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The Hodge bundle, the universal 0-section, and the log Chow ring of the moduli space of curves

Published online by Cambridge University Press:  14 February 2023

S. Molcho
Departement Mathematik, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland
R. Pandharipande
Departement Mathematik, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland
J. Schmitt
Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland
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We bound from below the complexity of the top Chern class $\lambda _g$ of the Hodge bundle in the Chow ring of the moduli space of curves: no formulas for $\lambda _g$ in terms of classes of degrees 1 and 2 can exist. As a consequence of the Torelli map, the 0-section over the second Voronoi compactification of the moduli of principally polarized abelian varieties also cannot be expressed in terms of classes of degree 1 and 2. Along the way, we establish new cases of Pixton's conjecture for tautological relations. In the log Chow ring of the moduli space of curves, however, we prove $\lambda _g$ lies in the subalgebra generated by logarithmic boundary divisors. The proof is effective and uses Pixton's double ramification cycle formula together with a foundational study of the tautological ring defined by a normal crossings divisor. The results open the door to the search for simpler formulas for $\lambda _g$ on the moduli of curves after log blow-ups.

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1. Introduction

1.1 The Hodge bundle

Let $\overline {\mathcal {M}}_g$ be the moduli space of Deligne–Mumford stable curves, and let

\[ \pi: \mathcal{C}_g \rightarrow \overline{\mathcal{M}}_g \]

be the universal curve with relative dualizing sheaf $\omega _\pi$. The rank $g$ Hodge bundle $\mathbb {E}_g$ on $\overline {\mathcal {M}}_g$ is defined by

\[ \mathbb{E}_g = \pi_* \omega_\pi. \]

The study of the Chern classes of the Hodge bundle goes back at least to Mumford's Grothendieck–Riemann–Roch calculation [Reference MumfordMum83] in the 1980s. Starting in the late 1990s, the connection of the Hodge bundle to the deformation theory of the moduli space of stable maps has led to an exploration of Hodge integrals in various contexts (see [Reference Aganagic, Klemm, Mariño and VafaAKMV05, Reference Ekdahl, Lando, Shapiro and VainshteinELSV01, Reference Faber and PandharipandeFP00a, Reference Graber and PandharipandeGP99, Reference Li, Liu, Liu and ZhouLLLZ09, Reference Liu, Liu and ZhouLLZ03, Reference Maulik, Oblomkov, Okounkov and PandharipandeMOOP11, Reference Okounkov and PandharipandeOP04, Reference PandharipandePan99]).

The top Chern classFootnote 1 of the Hodge bundle

\[ \lambda_g = c_g(\mathbb{E}_g) \in \mathsf{CH}^g(\overline{\mathcal{M}}_g) \]

plays a special role for several reasons.

  1. (i) Two vanishing properties hold:

    \[ \lambda_g^2=0 \in \mathsf{CH}^{2g}(\overline{\mathcal{M}}_g) \quad \text{and}\quad \lambda_g|_{\Delta_0}= 0 \in \mathsf{CH}^g(\Delta_0), \]
    where $\Delta _0\subset \overline {\mathcal {M}}_g$ is the divisor of curves with a non-separating node. The first vanishing follows from the highest graded part of Mumford's relation
    \[ c(\mathbb{E}_g)\cdot c(\mathbb{E}_g^*)= 1, \]
    proven in [Reference MumfordMum83, equations (5.4), (5.5)]. The second follows from the existence of a trivial quotientFootnote 2
    \[ \mathbb{E}_g \twoheadrightarrow \mathbb{C} \]
    determined by the residue at (a branch of) the node (see [Reference Faber and PandharipandeFP00b, § 0.4]).
  2. (ii) The class $(-1)^g \lambda _g$ appears in the virtual fundamental class of the moduli of contracted maps in the Gromov–Witten theory of target curves. Since the double ramification cycle in the degree 0 case is defined via contracted maps, we have

    \[ \mathsf{DR}_{g,(0,\ldots,0)} = (-1)^g\lambda_g \in \mathsf{CH}^g(\overline{\mathcal{M}}_{g,n}), \]
    where $\overline {\mathcal {M}}_{g,n}$ is the moduli space of stable pointed curves. See [Reference Janda, Pandharipande, Pixton and ZvonkineJPPZ17, §§ 0.5.3 and 3.1].

    Another basic consequence is the $\lambda _g$-formula [Reference Faber and PandharipandeFP03],

    \[ \int_{\overline{\mathcal{M}}_{g,n}} \psi_1^{k_1} \cdots \psi_n^{k_n} \lambda_g = \binom{2g+n-3}{k_1,\ldots, k_n} \cdot \int_{\overline{\mathcal{M}}_{g,1}} \psi_1^{2g-2} \lambda_g, \]
    predicted by the Virasoro constraints for degree 0 maps to curves [Reference Getzler and PandharipandeGP98]. Here
    \[ \psi_i = c_1(\mathbb{L}_i) \in \mathsf{CH}^1(\overline{\mathcal{M}}_{g,n}) \]
    is the Chern class of the cotangent line at the $i$th point. The $\lambda _g$-formula plays a central role in the study of the tautological ring $\mathsf {R}^\star (\mathcal {M}^{\mathsf {ct}}_{g,n})$ of the moduli space of curves of compact type [Reference PandharipandePan12].
  3. (iii) Again as an excess class, $(-1)^g\lambda _g$ appears fundamentally in the local Gromov–Witten theory of surfaces. For example, the Katz–Klemm–Vafa formula [Reference Katz, Klemm and VafaKKV99] proven in [Reference Maulik, Pandharipande and ThomasMPT10, Reference Pandharipande and ThomasPT16] concerns integrals

    \[ \int_{[\overline{\mathcal{M}}_g(S,\beta)]^{\mathsf{red}}} (-1)^g \lambda_g \]
    against the reduced virtual fundamental class of the moduli space of stable maps to $K3$ surfaces. For a recent study of the parallel problem for local log Calabi–Yau surfaces (with integrand $(-1)^g \lambda _g$); see [Reference BousseauBou20].
  4. (iv) The class $(-1)^g\lambda _g$ arises via the pull-back of the universal $0$-section of the moduli space of principally polarized abelian varieties (PPAVs). Over the moduli space of compact type curves, the connection to PPAVs shows a third vanishing property,

    \[ \lambda_g |_{\mathcal{M}_{g}^{\mathsf{ct}}} = 0 \]
    (see [Reference van der GeervdG99]). We will discuss PPAVs further in § 1.2 below.

Our main results here concern the complexity of the class $\lambda _g$ in the Chow ring. For $\overline {\mathcal {M}}_g$, we bound from below the complexity of formulas for

\[ \lambda_g\in \mathsf{CH}^\star(\overline{\mathcal{M}}_g). \]

As a consequence of the connection to the moduli of PPAVs, we also bound from below the complexity of formulas for the universal $0$-section.

The log Chow ring of $(\overline {\mathcal {M}}_g,\partial \overline {\mathcal {M}}_g)$ is defined as a colimit over all iterated blow-ups of boundary strata. The usual Chow ring is naturally a subalgebra

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

The main positive result of the paper is the simplicity of $\lambda _g$ in the log Chow ring. We prove

\[ \lambda_g \in \mathsf{divlogCH}^\star(\overline{\mathcal{M}}_g, \partial \overline{\mathcal{M}}_g), \]


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

is the subalgebra generated by logarithmic boundary divisors. While $\lambda _g$ in Chow is complicated, $\lambda _g$ in log Chow is as simple as possible! We present several related open questions.

1.2 The 0-section

Let $\mathcal {A}_g$ be the moduli space of PPAVs of dimension $g$, and let

\[ \pi: \mathcal{X}_g \rightarrow \mathcal{A}_g \]

be the universal abelian variety $\pi$ equipped with a universal $0$-section

\[ s: \mathcal{A}_g \rightarrow \mathcal{X}_g. \]

The image of the $0$-section determines an algebraic cycle class

\[ Z_g \in \mathsf{CH}^g(\mathcal{X}_g). \]

The second Voronoi compactification of $\mathcal {A}_g$ has been given a modular interpretation by Alekseev:

\[ \mathcal{A}_g \subset \overline{\mathcal{A}}^{\mathsf{Alekseev}}_g. \]

Olsson [Reference OlssonOls12] provided a modular interpretation for the normalization

\[ \overline{\mathcal{A}}^{\mathsf{Olsson}}\rightarrow \overline{\mathcal{A}}_g^{\mathsf{Alekseev}}. \]

Our approach here will be equally valid for both $\overline {\mathcal {A}}^{\mathsf {Olsson}}$ and $\overline {\mathcal {A}}_g^{\mathsf {Alekseev}}$. We will simply denote the compactification by

\[ \mathcal{A}_g \subset \overline{\mathcal{A}}_g, \]

where $\overline {\mathcal {A}}_g$ stands for either the space of Alekseev or the space of Olsson.

The four important propertiesFootnote 3 of the compactification $\overline {\mathcal {A}}_g$ which we will require are as follows.

  • The points of $\overline {\mathcal {A}}_g$ parameterize (before normalization) stable semiabelic pairs which are quadruples $(G,P,L,\theta )$ where $G$ is a semiabelian variety, $P$ is a projective variety equipped with a $G$-action, $L$ is an ample line bundle on $P$, and $\theta \in H^0(P,L)$. The data $(G,P,L,\theta )$ satisfy several further conditions (see § 4.2.16 of [Reference OlssonOls12]).

  • There is a universal semiabelian variety

    \[ \overline{\pi}: \overline{\mathcal{X}}_g \rightarrow \overline{\mathcal{A}}_g \]
    with a $0$-section
    \[ \overline{s}: \overline{\mathcal{A}}_g \rightarrow \overline{\mathcal{X}}_g \]
    corresponding to the semiabelian variety which is the first piece of data of a stable semiabelic pair (the rest of the pair data will not play a role in our study).
  • The usual Torelli map $\tau : \mathcal {M}_g \rightarrow \mathcal {A}_g$ extends canonically,

    \[ \overline{\tau}: \overline{\mathcal{M}}_g \rightarrow \overline{\mathcal{A}}_g \]
    (see [Reference AlekseevAle04]).
  • The $\overline {\tau }$-pull-back to $\overline {\mathcal {M}}_g$ of $\overline {\mathcal {X}}_g$ is the universal family

    \[ \mathsf{Pic}_\epsilon^0 \rightarrow \overline{\mathcal{M}}_g \]
    parameterizing line bundles on the fibers of the universal curve
    \[ \epsilon:\mathcal{C}_g \rightarrow \overline{\mathcal{M}}_g \]
    which have degree 0 on every component of any fiber [Reference AlekseevAle04].

The image of the $0$-section $\overline {s}$ determines an operational Chow class

\[ \overline{Z}_g\in \mathsf{CH}^g_{\mathsf{op}}(\overline{\mathcal{X}}_g) \]

since the image is an étale local complete intersection in $\overline {\mathcal {X}}_g$. The class $\overline {Z}_g$ is related to $(-1)^g\lambda _g$ via a pull-back construction. Let

\[ t:\overline{\mathcal{M}}_g \rightarrow \mathsf{Pic}_\epsilon^0 \]

be the $0$-section defined by the trivial line bundle. By the properties of

\[ \overline{\pi}: \overline{\mathcal{X}}_g \rightarrow \overline{\mathcal{A}}_g \]

discussed above,

\[ \overline{\tau}^*\overline{s}^*(\overline{Z}_g) = t^*(t_*[\overline{\mathcal{M}}_g]). \]

By the standard analysis of the vertical tangent bundle of $\mathsf {Pic}_\epsilon ^0$,

\[ t^*(t_*[\overline{\mathcal{M}}_g]) = (-1)^g \lambda_g \in \mathsf{CH}^g(\overline{\mathcal{M}}_g). \]

Indeed, by the excess intersection formula the class $t^*(t_*[\overline {\mathcal {M}}_g])$ equals the top Chern class of the normal bundle of the $0$-section of $\mathsf {Pic}_\epsilon ^0$. Over $[C]\in \overline {\mathcal {M}}_g$, the fiber of the normal bundle is the first-order deformation space of the trivial line bundle on $C$. The deformation space is given by

\[ H^1(C, \mathcal{O}_C) = H^0(C, \omega_C)^\vee, \]

the fiber of the dual of the Hodge bundle $\mathbb {E}_g^\vee$ with top Chern class $(-1)^g \lambda _g$. We conclude that

(1)\begin{equation} \overline{\tau}^*\overline{s}^*(\overline{Z}_g) = (-1)^g \lambda_g \in\mathsf{CH}^g(\overline{\mathcal{M}}_g). \end{equation}

1.3 Complexity of the $0$-section

The study the $0$-section over $\mathcal {A}_g$ is related to the double ramification cycle (especially over curves of compact type) (see Hain [Reference HainHai13] and Grushevsky-Zakharov [Reference Grushevsky and ZakharovGZ14a]). A central idea there is to use the beautiful formula

(2)\begin{equation} Z_g = \frac{\Theta^g}{g!}\in \mathsf{CH}^g(\mathcal{X}_g), \end{equation}

where $\Theta \in \mathsf {CH}^1(\mathcal {X}_g)$ is the universal symmetric theta divisor trivialized along the $0$-section. The proof of (2) in Chow uses the Fourier–Mukai transformation and work of Deninger and Murre [Reference Deninger and MurreDM91] (see [Reference Birkenhake and LangeBL04, Reference VoisinVoi14]). The article [Reference Grushevsky and ZakharovGZ14a] provides a more detailed discussion of the history of (2).

We are interested in the following question: to what extent is an equation of the form of (2) possible over $\overline {\mathcal {A}}_g$? A result by Grushevsky and Zakharov along these lines appears in [Reference Grushevsky and ZakharovGZ14b]. As before, let

\[ \overline{Z}_g \in \mathsf{CH}^g_{\mathsf{op}}(\overline{\mathcal{X}}_g) \]

be the class of the $0$-section $\overline {s}$. Grushevsky and Zakharov calculate the restriction $\overline {Z}_g|_{\mathcal {U}_g}$ of $\overline {Z}_g$ over a particular open setFootnote 4

\[ \mathcal{A}_g\subset \mathcal{U}_g \subset \overline{\mathcal{A}}_g \]

in terms of $\Theta$, a boundary divisor $D\in \mathsf {CH}^1(\overline {\mathcal {X}}_g|_{\mathcal {U}_g})$, and a class

\[ \Delta \in \mathsf{CH}^2(\overline{\mathcal{X}}_g|_{\mathcal{U}_g}). \]

The result of Grushevsky and Zakharov shows that while the naive extension of (2) does not hold over $\mathcal {U}_g$, the class $\overline {Z}_g|_{\mathcal {U}_g}$ lies in the subalgebra of $\mathsf {CH}^\star (\overline {\mathcal {X}}_g|_{\mathcal {U}_g})$ generated by classes of degrees 1 and 2. The formula of [Reference Grushevsky and ZakharovGZ14b] is a useful extension of (2).

The divisor classes $\mathsf {CH}^1_{\mathsf {op}}(\overline {\mathcal {X}}_g)$ generate a subalgebra

\[ \mathsf{divCH}^\star_{\mathsf{op}}(\overline{\mathcal{X}}_g) \subset \mathsf{CH}^\star_{\mathsf{op}}(\overline{\mathcal{X}}_g). \]

The first bound from below of the complexity of the class of the $0$-section is the following result.

Theorem 1 For all $g\geq 3$, we have $\overline {Z}_g\notin \mathsf {divCH}^\star _{\mathsf {op}}(\overline {\mathcal {X}}_g)$.

As a consequence, no divisor formula extending (2) is possible for $\overline {\mathcal {A}}_g$. Though not stated, the analysis of [Reference Grushevsky and ZakharovGZ14b] over $\mathcal {U}_g$ can be used to show that $\overline {Z}_g|_{\mathcal {U}_g}$ is not in the subalgebra of $\mathsf {CH}^\star (\overline {\mathcal {X}}_g|_{\mathcal {U}_g})$ generated by classes of degree 1. Theorem 1 can therefore also be obtained from [Reference Grushevsky and ZakharovGZ14b].Footnote 5

In fact, we can go further. Let

\[ \mathsf{CH}^\star_{\leq k}(\overline{\mathcal{X}}_g) \subset \mathsf{CH}^*_{\mathsf{op}}(\overline{\mathcal{X}}_g) \]

be the subalgebra generated by all elements of degree at most $k$, so that

\[ \mathsf{divCH}^\star_{\mathsf{op}}(\overline{\mathcal{X}}_g) = \mathsf{CH}^\star_{\leq 1}(\overline{\mathcal{X}}_g). \]

Theorem 2 For all $g\geq 7$, we have $\overline {Z}_g\notin \mathsf {CH}^\star _{\leq 2}(\overline {\mathcal {X}}_g)$.

By Theorem 2, the Grushevsky–Zakharov formula for $\overline {Z}_g |_{{\mathcal {U}}_g}$ will require corrections by higher-degree classes when extended over $\overline {\mathcal {A}}_g$. We propose the following conjecture about the complexity of the class $\overline {Z}_g$.

Conjecture A No extension of (2) over $\overline {\mathcal {A}}_g$ for all $g$ can be written in terms of classes of uniformly bounded degree.

The pull-back relation (1) relates the complexity of the class

\[ \lambda_g\in \mathsf{CH}^\star(\overline{\mathcal{M}}_g) \]

to the complexity of $\overline {Z}_g\in \mathsf {CH}^\star _{\mathsf {op}}(\overline {\mathcal {X}}_g)$. Theorems 1 and 2 will be the immediate consequence of parallelFootnote 6 complexity bounds for $\lambda _g$.

1.4 Complexity of $\lambda _g$

The divisor classes $\mathsf {CH}^1(\overline {\mathcal {M}}_g)$ generate a subalgebra

\[ \mathsf{divCH}^\star(\overline{\mathcal{M}}_g) \subset \mathsf{CH}^\star(\overline{\mathcal{M}}_g). \]

The first bound from below of the complexity of $\lambda _g$ is the following result.

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

Via the pull-back relation (1), Theorem 3 immediately implies Theorem 1. The proof of Theorem 3, presented in § 2, starts with explicit calculations in the tautological ring in genera $3$ and $4$ using the Sage package admcycles [Reference Delecroix, Schmitt and van ZelmDSvZ21]. A boundary restriction argument is then used to inductively control all higher genera.

For the analogue of Theorem 2, let

\[ \mathsf{CH}^\star_{\leq k}(\overline{\mathcal{M}}_g) \subset \mathsf{CH}^*(\overline{\mathcal{M}}_g) \]

be the subalgebra generated by all elements of degree at most $k$. A similar strategy (with a much more complicated initial calculation in genus 5) yields the following result which implies Theorem 2.

Theorem 4 For all $g\geq 7$, we have $\lambda _g\notin \mathsf {CH}^\star _{\leq 2}(\overline {\mathcal {M}}_g)$.

The proofs of Theorems 3 and 4 require new cases of Pixton's conjecture about the ideal of relations in the tautological ring

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

Proposition 5 Pixton's relations generate all relations among tautological classes in $\mathsf {R}^4(\overline {\mathcal {M}}_{4,1})$ and $\mathsf {R}^5(\overline {\mathcal {M}}_{5,1})$.

While the above arguments become harder to pursue in general for $\mathsf {CH}^\star _{\leq k}(\overline {\mathcal {M}}_g)$, we expect the following assertion to hold.

Conjecture B For fixed $k$, $\lambda _g \in \mathsf {CH}^\star _{\leq k}(\overline {\mathcal {M}}_g)$ holds only for finitely many $g$.

Of course, Conjecture B implies Conjecture A.

1.5 Log Chow

Theorems 14 about the classes $\overline {Z}_g$ and $\lambda _g$ are in a sense negative results since formula types are excluded. Our main positive result about $\lambda _g$ concerns the larger log Chow ring

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

The log Chow ring and the subalgebra

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

generated by logarithmic boundary divisors are defined carefully in § 3. Our perspective, using limits over log blow-ups, requires the least background in log geometry. A more intrinsic approach to the definitions can be found in [Reference BarrottBar18].

Theorem 6 For all $g\geq 2$, we have $\lambda _g \in \mathsf {divlogCH}^\star (\overline {\mathcal {M}}_g, \partial \overline {\mathcal {M}}_g)$.

Our proof of Theorem 6 is constructive: we start with Pixton's formula for the double ramification cycle for constant maps [Reference Janda, Pandharipande, Pixton and ZvonkineJPPZ17] and show that each term lies in $\mathsf {divlogCH}^\star (\overline {\mathcal {M}}_g)$. In principle, it is possible to obtain bounds for the necessary log blow-ups from the proof, but these will certainly not be optimal. Finding a minimal (or efficient) sequence of log-blows of $(\overline {\mathcal {M}}_g, \partial \overline {\mathcal {M}}_g)$ after which $\lambda _g$ lies in the subalgebra of logarithmic boundary divisors is an interesting question.

A crucial part of the proof of Theorem 6 is the study in § 5 of the logarithmic tautological ring,

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

defined by a normal crossings divisor $D\subset X$ in a non-singular variety $X$. Tautological classes are defined here using the Chern roots of the normal bundle of logarithmic strata $S\subset X$. The precise definitions are given in § 5.1.

We prove three main structural results about logarithmic tautological classes.

  1. (i) $\mathsf {R}^\star (X,D) \subset \mathsf {divlogCH}^\star (X,D)$.

  2. (ii) pull-backs of tautological classes under log blow-ups are tautological.

  3. (iii) push-forwards of tautological classes under log blow-ups are tautological.

Our first proof of (i) is presented in § 5.2 via an explicit analysis of explosions: sequences of blow-ups associated to logarithmic strata of $X$. A second approach to (i)–(iii), via the geometry of the Artin fan of $(X,D)$, is given in § 5.5. The Artin fan perspective, advocated by D. Ranganathan,Footnote 7 is theoretically more flexible.

After Pixton's formula for the double ramification cycle for constant maps is shown to lie in $\mathsf {R}^\star (\overline {\mathcal {M}}_g, \partial \overline {\mathcal {M}}_g)$, property (i) implies Theorem 6. Since Pixton's formula and the proof of (i) are both effective, it is possible in principle to compute divisor expressions for $\lambda _g$. The result reveals the essential simplicity of $\lambda _g$ and opens the door to the search for a simpler formula in divisors.

The proof of Theorem 6 yields a refined result: only logarithmic boundary divisors over

\[ \Delta_0\subset \overline{\mathcal{M}}_g \]

are needed to generate $\lambda _g$. The parallel result is also true for pointed curves:

\[ \lambda_g \in \mathsf{divlogCH}^\star(\overline{\mathcal{M}}_{g,n},\Delta_0) \]

for $2g-2+n>0$.

We have seen that $(-1)^g \lambda _g$ is a special case of the double ramification cycle. The general double ramification cycle

\[ {\mathsf{DR}}_{g,A} \in \mathsf{CH}^g(\overline{\mathcal{M}}_{g,n}) \]

is defined with respect to a vector of integers $A=(a_1,\ldots,a_n)$ satisfying

\[ \sum_{i=1}^n a_i =0. \]

In [Reference Holmes, Pixton and SchmittHPS19, Appendix A], the double ramification cycle was lifted to log Chow,Footnote 8

(3)\begin{equation} \widetilde{\mathsf{DR}}_{g,A} \in \mathsf{logCH}^g(\overline{\mathcal{M}}_{g,n}). \end{equation}

Motivated by Theorem 6, we conjectureFootnote 9 a uniform divisorial property of the lifted double ramification cycle (3).

Conjecture C For all $g$ and $A$, we have $\widetilde {\mathsf {DR}}_{g,A} \in \underline {\mathsf {div}}\mathsf {logCH}^\star (\overline {\mathcal {M}}_{g,n})$ where

\[ \underline{\mathsf{div}}\mathsf{logCH}^\star(\overline{\mathcal{M}}_{g,n}) \subset \mathsf{logCH}^\star(\overline{\mathcal{M}}_{g,n}) \]

is the subalgebra generated by logarithmic boundary divisors together with the cotangent line classes $\psi _1,\ldots, \psi _n$.

Finally, we return to the $\Theta$-formula (2) for $Z_g$. Is an extension of the $\Theta$-formula possible over $\overline {\mathcal {M}}_g$ in $\mathsf {logCH}^\star (\overline {\mathcal {M}}_g)$? More specifically, can we find

\[ \mathsf{T}\in \mathsf{logCH}^1(\overline{\mathcal{M}}_g) \]

which satisfies the following two properties?

  1. (i) The restriction of $\mathsf {T}$ over the moduli of curves $\mathcal {M}^{\mathsf {ct}}_g$ of compact type is $0$.

  2. (ii) $(-1)^g\lambda _g = \frac {\mathsf {T}^g}{g!} \in \mathsf {logCH}^g(\overline {\mathcal {M}}_g)$.

Property (i) is imposed since

\[ \Theta|_{Z_g} = 0 \in \mathsf{CH}^1(Z_g) \]

by the trivialization condition for $\Theta$. Unfortunately, the answer is no even for genus $2$.

Proposition 7 There does not exist a class $\mathsf {T}\in \mathsf {logCH}^1(\overline {\mathcal {M}}_2)$ satisfying the restriction property (i) and

\[ (-1)^2\lambda_2 = \frac{\mathsf{T}^2}{2!} \in \mathsf{logCH}^2(\overline{\mathcal{M}}_2). \]

The $\Theta$-formula for $(-1)^g\lambda _g$ can not be extended in a straightforward way in $\mathsf {CH}^g(\overline {\mathcal {M}}_g)$ or $\mathsf {logCH}^g(\overline {\mathcal {M}}_g)$. However,

\[ \lambda_g\in \mathsf{logCH}^g(\overline{\mathcal{M}}_g) \]

is a degree $g$ polynomial in the logarithmic boundary divisors over $\Delta _0\subset \overline {\mathcal {M}}_g$.

Question D Find a polynomial formula in logarithmic boundary divisors for $\lambda _g$ in log Chow (without using Pixton's formula).

The larger bChow ring of $\overline {\mathcal {M}}_g$ is defined as a limit over all blow-ups:

\[ \mathsf{CH}^\star(\overline{\mathcal{M}}_g) \subset \mathsf{logCH}^\star(\overline{\mathcal{M}}_g, \partial\overline{\mathcal{M}}_g) \subset \mathsf{bCH}^\star(\overline{\mathcal{M}}_g). \]

The bChow ring is by far the largest of the three Chow constructions. In § 7, we show that the main questions of the paper become trivial in bChow. In fact, for every non-singular variety $X$, we have

\[ \mathsf{divbCH}^\star(X) = \mathsf{bCH}^\star(X). \]

The logarithmic geometry of $\overline {\mathcal {M}}_g$ is therefore the natural place to study Question D for $\lambda _g$.

2. $\lambda _g$ in the Chow ring

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

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


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 8 If $\mathsf {Q}_g(r,k)$ is true, then $\mathsf {Q}_{g+1}(r,k)$ and $\mathsf {Q}_{g+1}(r+1,k)$ are true.

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. The log Chow ring

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. Relationship with logarithmic geometry

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. The divisor subalgebra of log Chow

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.

Theorem 11 Let $(X,D)$ be a non-singular variety with a normal crossings divisor. Let $S\subset X$ be an open stratum. Every normally decorated class associated to $S$ lies in $\mathsf {divlogCH}^\star (X,D)$.

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