished set of divisors

   \[ E_0, E_1, \ldots, E_{s-1} \in \mathsf{CH}^1(\mathsf{E}_S(S)). \]
   Here, $E_0$ is the pull-back to $\mathsf {E}_S(S)$ of
   \[ \mathcal{O}_{\mathsf{P}(\mathsf{N}_S)}(-1) \rightarrow \mathsf{P}(\mathsf{N}_S) \]
   determined by the zeroth stage of the construction of the explosion. Then $E_j \in \mathsf {CH}^1( \mathsf {E}_S(S))$ is the pull-back to $\mathsf {E}_S(S)$ of the exceptional divisor obtained from the blow-up of the strict transform of the $(j-1)$-dimensional coordinate linear spaces in the fibers of (21).

5. (v) Every class of the form

   \[ [\mathsf{E}_S(S)]\cdot \mathsf{F}(E_0,\ldots,E_{s-1}) \in \mathsf{CH}^*(\mathsf{E}_S(X,D)), \]
   where $\mathsf {F}$ is a polynomial, lies in the divisor ring of log Chow,
   \[ [\mathsf{E}_S(S)]\cdot \mathsf{F}(E_0,\ldots,E_{s-1}) \in \mathsf{divlogCH}^*(X,D). \]
   The claim follows from the geometric construction of the explosion. To start, $\mathsf {E}_S(S)$ is a component of the associated normal crossings divisor of $\mathsf {E}_S(X,D)$. For each $0\leq j \leq s-1$, $E_j$ comes from the pull-back of a divisor stratum of the blow-up at the $j$th stage.

To the explosion geometry, we can apply Fulton's excess intersection formula. We start with the zeroth stage:

\[ e_0: X_0 \rightarrow X \]

is the blow-up along $S$, and

\[ e_0^*[S] = [\mathsf{P}(\mathsf{N}_S)] \cdot c_{s-1} \bigg(\frac{\mathsf{N}_S}{ \mathcal{O}_{\mathsf{P}(\mathsf{N}_S)}(-1)}\bigg). \]

When we pull back $e_0^*[S]$ all the way to $\mathsf {E}_S(X,D)$, we obtainFootnote ¹⁹

\[ e^*[S] = [\mathsf{E}_S(S)] \cdot c_{s-1}\bigg(\frac{\mathsf{N}_S}{ \mathcal{O}_{\mathsf{P}(\mathsf{N}_S)}(-1)}\bigg). \]

By definition, we have

\[ c(\mathcal{O}_{\mathsf{P}(\mathsf{N}_S)}(-1)) = 1+ E_0. \]

By property (v) above for the explosion geometry, to prove

(22)\begin{equation} \epsilon_*[S]\in \mathsf{divlogCH}^\star(X,D), \end{equation}

we need only show that

(23)\begin{equation} c_{k}({\mathsf{N}_S}) = \mathsf{F}_k(E_0, \ldots, E_{s-1}) \in \mathsf{CH}^k(\mathsf{E}_S(S)) \end{equation}

for polynomials $\mathsf {F}_k$, $1\leq k \leq s-1$.

Claim (23) is established directly by the following basic formula of the explosion geometry. For $0\leq j\leq s-1$, let

\[ \mathsf{L_j} = \sum_{i=0}^j E_i. \]

Let $\sigma _k$ be the $k$th elementary symmetric polynomial. Then we claim that

(24)\begin{equation} c_{k}({\mathsf{N}_S}) = {\sigma}_k(\mathsf{L}_0, \ldots, \mathsf{L}_{s-1}) \in \mathsf{CH}^k(\mathsf{E}_S(S)). \end{equation}

Once we prove (24), this immediately shows (23) and thus, as explained above, establishes (22). We remind ourselves that (22) represents the base case $P_G=1$ of our inductive proof that $[S, P_G] \in \mathsf {divlogCH}^*(X)$.

Let $\mathsf {T}=(\mathbb {C}^*)^s$ and let $t_i : T \to \mathbb {C}^*$ be the projection to the $i$th factor, which we interpret as the weight of the standard representation of this $i$th factor. To show formula (24), we consider the universal $\mathsf {T}$-equivariant model where $S\subset X$ is

\[ \mathbf{0}\in \mathbb{C}^s \]

and the logarithmic boundary $H\subset \mathbb {C}^s$ is the union of the $s$ coordinate hyperplanes. Then the $\mathsf {T}$-action on

\[ e_{\mathbf{0}}:\mathsf{E}_{\mathbf{0}}(\mathbb{C}^s,H) \rightarrow {\mathbf{0}} \]

has $s!$ isolated $\mathsf {T}$-fixed points naturally indexed by elements of the symmetric group $\Sigma _s$. The weights of the divisors

\[ \mathsf{L}_0, \ldots, \mathsf{L}_{s-1} \]

with their canonical $\mathsf {T}$-equivariant lifts at the $\mathsf {T}$-fixed point $\gamma \in \Sigma _s$ are

\[ t_{\gamma(1)}, t_{\gamma(2)}, t_{\gamma(3)}, \ldots, t_{\gamma(s)} \]

respectively. Formula (24) then follows immediately for the $\mathsf {T}$-equivariant model. The general case of (24) is a formal consequence.

We now will establish the induction step. Let $S\subset X$ be a simple stratum of codimension $s$ with monodromy groupFootnote ²⁰ $G$ of the branches of $D$ cutting out $S$. We must prove

\[ [S, P_G] =\epsilon_*(P_G(c_1(N_1), \ldots,c_1(N_s)))\in \mathsf{divlogCH}^\star(X,D) \]

for every $G$-invariant polynomial $P_G$. By induction, we assume the truth of the statement for polynomials of lower degree.

Let $P_G$ be a $G$-equivariant polynomial in $c_1(N_1), \ldots, c_1(N_s)$ of degree $d>0$. We will prove a stronger property for the induction argument:

\[ \epsilon_*(P_G(c_1(N_1), \ldots,c_1(N_s))) \in \mathsf{divlogCH}^\star(X,D) \]

can be expressed as a linear combination of terms of the form

\[ \widehat{D}_1\widehat{D}_2 \cdots \widehat{D}_d \]

where the $\widehat {D}_i$ are components of the logarithmic boundary of an iterated blow-up of the explosion $\mathsf {E}_S(X,D)$ and $\widehat {D}_1$ lies over

\[ \mathsf{E}_S(S)\subset\mathsf{E}_S(X,D). \]

Our proof of the base of the induction establishes the stronger property.

We can assume $P_G$ is the summationFootnote ²¹ $M_G$ of the $G$-orbit of a degree $d$ monomial $M$,

\[ M_G=\frac{1}{|\mathsf{Stab}(M)|}\sum_{g\in G} g(M). \]

We will study the geometry of the exceptional divisor of the explosion

\[ e_S: \mathsf{E}_S(S) \rightarrow S \]

locally over an analytic open set $U_p \subset S$ of $p\in S$.

Over small enough $U_p$, we can separate all the branches $B_1,\ldots, B_s$ of $D$ which cut out $S$, and we can write

(25)\begin{equation} M = c_1(N_1)^{m_1}\cdots c_1(N_s)^{m_s} = B_1^{m_1} \cdots B_s^{m_s}. \end{equation}

Over $U_p$, we can separate all the exceptional divisors of all the blow-ups in the construction of

\[ \mathsf{E}_S(S) \rightarrow \mathsf{P}(\mathsf{N}_S) \]

explained in (iii) above. There are $2^{s}-2$ such exceptional divisors in bijective correspondence to all the proper non-zero coordinate linear subspaces of the fiber $\mathsf {N}_{S}|_p$ of $\mathsf {N}_S$ at $p$. We denote these $2^{s}-2$ exceptional divisors by $E_\Lambda$, where

\[ \Lambda \subset \mathsf{N}_{S}|_p \]

is a proper coordinate linear space. As before, we denote the pull-back of $\mathcal {O}_{\mathsf {P}(\mathsf {N}_S)}(-1)$ to $\mathsf {E}_S(S)$ by $E_0$.

Via the pull-back formula for $B_i$, we have

(26)\begin{equation} e^*(N_i) = E_0+ \sum_{\Lambda \subset H_i} E_\Lambda \in \mathsf{CH}^1( e^{-1}(U_p)), \end{equation}

where $H_i \subset \mathsf {N}_{S}|_p$ is the hyperplane associated to $B_i$. We now substitute formula (26) into (25) to find that

\[ M\in \mathbb{Q}[E_0,\{E_\Lambda\}_\Lambda]. \]

Of course, $M$ has degree $d$ in the divisors $E_0$ and $\{ E_\Lambda \}_\Lambda$.

Let $M^E$ be a monomial of degree $d$ in the divisors

(27)\begin{equation} E_0\quad \text{and}\quad \{ E_\Lambda\}_\Lambda. \end{equation}

The monodromy group $G$ actsFootnote ²² canonically on the set (27) leaving $E_0$ fixed. Let

\[ M^E_G=\frac{1}{|\mathsf{Stab}(M^E)|}\sum_{g\in G} g(M^E) \]

be the summation over the $G$-orbit of $M^E$. Since $M^E_G$ is $G$-invariant, $M^E_G$ is a well-defined class

\[ M^E_G \in \mathsf{CH}^d(\mathsf{E}_S(S)). \]

To prove the stronger induction step, we need only proveFootnote ²³ that

(28)\begin{equation} \iota_*\bigg(M^E_G\cdot c_{s-1}\bigg(\frac{\mathsf{N}_S}{ \mathcal{O}_{\mathsf{P}(\mathsf{N}_S)}(-1)}\bigg)\bigg) \in \mathsf{divlogCH}^*(X,D) \end{equation}

can be expressed as a linear combination of terms of the form

\[ \widehat{D}_1\widehat{D}_2 \cdots \widehat{D}_d, \]

where the $\widehat {D}_i$ are components of the logarithmic boundary of an iterated blow-up of the explosion $\mathsf {E}_S(X,D)$ and $\widehat {D}_1$ lies over $\mathsf {E}_S(S)$. To see why the claim for (28) is enough, we write

\begin{align*} e^*[S,M_G] &= \sum_{M^E_G} e^*[S] \cdot M^E_G\\ &= \sum_{M^E_G}[\mathsf{E}_S(S)]\cdot c_{s-1} \bigg(\frac{\mathsf{N}_S}{ \mathcal{O}_{\mathsf{P}(\mathsf{N}_S)}(-1)}\bigg)\cdot M^E_G\\ &= \sum_{M^E_G}\iota_*\bigg(M^E_G \cdot c_{s-1}\bigg(\frac{\mathsf{N}_S}{ \mathcal{O}_{\mathsf{P}(\mathsf{N}_S)}(-1)}\bigg)\bigg). \end{align*}

The first equality is written with the understanding that $e^*[S]$ is supported on $\mathsf {E}_S(S)$.

To study $M^E_G$, we take a geometric approach. If $M^E$ is just $E_0^d$, then (28) is already of the claimed form by our analysis in the base case. Otherwise, $M^E$ has at least one factor $E_\Lambda$. Since $\{E_\Lambda \}_\Lambda$ is a set of simple normal crossings divisors on $\mathsf {E}_S(S)$, we claim that we can write $M^E$ (if non-zero) as

\[ M^E=E_{\Lambda_1}\cdots E_{\Lambda_t} \cdot \widetilde{M}^E, \]

where $E_{\Lambda _1},\ldots,E_{\Lambda _t}$ are distinct divisors with a non-empty transverse intersection

\[ I_{U_p} = E_{\Lambda_1}\cap \cdots\cap E_{\Lambda_t}\ \text{over}\ U_p. \]

Moreover, we can assume every divisor of the monomial $\widetilde {M}^E$ contains $I_{U_p}$. Indeed, we construct inductively for $i=1,2, \ldots$ a representation

\[ M^E = E_{\Lambda_1} \cdots E_{\Lambda_i} \cdot \widetilde{M}_i^E \]

such that the $E_{\Lambda _j}$ are distinct and have non-zero transverse intersection. For $i=1$ this is just our assumption that $M^E$ has some factor $E_{\Lambda }=:E_{\Lambda _1}$. On the other hand, given the representation above for some $i$, if all factors $E_{\Lambda '}$ of $\widetilde {M}_i^E$ contain $E_{\Lambda _1} \cap \cdots \cap E_{\Lambda _i}$, we are done, setting ${t=i}$. If there is an $E_{\Lambda '}$ not satisfying this, we set $E_{\Lambda _{i+1}}=E_{\Lambda '}$. If the intersection $E_{\Lambda _1} \cap \cdots \cap E_{\Lambda _{i+1}}$ was empty, then $M^E=0$, giving a contradiction. Thus, the intersection is non-empty, and transverse by the fact that the $E_{\Lambda }$ are a normal crossings divisor. We continue inductively, and this construction concludes after at most $d$ steps.

When the monodromy invariant $M^E_G$ is considered, we obtain a non-singular subvariety of $\mathsf {E}_S(S)$ of codimension $t$,

\[ V \subset \mathsf{E}_S(S), \]

which is a simple stratum of $\mathsf {E}_S(X,D)$,

\[ \epsilon^V: V \rightarrow \mathsf{E}_S(X,D). \]

Over $U_p$, the subvariety $V$ restricts to the unionFootnote ²⁴ of the distinct $G$-translates of $I_{U_p}$. The crucial geometric observation is

\[ \iota_*(M^E_G) = \epsilon^V_*(\widetilde{P}) \in \mathsf{CH}^\star(\mathsf{E}_S(X,D), \]

where $\widetilde {P}$ is defined by $\widetilde {M}^E$ and is of degree at most $d-1$.

We can apply the strong induction property: the class

\[ \epsilon^V_*(\widetilde{P})\in \mathsf{divlogCH}^\star(X,D) \]

can be expressed as a linear combination of terms of the form

\[ \widehat{D}_1\widehat{D}_2 \cdots \widehat{D}_d \]

where the $\widehat {D}_i$ are components of the logarithmic boundary of an iterated blow-up of the explosion of $V$ in $\mathsf {E}_S(X,D)$ and $\widehat {D}_1$ lies over

\[ \mathsf{E}_V(V)\subset\mathsf{E}_S(X,D). \]

Then the claim

(29)\begin{equation} \iota_*\bigg(M^E_G \cdot c_{s-1} \bigg(\frac{\mathsf{N}_S}{ \mathcal{O}_{\mathsf{P}(\mathsf{N}_S)}(-1)}\bigg)\bigg) \in \mathsf{divlogCH}^*(X,D) \end{equation}

holds by the analysis of

\[ c_{s-1}\bigg(\frac{\mathsf{N}_S}{ \mathcal{O}_{\mathsf{P}(\mathsf{N}_S)}(-1)}\bigg) \]

on $\mathsf {E}_S(S)$ in the base case of the induction. Since each monomial

\[ \widehat{D}_1\widehat{D}_2 \cdots \widehat{D}_d \]

of $\epsilon ^V_*(\widetilde {P})$ lies over $\mathsf {E}_V(V)$, which in turn lies over $\mathsf {E}_{S}(S)$, the analysis of the base case yields the desired result (29).

The induction argument is complete, so we have proven Theorem 11 in the case where $S$ is a simple stratum of $(X,D)$. The general case follows by repeated application of the result for a simple stratum.

Let $S\subset X$ be a stratum with a singular closure

\[ \overline{S} \subset X. \]

The first step is to blow up simple strata in $\overline {S}$,

\[ \widehat{X} \rightarrow X, \]

until the strict transform of $\overline {S}$,

\[ \widehat{S} \subset \widehat{X}, \]

is non-singular. Since $S$ is simple stratum of the blow-up $\widehat {X}$, we can apply Theorem 11 to ${S} \subset \widehat {X}$.

Via the blow-down map, we have

\[ \widehat{S} \rightarrow \overline{S}. \]

There are two discrepancies to handle before deducing Theorem 11 for normally decorated classes associated to $S\subset X$ from the result for normally decorated classes associated to $S\subset \widehat {X}$.

1. (i) The fundamental class $[\widehat {S}]\in \mathsf {CH}^\star (\widehat {X})$ is not the pull-back of $[\overline {S}]\in \mathsf {CH}^\star ({X})$.

2. (ii) The normal directions of $\widehat {S}\subset \widehat {X}$ differ from the pull-backs of the normal directions of $\overline {S}\subset {X}$.

However, both discrepancies are corrected by applying the simple stratum result to the lower-dimensional strata occurring in $\widehat {S}\,{\setminus}\, S$.

5.3 Explosion geometry and barycentric subdivision

The explosion operation $E(X,D)$ along a simple stratum $S\subset X$, which appeared in the proof § 5.2, is an essentially combinatorial operation that has a natural interpretation in terms of the geometry of the cone complex $C(X,D)$.

Consider first a cone $\sigma$ of dimension $n$ in a lattice $N$, and let $A_\sigma$ be the associated toric variety. Let $\mathcal {A}_\sigma$ be the associated Artin fan, which is simply the stack quotient of $A_\sigma$ by the corresponding dense torus $T_\sigma$. The logarithmic stratification of $A_\sigma$ is precisely the stratification defined by the orbits of $T_\sigma$, and there is a bijective dimension-reversing correspondence between faces of $\sigma$ and strata. We write $\sigma (k)$ for the $k$-dimensional faces of $\sigma$ and thus the codimension $k$ strata of $A_\sigma$.

For each face $\tau$ of $\sigma$, the barycenter $b_\tau$ of $\tau$ is the sum

\[ b_\tau=\sum_{v_i \in \tau \cap \sigma(1)} v_i \]

of the primitive vectors along the extremal rays of $\tau$. For any flag

\[ \tau_0 \subset \tau_1 \subset \cdots \subset \tau_k \]

of faces of $\sigma$, the barycenters $b_{\tau _0}, \ldots, b_{\tau _k}$ span a cone. The set of all such cones, for all flags in $\sigma$, forms a subdivision of $\sigma$, which we call the barycentric subdivision $\widetilde {\sigma }$ of $\sigma$.

Alternatively, we can build the barycentric subdivision inductively: at step $1$, we start with the star subdivision over the barycenter of faces in $\sigma (n)$ (where $\sigma$ has dimension $n$), then take the star subdivision over faces in $\sigma (n-1)$, and so on, terminating after $n-1$ steps with $\sigma (2)$, after which the operation no longer has any effect. We thus produce a sequence of $n-1$ subdivisions

\[ \widetilde{\sigma} = \sigma_{n-1} \to \sigma_{n-1} \cdots \to \sigma_1 \to \sigma_0=\sigma \]

When $\sigma = \mathbb {R}_{\ge 0}^n$, which is our main case of interest, the barycentric subdivision has $n!$ maximal cones.

The barycentric subdivision of $\sigma$ produces a log modification

\[ \widetilde{A}_\sigma \to A_\sigma, \]

which is in fact a log blow-up. More precisely, we have constructed the subdivision $\widetilde {A}_\sigma \to A_\sigma$ as a sequence

\[ \widetilde{A}_\sigma = A_{n-1} \to A_{n-2} \to \cdots A_1 \to A_0 = A_\sigma \]

and the map $A_k \to A_{k-1}$ is determined by the subdivision $\sigma _k \to \sigma _{k-1}$, which is the subdivision corresponding to the domains of linearity of a piecewise linear function (see [Reference Kempf, Knudsen, Mumford and Saint-DonatKKMS73] for the construction). In the case of interest,

\[ \sigma = \mathbb{R}_{\ge 0}^n, \]

the map $A_1 \to A_0$ is the blow-up of $\mathbb {A}^n$ at the origin, $A_2 \to A_1$ is the blow-up along the strict transforms of the coordinate lines, and in general $A_k \to A_{k-1}$ is the blow-up along the strict transforms of the dimension $k-1$ hyperplanes of $\mathbb {A}^n$ in $A_{k-1}$. Thus, the barycentric subdivision of $\mathbb {A}^n$ is precisely the explosion of $\mathbb {A}^n$ along the origin.

The barycentric subdivision construction is clearly equivariant and therefore descends to the Artin fan $\mathcal {A}_\sigma$ of $A_\sigma$. Furthermore, the subdivision is the same on isomorphic faces of $\sigma$ and invariant with respect to automorphisms of $\sigma$. Consequently, given any cone complex $C$, the barycentric subdivisions of individual cones glue to a global subdivision of $C$, and that is true even if faces of $C$ are identified or if there is monodromy in $C$. Thus, for a normal crossings pair $(X,D)$, we can define the barycentric subdivision $\widetilde {C}(X,D)$ of the cone complex $C(X,D)$, and equivalently, a log blow-up

\[ \widetilde{\mathcal{A}}_X\to\mathcal{A}_X \]

of the Artin fan. We also obtain globally a log blow-up

\[ (\widetilde{X},\widetilde{D})= X \times_{\mathcal{A}_X} \widetilde{\mathcal{A}}_X \to (X,D) \]

with Artin fan $\mathcal {A}_{\widetilde {X}} = \widetilde {\mathcal {A}}_X$.

The explosion of § 5.2 can only be defined locally around a simple stratum $S$. A quasi-projective stratum $S$ (not necessarily simple) of a normal crossings pair $(X,D)$ corresponds to a cone $\sigma$ of $C(X,D)$. More precisely, the quasi-projective stratum $S$ corresponds to the interior of $\sigma$, and the whole of $\sigma$ corresponds to a canonical open set $U$ in $X$ that contains $S$ as its minimal stratum: the open set $U$ consists of all quasi-projective strata whose closure contains $S$. The explosion $\mathsf {E}_S(U, D|_U))$ is well defined.

The cone $\sigma$ has a cover by $\mathbb {R}^n_{\ge 0}$, and, more precisely, by a quotient of $\mathbb {R}^n_{\ge 0}$ obtained by potentially identifying faces and taking a quotient by a group $G$. The group $G$ is precisely the monodromy group of the divisors $D$ that cut out $S$ considered in § 5, and the interior $\sigma ^\circ$ of $\sigma$ is in fact the stack quotient $[\mathbb {R}_{>0}^n/G]$. Similarly, the Artin fan $\mathcal {U}$ of $U$ has an analogous étale cover by the groupoid quotient of $[\mathbb {A}^n/\mathbb {G}_m^n \rtimes G]$, with $S$ corresponding to the minimal stratum

\[ B(\mathbb{G}_m^n \rtimes G) \subset \mathcal{U}. \]

The cover is not representable, but is representable over $S$. From the discussion of the barycentric subdivision of $\mathbb {A}^n$, we see that $\mathsf {E}_S(U,D_U)$ is precisely the barycentric subdivision $\widetilde {X} \to X$ restricted to $U$. We may thus view the barycentric subdivision as globalizing the explosion geometry.

If the stratum $S$ is simple, the explosion of § 5.2 is defined over a neighborhood of $\overline {S}$. However, the extension no longer coincides with the barycentric subdivision. The barycentric subdivision performs additional blow-ups, first blowing up all minimal strata in the closure of $S$ (and also strata around $\overline {S}$ whose closure does not necessarily meet $S$).

We illustrate the concepts discussed above through an example. Let $(X,D)$ be a log scheme whose cone complex is the cone over an equilateral triangle, with all edges identified and with monodromy $\mathbb {Z}/3\mathbb {Z}$. For example, we can construct $(X,D)$ by taking

\[ X \to B \]

to be a family with fiber $\mathbb {A}^3$ over a non-singular base $B$ satisfying $\pi _1(B) = \mathbb {Z}$, so that the generator of $\pi _1(B)$ cyclically permutes the coordinate hyperplanes of $\mathbb {A}^3$. The divisor $D\subset X$ is then the union of these coordinate hyperplanes over $B$.

The log scheme $(X,D)$ has four strata: the open set $X-D$, corresponding to the empty face of the triangle (or, equivalently, the vertex of the cone over the triangle); the interior of the divisor $D$ corresponding to the vertex

\[ e_1=e_2=e_3; \]

the locus which is étale locally the intersection of exactly two irreducible components of $D$ corresponding to edge

\[ \overline{e_1e_2} = \overline{e_1e_3}= \overline{e_2e_3}; \]

and the triple point singularity of $D$ corresponding to the whole triangle. We name the strata $Q,R,S,T$, respectively. While $T$ is simple, $S$ is not, since

\[ \overline{S}=S \cup T \]

is not normal. The strata are taken bijectively to points of the Artin fan via the map

\[ \alpha: X \to \mathcal{A}_X \]

We depict the Artin fan as four points, each isomorphic to $B\mathbb {G}_m^k \rtimes G$ as indicated, with points drawn increasingly bigger to describe the topology (the closure contains all smaller points).

Consider the explosion of the quasi-projective stratum $S$ depicted by the open line segment $\overline {e_1e_2}$. The open set $U$ over which the explosion is defined is $Q \cup R \cup S$. The explosion of $S$ is the barycentric subdivision of $\overline {e_1e_2}$:

However, the above explosion does not extend away from $U$. The blow-up of $\overline {S}$, over an étale cover of $X$ is depicted as

But the blow-up does not descend to $X$ as it does not respect the face identifications/ automorphisms of $C(X,D)$. The barycentric subdivision is depicted as

The corresponding log blow-up restricts to the explosion over $U$. Over $X$, the log blow-up is not the blow-up of $\overline {S}$, but the explosion of $T$.

5.4 Tautological classes

Let $(X,D)$ be a non-singular variety with a normal crossings divisor. We define the logarithmic tautological ring

\[ \mathsf{R}^\star(X,D) \subset \mathsf{CH}^\star(X) \]

to be the $\mathbb {Q}$-linear subspace spanned by all normally decorated strata classes (which is easily seen to be closed under the intersection product). Theorem 11 can then be written as

\[ \mathsf{R}^\star(X,D) \subset\mathsf{divlogCH}^\star(X,D). \]

The logarithmic tautological ring of $(X,D)$ depends strongly on the divisor $D$. For example, if $X$ is irreducible and $D=\emptyset$, then there is only one stratum and

\[ \mathsf{R}^\star(X,\emptyset) = \mathbb{Q}. \]

For the moduli space of curves, the inclusion

\[ \mathsf{R}^\star(\overline{\mathcal{M}}_g, \Delta_0) \subset \mathsf{R}^\star(\overline{\mathcal{M}}_g, \partial \overline{\mathcal{M}}_g) \]

is proper for $g\geq 2$. Furthermore, the inclusion

\[ \mathsf{R}^\star(\overline{\mathcal{M}}_g, \partial \overline{\mathcal{M}}_g) \subset \mathsf{R}^\star(\overline{\mathcal{M}}_g) \]

in the standard tautological ringFootnote ²⁵ is proper for $g\geq 3$ since $\mathsf {R}^\star (\overline {\mathcal {M}}_g)$ contains $\kappa$ and $\psi$ classes which do not appear in the logarithmic constructions.

Let $(X,D)$ be a non-singular variety with a normal crossings divisor. Let

\[ \pi: \widetilde{X} \rightarrow X \]

be a simple blow-up of $(X,D)$. Let $\widetilde {D}\subset \widetilde {X}$ be the associated normal crossings divisor. We will prove the following two basic properties of logarithmic tautological rings.

Theorem 12 The pull-back

\[ \pi^* : \mathsf{R}^\star({X},{D}) \rightarrow \mathsf{CH}^\star(\widetilde{X}) \]

has image in $\mathsf {R}^\star (\widetilde {X}, \widetilde {D})$.

Theorem 13 The push-forward

\[ \pi_* :\mathsf{R}^\star(\widetilde{X},\widetilde{D}) \rightarrow \mathsf{CH}^\star(X) \]

has image in $\mathsf {R}^\star (X,D)$.

By Theorems 12 and 13, we can simply write

\[ \pi^* :\mathsf{R}^\star({X},{D}) \rightarrow \mathsf{R}^\star(\widetilde{X},\widetilde{D}),\quad \pi_* :\mathsf{R}^\star(\widetilde{X},\widetilde{D}) \rightarrow \mathsf{R}^\star(X,D). \]

Theorems 12 and 13 will proven in § 5.6 via the geometry of the Artin fan. As a consequence, we will present a more conceptual (but less constructive) proof of Theorem 11.

5.5 The Chow ring of the Artin fan

Let $(X,D)$ be a non-singular variety with a normal crossings divisor. We relate here the normally decorated strata classes of $(X,D)$ to Chow classes on the Artin fan $\mathcal {A}_X$ of $(X,D)$. Here, since $\mathcal {A}_X$ is a smooth, finite type algebraic stack stratified by quotient stacks, it has well-defined Chow groups $\mathsf {CH}^\star (\mathcal {A}_X)$ with an intersection product as defined in [Reference KreschKre99]. Note that for our proof below it will not be necessary to recall the precise definition from [Reference KreschKre99], since we only use some properties and examples of these Chow groups (such as the existence of an excision sequence) that we recall when needed. Also, we stress again that all Chow groups below are with $\mathbb {Q}$-coefficients.

As we explain in § 4.4, there is a smooth morphism to the Artin fan,

\[ \alpha: X \rightarrow \mathcal{A}_X. \]

Theorem 14 There is a canonical isomorphism

\[ \mathsf{CH}^\star(\mathcal{A}_X) \cong {\rm PP}^\star(C(X,D)) \]

between the Chow ring of $\mathcal {A}_X$ and the algebra of piecewise polynomial functions on the cone complex $C(X,D)$.

Proof. By construction, the Artin fan $\mathcal {A}_X$ has a presentation as a colimit

\[ \mathcal{A}_X =\varinjlim_{x \in \mathcal{S}} \mathcal{A}_{x}, \]

where $\mathcal {S}$ is a finite diagram, each map $\mathcal {A}_x$ is a stack of the form $[\mathbb {A}^n/\mathbb {G}_m^n]$, and all maps in the diagram are étale. First, we note that for the individual stacks $\mathcal {A}_x = [\mathbb {A}^n/\mathbb {G}_m^n]$ we have

(30)\begin{equation} \mathsf{CH}^\star([\mathbb{A}^n/\mathbb{G}_m^n]) \cong \mathsf{CH}^\star([\mathsf{Spec}(\mathbb{C})/\mathbb{G}_m^n]) \cong \mathbb{Q}[x_1, \ldots, x_n]. \end{equation}

The first equality is because

\[ [\mathbb{A}^n/\mathbb{G}_m^n] \to [\mathsf{Spec}(\mathbb{C})/\mathbb{G}_m^n] \]

is a vector bundle and induces an isomorphism of Chow groups by [Reference KreschKre99, Theorem 2.1.12 (vi)]. The second equality is because the equivariant Chow ring of a product of tori is a polynomial algebra [Reference Edidin and GrahamEG98, § 3.2], which can be identified with polynomials on the cone $\sigma _{X,x}$ associated to $\mathcal {A}_x$ (appearing in the colimit presentation of $C(X,D)$).

For the entire Artin fan $\mathcal {A}_X$, we claim

(31)\begin{equation} \mathsf{CH}^\star{\mathcal{A}_X} = \varprojlim_{x \in \mathcal{S}} \mathsf{CH}^\star{\mathcal{A}_{x}}. \end{equation}

If we can show equality (31), then Theorem 14 will follow since the result holds for each term on the right-hand side by (30). Piecewise polynomial functions on $C(X,D)$ are defined by the corresponding limit presentation.

All the stacks appearing in (31) are very special: they are non-singular and have a stratification with strata isomorphic to

\[ B(\mathbb{G}_m^n \rtimes G) \]

where $G$ is a finite group. For the argument below, it will be more convenient to index Chow groups by the dimension of the cycles (instead of the codimension) and proveFootnote ²⁶

(32)\begin{equation} \mathsf{CH}_{\star}(\mathcal{A}_X) = \varprojlim_{x \in \mathcal{S}} \mathsf{CH}_{\star}(\mathcal{A}_x). \end{equation}

Let $\mathcal {C}$ denote the full 2-subcategory of the 2-category of algebraic stacks with Ob($\mathcal {C}$) given by algebraic stacks $\mathcal {A}$ with a stratification by stacks of the form $B (\mathbb {G}_m^n \rtimes G)$, with $G$ a finite group. Similarly, let $\mathcal {C}^\circ$ be the full 2-subcategory of $\mathcal {C}$ with objects given by stacks of the form $B \mathbb {G}_m^n$. We start with a stackFootnote ²⁷ $\mathcal {A}_X \in \mathcal {C}$ with a colimit presentation

\[ \mathcal{A}_X = \varinjlim_{x \in \mathcal{S}} \mathcal{A}_x = \mathcal{A}_X, \]

where $\mathcal {A}_x \in \mathcal {C}^\circ$ and all maps in the diagram are étale. We will prove (32) by induction on the number of strata of $\mathcal {A}_X$.

Assume first that there is a unique stratum,

\[ \mathcal{A}_X = B(\mathbb{G}_m^n \rtimes G), \]

and all maps in the diagram $\mathcal {S}$ are isomorphisms. Then the groupoid

\[ \varinjlim_{x \in \mathcal{S}} \mathcal{A}_x \]

is equivalent to the quotient $B\mathbb {G}_m^n/G$, and the statement is equivalent to

\[ \mathsf{CH}_*(B(\mathbb{G}_m^n \rtimes G)) = \mathsf{CH}_*(B\mathbb{G}_m^n)^G, \]

which is true (see [Reference Bae and SchmittBS21, Lemma 2.20]). In general, we pick an open stratum $U \in \mathcal {A}_X$ with preimage $U_x \in \mathcal {A}_x$. Then, by [Reference KreschKre99, Proposition 4.2.1], we have an exact sequence

with $Z = \mathcal {A}_X - U$. Since $U$ is of the form $U = B(\mathbb {G}_m^n \rtimes G)$, we can use [Reference Bae and SchmittBS21, Proposition 2.14, Remark 2.21] to see that

\[ \mathsf{CH}(U,1) = \mathsf{CH}(U) \otimes_{\mathbb{Q}} \mathsf{CH}(\mathrm{Spec}(\mathbb{C}), 1). \]

Then, by [Reference Bae and SchmittBS21, Remark 2.18], the connecting homomorphism $\mathsf {CH}(U,1) \to \mathsf {CH}(Z)$ vanishes. So we obtain an exact sequence

and the same sequence holds with $\mathcal {A}_X$ replaced by $\mathcal {A}_x$, $U$ by $U_x$, and $Z$ by $Z_x = \mathcal {A}_x - U_x$. As projective limits are left exact, we obtain the following diagram.

By induction, the left and right vertical arrows are isomorphisms. But the bottom row is exact as well: the composed map

\[ \mathsf{CH}(\mathcal{A}_X) \to \mathsf{CH}({U}) \cong \varprojlim_{x \in \mathcal{S}}\mathsf{CH}(U_x) \]

is surjective and factors through $\varprojlim _{x \in \mathcal {S}}\mathsf {CH}(\mathcal {A}_x)$. Thus, the map

\[ \mathsf{CH}(\mathcal{A}_X) \to \varprojlim_{x \in \mathcal{S}}\mathsf{CH}(\mathcal{A}_x) \]

is an isomorphism as well.

Theorem 14 has clear precursors in the toric context by Payne [Reference PaynePay06] and Brion [Reference BrionBri94]. In the logarithmic context, we were directly motivated by ideas of Ranganathan. A development of the theory for general log schemes can be found in [Reference Molcho and RanganathanMR21].

Theorem 15 The logarithmic tautological ring

\[ \mathsf{R}^\star(X,D) \subset \mathsf{CH}^\star(X) \]

coincides with the image $\alpha ^*\mathsf {CH}^\star (\mathcal {A}_X) \subset \mathsf {CH}^\star (X)$.

Proof. Fix a stratum $S\subset X$ with closure $\overline {S}\subset X$, and normalization

\[ \epsilon: \widetilde{S} \rightarrow \overline{S} \subset X. \]

Consider the cone complex $C(X,D)$ and the Artin fan $\mathcal {A}_X$ of $(X,D)$ with

\[ \alpha:X \to \mathcal{A}_X. \]

Let $\widetilde {P}$ be the total space of the principal $G$-bundle over the normalization $\widetilde {S}$ defined by the branches of $D$ in § 5.1,

\[ \mu: \widetilde{P} \to \widetilde{S},\quad \mu_X = \epsilon \circ \mu : \widetilde{P} \to X. \]

We observe that all the relevant geometry is pulled back from the Artin fan $\mathcal {A}_X$: the stratum $S$ corresponds to the stratum

\[ \alpha(S) = \mathcal{S} \subset \mathcal{A}_X \]

with closure $\overline {\mathcal {S}} = \alpha (\overline {S})$. Let $\widetilde {\mathcal {S}}$ be the normalization of $\overline {\mathcal {S}}$, and let

\[ \mu:\widetilde{\mathcal{P}} \to \widetilde{\mathcal{S}},\quad \mu_{\mathcal{A}} = \widetilde{\mathcal{P}} \to {\mathcal{A}}_X \]

be the total space of the principal $G$-bundle over $\widetilde {\mathcal {S}}$. Then

\[ S = \mathcal{S} \times_{\mathcal{A}_X} X,\quad \overline{S} = \overline{\mathcal{S}} \times_{\mathcal{A}_X} X,\quad \widetilde{S} = \widetilde{\mathcal{S}} \times_{\mathcal{A}_X} X,\quad \widetilde{P} = \widetilde{\mathcal{P}} \times_{\mathcal{A}_X} X. \]

Furthermore, since the map $\alpha$ is smooth, we find that $N_{\widetilde {S}/X}$ is the pull-back of $N_{\widetilde {\mathcal {S}}/\mathcal {A}_X}$, and the splitting of $N_{\widetilde {S}/X}$ on $\widetilde {P}$ into line bundles is pulled back from the splitting of $N_{\widetilde {\mathcal {S}}/\mathcal {A}_X}$ on $\widetilde {\mathcal {P}}$. In other words, we have the following Cartesian diagram.

Normally decorated strata classes on $\overline {S}$ have the form $\mu _{X*}\alpha _P^*(\gamma )$ for $\gamma \in \mathsf {CH}^\star (\widetilde {\mathcal {P}})$. As $\alpha,\alpha _P$ are smooth, $\mu _{X*}\alpha _P^*= \alpha ^*\mu _{\mathcal {A}*}$. Therefore,

\[ R(X,D) \subset \alpha^*\mathsf{CH}^\star(\mathcal{A}_X). \]

In fact, the argument shows more precisely that

\[ R(X,D) = \alpha^*R(\mathcal{A}_X,\mathcal{D}) \]

for $\mathcal {D} = \alpha (D)$ the corresponding divisor in $\mathcal {A}_X$. In other words, the logarithmic tautological ring of $(X,D)$, which is generated by the Chern roots of the normal bundles on the various monodromy torsors of the strata of $(X,D)$, is the pull-back of the logarithmic tautological ring of $\mathcal {A}_X$, generated by the analogous constructions over the strata of $\mathcal {A}_X$. Thus, it suffices to show that the normally decorated strata classes of $\mathcal {A}_X$ generate the Chow ring of $\mathcal {A}_X$. We may thus reduce to proving the theorem for $\mathcal {A}_X$.

So let $\gamma$ be a class in $\mathsf {CH}^\star (\mathcal {A}_X)$. We must show that

\[ \gamma \in R(\mathcal{A}_X,\mathcal{D}). \]

We may assume that $\gamma$ is supported on $\overline {\mathcal {S}}$ for some stratum $\mathcal {S}\subset \mathcal {A}_X$. Suppose, by induction, we have shown that every such class supported on a stratum $\overline {\mathcal {S}'}$ with

\[ \dim \mathcal{S}' < \dim \mathcal{S} \]

is in $R(\mathcal {A}_X,\mathcal {D})$. Suppose further that we can find a class $\delta \in R(\mathcal {A}_X,\mathcal {D})$ such that $\gamma$ equals $\delta$ on $\mathcal {S}$. Then

\[ \gamma-\delta\in \mathsf{CH}^\star(\mathcal{A}_X) \]

is supported on lower-dimensional strata and therefore lies in $R(\mathcal {A}_X,\mathcal {D})$, so that we have $\gamma \in R(\mathcal {A}_X,\mathcal {D})$ as well. Thus, the induction hypothesis ensures that, for a given dimension $\dim \mathcal {S}$, we can remove strata $\mathcal {S}'$ with $\dim \mathcal {S}' < \dim \mathcal {S}$, and thus it suffices to prove the statement with the additional assumption that $\mathcal {S}$ is closed in $\mathcal {A}_X$. Note that this reduction also suffices to handle the base of the induction: the minimal dimensional strata of $\mathcal {A}_X$ are automatically closed.

Suppose then that $\gamma$ is a class supported on $\mathcal {S}$, and $\mathcal {S}$ is closed of codimension $n$ in $\mathcal {A}_X$. Then $\mathcal {S} \cong B(\mathbb {G}_m^n \rtimes G)$, $\mathcal {A}_X$ is a quotient of $[\mathbb {A}^n/\mathbb {G}_m^n]$ by an étale equivalence relation in a neighborhood of $\mathcal {S}$, and the monodromy torsor $\mu : \tilde {\mathcal {P}} \to \mathcal {S}$ is isomorphic to $B\mathbb {G}_m^n$.

We can use this to describe the normal bundle $N_{\mathcal {S}/\mathcal {A}_X}$ on $\mathcal {S}$: the data of this vector bundle on $\mathcal {S}$ are equivalent to specifying the bundle $\mu ^* N_{\mathcal {S}/\mathcal {A}_X}$ on $\tilde {\mathcal {P}}$ together with a $G$-action. It is given by

\[ \mu^* N_{\mathcal{S}/\mathcal{A}_X} = \oplus_{i=1}^n N_i := \oplus_{i=1}^n \mathcal{O}(\mathcal{D}_i)|_{\tilde{\mathcal{P}}} \]

where $\mathcal {D}_i$ is the $i$th hyperplane divisor in $[\mathbb {A}^n/\mathbb {G}_m^n]$. The monodromy group $G$ acts by permuting the hyperplanes $\mathcal {D}_i$ cutting out $S$, and this action lifts to a corresponding action permuting the direct summands $N_i$ above. In particular, while the pull-back $\mu ^* N_{\mathcal {S}/\mathcal {A}_X}$ is a direct sum, the individual direct summands are in general not invariant under the $G$-action, and thus $N_{\mathcal {S}/\mathcal {A}_X}$ is not actually split on $\mathcal {S}$.

Still, on $\tilde {\mathcal {P}}$ we have that the classes $x_i:=c_1(N_i)$ form a generating set for the algebra

\[ \mathsf{CH}(\tilde{\mathcal{P}}) \cong \mathbb{Q}[x_1,\ldots,x_n]. \]

On the other hand, the map $\mu$ gives an isomorphism

\[ \mu^*: \mathsf{CH}(\mathcal{S}) \cong \mathsf{CH}(\tilde{\mathcal{P}})^G \]

with inverse $ ({1}/{|G|})\mu _*$, since we are working with rational Chow groups. Thus, $\gamma$ is the image of $ ({1}/{|G|})\mu ^*\gamma$ under $\mu _*$, which is a $G$-invariant polynomial in the $x_i$. This shows that $\gamma$ is a normally decorated strata class, completing the proof.

Theorem 15 immediately implies that $\mathsf {R}^\star (X,D)\subset \mathsf {CH}^\star (X)$ is closed under the intersection product (a claim which was left to the reader in § 5.4). On the other hand, it is not immediate to see which piecewise polynomial corresponds to which normally decorated strata class. The precise correspondence between piecewise polynomials and normally decorated strata classes has now been established in [Reference Holmes, Molcho, Pandharipande, Pixton and SchmittHMPPS22, § 6].

5.6 Proofs of Theorems 12 and 13

Fix a normal crossings pair $(X,D)$ with Artin fan $\mathcal {A}_X$ and map

\[ \alpha:X \to \mathcal{A}_X. \]

Consider an arbitrary smooth log modification

\[ f: \widetilde{X} \to X, \]

necessarily of the form $(\widetilde {X},\widetilde {D})$, with an associated map

\[ \widetilde{\alpha}: \widetilde{X} \to \mathcal{A}_{\widetilde{X}}. \]

By definition, the log modification $\widetilde {X} \to X$ is pulled back to $X$ from a log modification $\widetilde {\mathcal {A}}_{X} \to \mathcal {A}_X$ of Artin fans, and we have a diagram

with the square being Cartesian, the map $c$ proper, Deligne-Mumford type, étale and bijective, and $\widetilde {\alpha } = c \circ \beta$. By Theorem 15,