2.1 The compact case and the Riemann-Roch theorem

We start with the proof of the special case of the main theorem.

Proposition 2.2. If $B = \overline {B}$ is smooth, compact and k-dimensional, then

(9) $$ \begin{align} \chi(B) = \int_{\overline{B}} \, c_k(T_B) \,. \end{align} $$

We start by recalling some intersection theory. Let $\mathcal {E}$ be a holomorphic vector bundle on B. Denote by the ith Chern class of E. Recall that $c_0 = 1$ and $c_i = 0$ for . The total Chern class of $\mathcal {E}$ is the formal sum $\mathrm {c}(\mathcal {E}) = 1 + c_1 + \cdots + c_r$ in $\mathrm {CH}(B)$ . Splitting formally $\mathrm {c}(\mathcal {E}) = \prod _{i=1}^r (1+\alpha _i)$ into the Chern roots, the Chern character is defined as the formal power series

$$\begin{align*}\mathrm{ch}(E) = \sum_{i=1}^r \exp(\alpha_i) = \sum_{s\geq 0}\frac{1}{s!} \sum_{i=1}^r \alpha_i^s = \mathrm{rk}(E) + c_1 + \frac{1}{2}(c_1^2 - 2c_2) + \cdots .\end{align*}$$

Furthermore, the Todd class is defined as

$$\begin{align*}\mathrm{td}(E) = \prod_{i=1}^r \frac{\alpha_i}{1-\exp(-\alpha_i)} = 1 + \frac{1}{2} c_1 + \frac{1}{12}(c_1^2 + c_2) + \frac{1}{24} c_1c_2 + \cdots .\end{align*}$$

The Grothendieck-Riemann-Roch theorem in the case of a map $f:X \to Y$ , and for the special case that the higher direct images $R^if_* \mathcal {E}$ vanish, states that

(10) $$ \begin{align} \mathrm{ch}(f_* \mathcal{E})\cdot \mathrm{td}(T_Y) = f_*(\mathrm{ch}(\mathcal{E})\cdot\mathrm{td}(T_X))\,. \end{align} $$

Proof of Proposition 2.2.

For a topological proof, see, for example, [Reference Bott and TuBT82, Proposition 11.24]. Using the notations already set up, we give a quick proof if moreover B is Kähler. From the Borel-Serre identity ([Reference FultonFul98, Example 3.2.5]) on a k-dimensional manifold,

$$ \begin{align*} c_k(T_B) = \mathrm{ch}\Bigl(\sum_{j=1}^k (-1)^j \Omega_B^j\Bigr) \cdot \mathrm{td}(T_B) \end{align*} $$

and the application

$$ \begin{align*} \int_{\overline{B}} \mathrm{ch}((-1)^j \Omega_B^j) \cdot \mathrm{td}(T_B) = \sum_{\ell \geq 0} (-1)^{\ell+j} h^\ell(B,\Omega^j_B) \end{align*} $$

of the Grothendieck-Riemann-Roch theorem for the map from B to a point, we get

$$ \begin{align*} \int_{\overline{B}} \,c_k(T_B) = \sum_{\ell,j \geq 0} (-1)^{\ell+j} h^\ell(B,\Omega^j_B) = \chi(B) \end{align*} $$

by the Hodge decomposition.

2.2 The noncompact case and log differential forms

We suppose throughout that $D = \cup _{j=1}^s D_j$ is a reduced normal crossing divisor: that is, with distinct irreducible components $D_i$ intersecting each other transversally. In this situation, $\Omega ^1_{\overline {B}}(\log D)$ is defined to be the vector bundle of rank n with the following local generators. In a neighbourhood U of a point where (say) the first $r \leq s$ divisors meet and $x_1,\ldots ,x_k$ is a local coordinate system with $D_j = \{x_j = 0\}$ , then the logarithmic cotangent bundle is defined by

(11) $$ \begin{align} \Omega^1_{\overline{B}}(\log D)(U) = \Big\langle \frac{dx_1}{x_1}, \ldots, \frac{dx_r}{x_r}, dx_{r+1}, \ldots, dx_{k} \Big \rangle\, \end{align} $$

as an $\mathcal {O}_{\overline B}(U)$ -module. There is a fundamental exact sequence for log differential forms, namely

(12) $$ \begin{align}\begin{aligned} 0&\to \Omega^1_{\overline B} \to \Omega^1_{\overline B}(\log D) \to \oplus_{j=1}^s (\mathfrak{i}_{j})_* \mathcal{O}_{D_j} \to 0\,, \end{aligned}\end{align} $$

where $\mathfrak {i}_j: D_j \to \overline B$ is the inclusion map. More details can be found, for example, in [Reference Esnault and ViehwegEV92, Proposition 2.3].

Proof of Proposition 2.1.

We first reduce to the case that D has simple normal crossings: that is, to the case that the $D_j$ are all smooth. This can always be achieved by an étale covering. Since both sides of equation (8) are multiplied by the degree under such a covering, we can assume simple normal crossings. Our goal is to prove

$$ \begin{align*} \int_{\overline B} c_k\Bigl(\Omega^1_{\overline{B}} \Bigl(\log \sum_{i \geq 2} D_i\Bigr)\Bigr) = \int_{\overline B} c_k\Bigl(\Omega^1_{\overline{B}}(\log D)\Bigr) - \int_{D_1} c_{k-1}\Bigl(\Omega^1_{D_1} \Bigl(\log \Bigl(\sum_{i \geq 2} D_i \cap D_1 \Bigr)\Bigr)\Bigr). \end{align*} $$

The claim follows then from the additivity $\chi (B) + \chi (D) = \chi (\overline {B})$ of the Euler characteristic, Proposition 2.2 and an application of the preceding identity to $B_j = \overline {B} \smallsetminus \cup _{i=j}^s D_j$ .

We consider the inclusion of the boundary divisor $D_1$ and deduce from the ideal sheaf sequence that $c((\mathfrak {i}_1)_*\mathcal {O}_{D_1}) =(1-[D_1])^{-1}$ and that $c(\mathcal {N}_{D_1}) = 1 + \mathfrak {i}_1^*[D_1]$ . Moreover the normal bundle sequence $0\to \mathcal {T}_{D_1} \to \mathfrak {i}_1^*\mathcal {T}_{\overline {B}} \to \mathcal {N}_{D_1} \to 0$ implies

(13) $$ \begin{align} c(\Omega^1_{D_1}) = \mathfrak{i}_1^* \Bigl( c(\Omega^1_{\overline{B}}) \cdot \frac{1}{1-[D_1]} \Bigr)\,. \end{align} $$

On the other hand, the sequence in equation (12) gives

(14) $$ \begin{align} c(\Omega^1_{\overline{B}}(\log D)) = c(\Omega^1_{\overline{B}}) \cdot \frac{1}{1-[D_1]} \cdot \prod_{j=2}^s \frac{1}{1-[D_j]} \end{align} $$

and also

(15) $$ \begin{align} \quad c\Bigl(\Omega^1_{D_1}\Bigl(\log \Bigl(\sum_{i \geq 2} D_i \cap D_1 \Bigr)\Bigr)\Bigr) = c(\Omega^1_{D_1}) \cdot \prod_{j=2}^s \frac{1}{1-[ D_1\cap D_j]}. \end{align} $$

Hence, comparing with equation (13), we get

(16) $$ \begin{align} c\Bigl(\Omega^1_{D_1} \Bigl(\log \Bigl(\sum_{i \geq 2} D_i \cap D_1 \Bigr)\Bigr) \Bigr) = \mathfrak{i}_1^*c\big(\Omega^1_{\overline{B}}(\log D)\big)\,. \end{align} $$

Finally, from equation (14) and the appropriate version of the sequence in equation (12), we also get

(17) $$ \begin{align} c(\Omega^1_{\overline{B}}(\log D)) = \frac{1}{1-[D_1]} c\Bigl(\Omega^1_{\overline{B}} \Bigl(\log \sum_{i \geq 2} D_i\Bigr)\Bigr). \end{align} $$

The claim now follows by multiplying this last expression with $1-[D_1]$ , integrating and taking the kth coefficient, using that $\int _{\overline {B}} [D_1] \cdot c_{k-1} (\Omega ^1_{\overline {B}} (\log D)) = \int _{D_1} \mathfrak {i}_1^*c_{k-1} (\Omega ^1_{\overline {B}}(\log D))$ .

3.1 Enhanced level graphs

To define strata and the ambient space in the meromorphic case, we assume that there are r positive m’s, s zeroes and l negative m’s, with $r+s+l=n$ : that is, we have $m_1\ge \dots \ge m_r>m_{r+1}=\dots =m_{r+s} = 0>m_{r+s+1}\ge \dots \ge m_{n}$ . Note that $m_j=0$ is allowed, representing an ordinary marked point. A pointed flat surface is usually denoted by $(X,\omega ,\mathbf {z})$ , where $\mathbf {z} = (z_1,\ldots ,z_n)$ are the marked points corresponding to the zeros, ordinary marked points and poles of $\omega $ . The sections over ${\Omega \overline {\mathcal M}}_{g,n}(\mu )$ corresponding to those marked points are denoted by $\mathcal {Z}_i$ . We denote the polar part of $\mu $ by $\tilde \mu =(m_{r+s+1},\dots ,m_n)$ . The strata of meromorphic differentials are then naturally defined inside the twisted Hodge bundle

$$ \begin{align*}K{\overline{\mathcal M}}_{g,n}(\tilde{\mu}) = f_* \Bigl(\omega_{\mathcal{X}/{\overline{\mathcal M}}_{g,n}} \Bigl(-\sum_{j=r+s+1}^n m_j \mathcal{Z}_j\Bigr)\Bigr).\end{align*} $$

The strata are smooth complex substacks ${\Omega \mathcal M}_{g,n}(\mu )$ of dimension $N = 2g-1+n$ in the holomorphic case $l=0$ and $N=2g-2+n$ in the meromorphic case.

To each boundary point in , there is an associated enhanced level graph, and D is stratified by the type of this associated graph. Here a level graph is defined to be a stable graph $\Gamma = (V,E,H)$ , with half-edges in H that are either paired to form edges E or correspond to the n marked points, together with a total order on the vertices (with equality permitted). The graph $\Gamma $ is supposed to be connected here; from Section 8 on its components are in bijection with the components of the flat surfaces the generalised stratum parametrises. For convenience, we usually define the total order using a level function $\ell : V(\Gamma ) \to \mathbb {Z}$ , usually normalized to take values in $\{0,-1,\ldots ,-L\}$ . We usually write $H_m = H \smallsetminus E$ for the half-edges corresponding to the marked points. Moreover, an enhancement (in [Reference Farkas and PandharipandeFP18] or [Reference Chen, Möller, Sauvaget and ZagierCMSZ20], this number is called a twist) is an assignment of a number $\kappa _e \geq 0$ to each edge e, so that $\kappa _e = 0$ if and only if the edge is horizontal. The triple $(\Gamma , \ell , \{\kappa _e\}_{e \in E(\Gamma )})$ is called an enhanced level graph. We denote the closure of the boundary stratum parametrising multi-scaled differentials (as defined below) compatible with $(\Gamma , \ell , \{\kappa _e\})$ by $D_{(\Gamma , \ell , \{\kappa _e\})}$ or usually simply by $D_\Gamma $ .

In particular, the boundary divisors consist of the divisor $D_{\text {h}}$ parametrising level graphs on just one level and with just one horizontal edge and the (‘vertical’) boundary divisors indexed by two-level graphs without horizontal edges. (For holomorphic types $\mu $ , the divisor $D_{\text {h}}$ is irreducible and parametrises graphs with a nonseparating edge; but for general $\mu $ , this divisor is reducible.) We give local coordinates near the boundary divisors in Section 6.2.

Note that the boundary strata $D_\Gamma $ may be empty for some enhanced level graphs. Deciding nonemptiness is the same as the realisability question that was addressed in [Reference Möller, Ulirsch and WernerMUW21] purely in terms of graphs. The general version, taking into account the residue conditions, is stated in the algorithmic part of [Reference Costantini, Möller and ZachhuberCMZ20]. Note that these boundary strata may also be disconnected; see the discussion of prong-matching equivalence classes below.

Recall that the construction of in [Reference Bainbridge, Chen, Gendron, Grushevsky and MöllerBCGGM3] gives a morphism to the projectivised twisted Hodge bundle over the Deligne-Mumford compactification. The line bundle $\mathcal {O}_{\overline {B}}(-1)$ is the pullback of the tautological bundle from there.

3.2 Twisted differentials and multi-scale differentials

The space is a moduli stack for families of a certain collection of differentials, called multi-scale differentials, and this modular interpretation will be used, for example, in Section 4 to define clutching maps and projection maps at the boundary. However, we will refer to as a moduli space to stick to the commonly used terminology. We recall the definition of a single multi-scale differentials, referring for full details of the definition for families to [Reference Bainbridge, Chen, Gendron, Grushevsky and MöllerBCGGM3]. We will recall further details where needed.

When referring to prongs, we fix a direction in $S^1$ throughout, say the horizontal direction. Suppose that a differential $\omega $ has a zero of order $m \geq 0$ at $q \in X$ . The differential $\omega $ selects inside the real projectivised tangent space $P_q = T_pX/ \mathbb {R}_{>0}$ a collection of $\kappa = m+1$ horizontal (outgoing) prongs at q, the tangent vectors $\mathbb {R}_{>0} \cdot \zeta ^i_\kappa \partial /\partial z$ in a chart where $\omega = z^m dz$ is in standard form and $\zeta _\kappa $ is a primitive $\kappa $ th root of unity. We denote them by $P_q^{\mathrm {out}} \subset P_q$ . The prongs are equivalently the tangent vectors to the outgoing horizontal rays. Dually, if $\omega $ has a pole of order $m \leq -2$ , then $\omega $ has $\kappa = -m-1$ horizontal (incoming) prongs at q, denoted by $P_q^{\mathrm {in}} \subset P_q$ , the tangent vectors $\mathbb {R}_{>0} \cdot -\zeta ^i_\kappa \partial /\partial z$ in a chart where $\omega = z^m dz$ .

We start with an auxiliary notion of differentials from [Reference Bainbridge, Chen, Gendron, Grushevsky and MöllerBCGGM1]. Given a pointed stable curve $(X,\mathbf {z})$ , a twisted differential is a collection of differentials $\eta _v$ on each component $X_v$ of X: that is, compatible with a level structure on the dual graph $\Gamma $ of X – it vanishes as prescribed by $\mu $ at the marked points z and satisfies the matching order condition at vertical nodes, the matching residue condition at horizontal nodes and the global residue condition of [Reference Bainbridge, Chen, Gendron, Grushevsky and MöllerBCGGM1]. We usually group the differentials on the components of level i of X to form the collection $\eta _{(i)}$ and refer to a twisted differential by ${\boldsymbol {\eta }} = (\eta _{(i)})$ .

A multi-scale differential of type $\mu $ on a stable curve X consists of an enhanced level structure $(\Gamma ,\ell ,\{\kappa _e\})$ on the dual graph $\Gamma $ of X, a twisted differential of type $\mu $ compatible with the enhanced level structure and a prong-matching for each node of X joining components of nonequal levels. Here the compatibility with the enhanced level structure requires that at each of the two points $q^\pm $ glued to form the node corresponding to the edge $e \in E(\Gamma )$ the number of prongs of the differential ${\boldsymbol {\eta }}$ is equal to $\kappa _e$ . Moreover, a prong-matching is an order-reversing isometry $\sigma _q: P_{q^-} \to P_{q^+}$ that induces a cyclic order-reversing bijection $\sigma _q: P^{\mathrm {in}}_{q^-} \to P^{\mathrm {out}}_{q^+}$ between the incoming prongs at $q^-$ and the outgoing prongs at $q^+$ .

Finally, we state the equivalence relation on multi-scale differentials used to construct . Multi-scale differentials only retain the information on the lower levels up to projectivisation. This rescaling of the lower levels is roughly given by a multiplicative torus $T^{L(\Gamma )}$ . More precisely, the group $(\mathbb {C}^*)^{L(\Gamma )}$ acts by rescaling plainly the differentials on each level. The universal cover $\mathbb {C}^{L(\Gamma )} \to (\mathbb {C}^*)^{L(\Gamma )}$ acts by rescaling the differentials on each level and simultaneously by fractional Dehn twists on the prong-matching. In fact, a subgroup acts trivially: the twist group $\mathrm {Tw}_{\Gamma }$ that we describe in detail in Section 3.4. So the action factors through the action of the quotient $T_\Gamma = \mathbb {C}^{L(\Gamma )}/\mathrm {Tw}_{\Gamma }$ called the level rotation torus, and two multi-scale differentials are defined to be equivalent if they differ by the action of $T_\Gamma $ .

The projectivised space parametrises projectivised multi-scale differentials, where $\mathbb {C}^*$ acts by simultaneously rescaling the differentials on all levels and leaving the prong-matchings untouched.

3.3 Divisors, degeneration, undegeneration

We let $\mathrm {LG}_L(B)$ be the set of all enhanced $(L+1)$ -level graphs without horizontal edges. Recall that boundary divisors of $\overline {B}$ are $D_{\text {h}}$ and $D_\Gamma $ for $\Gamma \in \mathrm {LG_1(B)}$ . For later use, we define

(18) $$ \begin{align} D = D_{\text{h}} + \sum_{\Gamma\in \mathrm{LG_1(B)}} D_\Gamma \end{align} $$

to be the total boundary divisor. The structure of the normal crossing boundary of is encoded by undegenerations. Given a nonhorizontal level graph $\Gamma $ with $L+1$ levels, the associated boundary stratum $D_{\Gamma }$ is contained in the intersection of L boundary divisors $D_{\Gamma _i}$ for $i=1,\ldots ,L$ , and we can describe this inclusion as follows. View the ith level passage as a horizontal line just above level $-i$ . Contract in $\Gamma $ all edges that do not cross this horizontal line to obtain a contraction map $\delta _i: \Gamma \to \Gamma _i$ of enhanced level graphs, where $\Gamma _i$ obtains a two-level structure with the top level corresponding to the components above the horizontal line and the bottom level those below that line. We call this the ith undegeneration of $\Gamma $ . This can be generalised for any subset $I = \{i_1,\dots ,i_n\} \subseteq \{1,\dots ,L\}$ and results in the undegeneration map

$$ \begin{align*} \delta_{i_1,\dots,i_n} \colon \mathrm{LG}_L(B)\to \mathrm{LG}_{n}(B)\,, \end{align*} $$

which contracts all the passage levels of a nonhorizontal level graph $D_{\Gamma }$ except for the passages between levels $-i_{k}+1$ and $-i_{k}$ , for those $i_k \in I$ . For notational convenience, we define $\delta _I^\complement = \delta _{I^\complement }$ .

A degeneration of level graphs is simply the inverse of an undegeneration. It is convenient to have a symbol to express this dual process, and we write

(19) $$ \begin{align} \Gamma {\rightsquigarrow} \widehat{\Delta} \qquad \text{or} \qquad \Gamma \overset{[i]}{\rightsquigarrow} \widehat{\Delta} \end{align} $$

for a general undegeneration, respectively, specifically for an undegeneration where the ith level is split into two levels.

Note that the map of graphs $\delta _I$ is only well-defined up to post-composition by automorphisms of the enhanced level graph $\delta _I(\Gamma )$ . Taking this into account will be important for intersection theory; see Section 8.1 and [Reference Costantini, Möller and ZachhuberCMZ20, Section 7].

3.4 Prong-matchings and their equivalence classes

In this section, we illustrate the amount of combinatorial information encoded in the notion of prong-matching, given that we also have to take into account the action of the level rotation torus. We start with a recurrent example.

Case of a level graph $\Gamma \in \mathrm {LG_1(B)}$ : that is, a divisor $D_\Gamma $ different from $D_{\text {h}}$ . Such an enhanced level graph has $|E(\Gamma )|$ edges, each of which carries the information of the prongs; consequently there are $K_\Gamma = \prod _{e \in E(\Gamma )} \kappa _e$ prong-matchings. However, this does not imply that locally $D_\Gamma $ is a degree $K_\Gamma $ -cover of the product of the moduli spaces corresponding to the upper and lower levels. Instead, the effect of projectivisation of the lower level on prong-matchings must be considered. This effect is given by the action of the level rotation group $R_\Gamma \cong \mathbb {Z} \subset \mathbb {C}$ in the universal cover of the level rotation torus. This group $R_\Gamma $ acts diagonally, turning the prong-matching at each edge by one (in a fixed direction). The stabiliser of a prong-matching is the twist group $\mathrm {Tw}_{\Gamma }$ referred to above. It is isomorphic to $\ell _\Gamma \mathbb {Z}$ as a subgroup of $R_\Gamma $ , where

(20) $$ \begin{align} \ell_\Gamma = \mathrm{lcm}(\kappa_e \colon e \in E(\Gamma)) \,. \end{align} $$

Orbits of $R_\Gamma $ are also called equivalence classes of prong-matchings. For two-level graphs, there are $g_{\Gamma } := K_{\Gamma }/\ell _{\Gamma }$ such equivalence classes.

For a general level graph $\Delta $ the situation is more complicated and the compactification acquires a nontrivial quotient stack structure that can be computed as follows. As above, there are $K_\Delta = \prod _{e \in E(\Delta )} \kappa _e$ prong-matchings. Now the level rotation group is $R_\Delta \cong \mathbb {Z}^{L}$ , where the ith factor twists by one all prong-matchings that cross the horizontal line above level $-i$ . The stabiliser of a prong-matching is still called the twist group $\mathrm {Tw}_{\Delta }$ . However, this group is no longer a product of the level-wise factors. In fact, for each $i \in \mathbb {N}$ , the twist group of the level-undegeneration $D_{\delta _i(\Delta )}$ is a subgroup of $\mathrm {Tw}_{\Delta }$ , and we call the sum of these subgroups the simple Twist group $\mathrm {Tw}_{\Delta }^s$ . The generic stack structure of $D_\Delta $ is given by the product of the action of the group $\mathrm {Aut}(\Delta )$ of enhanced level graphs automorphisms and a cyclic group of order

$$\begin{align*}e_{\Delta}=[\mathrm{Tw}_{\Delta}: \mathrm{Tw}_{\Delta}^s].\end{align*}$$

The number of prong-matching equivalence classes is

(21) $$ \begin{align} g_\Delta \,:= \,|R_\Delta-\text{orbits on the set}\ K_\Delta| = K_\Delta/[R_\Delta:\mathrm{Tw}_{\Delta}]\,. \end{align} $$

These indices can easily be computed using the elementary divisor theorem.

We discuss a simple case below where $\mathrm {Tw}_{\Delta } \neq \mathrm {Tw}_{\Delta }^s$ in preparation for the examples in Section 10. Finally, we generalise for later use the lcm defined above. We define $\ell _\Delta = \prod _{i=1}^L \ell _{\Delta ,i}$ and we use from now on the notation

(22) $$ \begin{align} \ell_{\Delta,i} = \mathrm{lcm}\Bigl(\kappa_e \colon e \in E(\Gamma)^{>-i}_{\leq -i}\Bigr) = \ell_{\delta_i(\Delta)} \end{align} $$

as an abbreviation of the one defined in the introduction, where $E(\Gamma )^{>-i}_{\leq -i}$ are the edges starting at level $-i+1$ or above and ending at level $-i$ or below.

Example 3.3. In the graph $\Delta $ in Figure 1 (left), there are three edges $e_1$ , $e_2$ and $e_3$ with enhancements a, b and c. The group $R_\Delta \cong \mathbb {Z}^{2}$ acts on $ \mathbb {Z}/a\oplus \mathbb {Z}/b\oplus \mathbb {Z}/c $ by mapping

$$ \begin{align*} (1,0) \mapsto (1,1,0) \quad \text{and} \quad (0,1) \mapsto (0,1,1)\,. \end{align*} $$

Consequently, there are $\gcd (a,b,c)$ orbits: that is, that many equivalence classes of prong-matchings near such a boundary point. The index of the twist group $\mathrm {Tw}_{\Delta }$ in $R_\Delta $ is thus $abc/\gcd (a,b,c)$ . On the other hand, as a consequence of the discussion in the divisor case, the index of the simple twist group $\mathrm {Tw}_{\Delta }^s$ in $R_\Delta $ is $ab/\gcd (a,b)\cdot bc/\gcd (b,c)$ . Since $\Delta $ has no level graphs automorphisms – that is, $\mathrm {Aut}(\Delta )$ is trivial – we conclude that in this case $D_\Delta $ is a quotient stack by a group of order

(23) $$ \begin{align} e_\Delta = \frac{\gcd(a,b,c)\,\mathrm{lcm}(a,b)\,\mathrm{lcm}(b,c)}{abc}\,. \end{align} $$

4.1 The compactification of generalised strata

We start with the definition of strata in the generality that we need. First, we allow for disconnected surfaces. Throughout, $\mu _i=(m_{i,1},\dots ,m_{i,n_i})\in \mathbb {Z}^{n_i}$ is the type of a differential: that is, we require that $\sum _{j=1}^{n_i} m_{i,j}=2g_i-2$ for some $g_i \in \mathbb {Z}$ for $i=1,\ldots ,k$ . For a tuple $\mathbf {g} = (g_1,\ldots ,g_k)$ of genera and a tuple $\mathbf {n} = (n_1,\ldots ,n_k)$ together with ${\boldsymbol {\mu }} = (\mu _1,\ldots , \mu _k)$ , we define the disconnected stratum

(24) $$ \begin{align} {\Omega\mathcal M}_{\mathbf{g},\mathbf{n}}({\boldsymbol{\mu}}) = \prod_{i=1}^k {\Omega\mathcal M}_{g_i,n_i}(\mu_i)\,. \end{align} $$

The projectivised stratum $\mathbb {P}\kern1pt{\Omega \mathcal M}_{\mathbf {g},\mathbf {n}}({\boldsymbol {\mu }})$ is the quotient by the diagonal action of $\mathbb {C}^*$ , not the quotient by the action of $(\mathbb {C}^*)^k$ .

Next, we prepare for global residue conditions. Let $H_p\subseteq \cup _{i=1}^k \{(i,1),\cdots (i,n_i)\}$ be the set of marked points such that $m_{i,j} < -1$ . Now consider vector spaces $\mathfrak {R}$ of the following special shape, modeled on the global residue condition from [Reference Bainbridge, Chen, Gendron, Grushevsky and MöllerBCGGM1]. Let $\lambda $ be a partition of $H_p$ with parts denoted by $\lambda ^{(k)}$ and a subset $\lambda _{\mathfrak {R}}$ of the parts of $\lambda $ such that

The subspace of surfaces with residues in $\mathfrak {R}$ will be denoted by

, and we will refer to them as generalised strata, too.

To compute dimensions, for example, it is convenient to define the residue subspace

(25) $$ \begin{align} R = \prod_{i=1}^{k} R_i \, \subseteq \, \prod_{i=1}^{k} \mathbb{C}^{l_i} \end{align} $$

of differentials of the generalised stratum ${\Omega \mathcal M}_{\mathbf {g},\mathbf {n}}({\boldsymbol {\mu }})$ , where $l_i$ is the number of negative entries in $\mu _i$ . Here $R_i$ is the vector subspace cut out by the residue theorem in the ith component in the space generated by the vectors $r_{i,j}$ for each $(i,j)$ with $m_{i,j} \leq -1$ . When writing $\mathfrak {R} \cap R$ , we consider the intersection inside $ \prod _{i=1}^{k} \mathbb {C}^{l_i}$ .

Remark 4.1. The dimension of the generalised stratum is

(26)

where $l=\sum l_i$ is the total number of poles: that is, marked points with $m_{i,j}<0$ .

We claim that the construction in [Reference Bainbridge, Chen, Gendron, Grushevsky and MöllerBCGGM3] can be carried out for disconnected surfaces and surfaces with an assigned residue subspace. We only have to replace in the definition of the twisted differentials $(X=(X_v)_{v \in V(G)}, \eta = (\eta _v)_{v \in V(G)})$ compatible with an enhanced level graph $\Gamma $ the global residue condition by the following condition. We construct a new auxiliary level graph $\widetilde {\Gamma }$ by adding a new vertex $v_{\lambda ^{(k)}}$ to $\Gamma $ at level $\infty $ for each element $\lambda ^{(k)}\in \lambda _{\mathfrak {R}}$ and converting a tuple $(i,j)\in \lambda ^{(k)}$ into an edge from the marked point $(i,j)$ to the vertex $v_{\lambda ^{(k)}}$ .

• ○ $\mathfrak {R}$ -global residue condition ( $\mathfrak {R}$ -GRC). The tuple of residues at the poles in $H_p$ belongs to $\mathfrak {R}$ , and for every level $L<\infty $ of $\widetilde {\Gamma }$ and every connected component Y of the subgraph $\widetilde {\Gamma }_{>L}$ , one of the following conditions holds:

  1. i) The component Y contains a marked point with a prescribed pole that is not in $\lambda _{\mathfrak {R}}$ .

  2. ii) The component Y contains a marked point with a prescribed pole $(i,j) \in H_p$ , and there is an $r \in \mathfrak {R}$ with $r_{(i,j)} \neq 0$ .

  3. iii) Let $q_1,\ldots ,q_b$ denote the set of edges where Y intersects $\widetilde {\Gamma }_{=L}$ . Then

     $$ \begin{align*}\sum_{j=1}^b\mathrm{Res}_{q_j^-}\eta_{v^-(q_j)} = 0\,,\end{align*} $$
     where $v^-(q_j)\in \widetilde {\Gamma }_{=L}$ .

This differs from the global residue condition in [Reference Bainbridge, Chen, Gendron, Grushevsky and MöllerBCGGM1] only in the subdivision of cases in i) and ii). As for the normal GRC (see [Reference Möller, Ulirsch and WernerMUW21]), the $\mathfrak {R}$ -GRC also has an algorithmic graph theoretic description; see [Reference Costantini, Möller and ZachhuberCMZ20].

Again, as in the usual situation, the divisors $D_\Gamma $ of generalised strata may also be disconnected or empty.

Example 4.3. To illustrate the $\mathfrak {R}$ -global residue condition, we consider the generalised stratum $\overline {B} = \mathbb {P}\left ( {\Omega \mathcal M}_{0,{3}}(-2,-2,2)\times {\Omega \mathcal M}_{0,{4}}(-2,-2,1,1)\right )^{\mathfrak {R}} $ , where the special legs are given by the first two marked points of the first component and the first two marked points of the second components,

$$\begin{align*}H_p = \{(1,1),(2,1),(1,2),(2,2)\}, \end{align*}$$

and the residue space is given by the partition

$$\begin{align*}\lambda_{\mathfrak{R}} = \{\{(1,1),(1,2)\},\{(2,1),(2,2)\}\}.\end{align*}$$

This means

$$\begin{align*}\mathfrak{R} = \{r_{(1,1)}+r_{(2,1)} = 0,\ r_{(1,2)}+r_{(2,2)} = 0\} \subset \mathbb{C}^4,\end{align*}$$

and R is the subspace defined by the residue theorem on each of the two components, namely

$$\begin{align*}R = \{r_{(1,1)}+r_{(1,2)} = 0,\,\,\ r_{(2,1)}+r_{(2,2)} = 0\}. \end{align*}$$

By Remark 4.1, the above generalised stratum has dimension $1$ . We want to show that the $\mathfrak {R}$ -GRC implies that there is only one $2$ -level boundary divisor in the compactification defined in Proposition 4.2. This divisor is given by the $2$ -level graph with the 4-marked component on level $0$ and the other component on level $-1$ .

The only two possible level graphs that could occur are the 2-level graph $\Gamma _1$ described above and the 2-level graph $\Gamma _2$ , where the two components are inverted. Consider the auxiliary level graphs $\widetilde {\Gamma _1}$ and $\widetilde {\Gamma _2}$ needed to check the $\mathfrak {R}$ -GRC given in Figure 2.It is easy to see that condition (iii) of the $\mathfrak {R}$ -GRC implies that the graph $\Gamma _1$ is illegal since both residues on the genus 0 component with the single zero of order 2 on the top level are zero, and this cannot happen.

4.2 Level projections and clutching

Consider a boundary stratum $D_\Gamma $ given by an enhanced level graph $\Gamma $ . It parametrises multi-scale differentials, a differential on each level together with a prong-matching. However, there are no well-defined projection morphisms to the generalised strata on each level. For example, $D_\Gamma $ might have generically trivial quotient stack structure, and the generalised strata on its levels might have everywhere trivial stack structure, and yet special points of $D_\Gamma $ have nontrivial quotient structure. A graph $\Gamma $ with two edges and two levels degenerating to a triangle (Figure 1, left) provides an example. This is due to the fact that the equivalence relation in the notion of multi-scale differentials involves the twist group, which in the presence of edges across multiple levels intertwines what happens at the levels. Our goal here is to define a cover of $D_\Gamma $ that has such projection maps.

To define the generalised strata at the levels of $D_\Gamma $ , we let $(\mathbf {g}^{[i]}, \mathbf {n}^{[i]},{\boldsymbol {\mu }}^{[i]})$ for $i=0,\ldots ,-L$ be the discrete parameters genus, number of points and type at level i, and we let $\mathfrak {R}^{[i]}$ be the residue condition imposed at level i. These residue conditions are constructed via the $\mathfrak {R}$ -GRC described before. Our goal is:

Proposition 4.4. There exists a stack $D_\Gamma ^s$ , called the simple boundary stratum of type $\Gamma $ that admits a finite map $c_\Gamma : D_\Gamma ^s \to D_\Gamma $ and finite forgetful maps

(27)

for each $i=0,\ldots ,-L$ .

We denote by $p_\Gamma = \prod _{i=0}^{-L} p_\Gamma ^{[i]}$ the product of all level projections. In the case that $D_\Gamma $ is a divisor, we will also denote the two projections by

With the help of the finite coverings $c_\Gamma $ and the inclusion of the boundary strata $\mathfrak {i}_\Gamma : D_\Gamma \to \overline {B}$ , we have now the clutching maps $\zeta _\Gamma = \mathfrak {i}_\Gamma \circ c_\Gamma $ at our disposal to define the generators of the tautological ring appearing in Theorem 1.5.

The strategy of the proof of the proposition is to construct $D_\Gamma ^s$ as a cover dominating the local covers of neighbourhoods of more degenerate boundary strata, following the strategy already used in [Reference MumfordMum83]. We do not attempt to analyse whether $D_\Gamma ^s$ is smooth, but the covering we construct is branched at worst over the boundary divisors (hence locally over the coordinate axis, since the boundary is normal crossing); so $D_\Gamma ^s$ has at worst Cohen-Macaulay singularities (Proposition 2.2 in [Reference MumfordMum83]), which allows us to perform intersection theory as in [Reference MumfordMum83]. The objects of the construction are summarised in the following diagram that we now explain.

For any level graph $\Delta $ that is a degeneration of $\Gamma $ , we let $U_\Delta \subset D_\Gamma $ be the open subset parametrising multi-scale differentials compatible with an undegeneration of $\Delta $ . In particular, $U_\Gamma \subset D_\Gamma $ is the complement of all boundary strata where $\Gamma $ degenerates further. These $U_\Delta $ are covered by open subsets of the projectivised Dehn space [Reference Bainbridge, Chen, Gendron, Grushevsky and MöllerBCGGM3, Section 12], which is given as a set by

(28) $$ \begin{align} \mathbb{P}\Xi\mathcal{D}_\Delta = \Bigl(\coprod_{\Gamma {\rightsquigarrow} \Pi {\rightsquigarrow} {\Delta}} \big(\mathfrak{W}_{\operatorname{pm}}(\Pi)/T_{\Pi}^s\big)/\mathrm{Tw}_{\Delta} \Bigr) /\mathbb{C}^*\,, \end{align} $$

where $\mathfrak {W}_{\operatorname {pm}}(\Pi )$ is the space of prong-matched twisted differentials compatible with a $\mathrm {Tw}_{\Pi }^s$ -marking [Reference Bainbridge, Chen, Gendron, Grushevsky and MöllerBCGGM3, Sections 5 and 8], where the (simple) twist groups are defined in Section 3.4 and $T_{\Pi }^s = \mathbb {C}^{L(\Pi )}/\mathrm {Tw}_{\Pi }^s$ is the simple level rotation torus, a finite cover of the level rotation torus defined there. The complex structure of this Dehn space is provided in [Reference Bainbridge, Chen, Gendron, Grushevsky and MöllerBCGGM3, Sections 5 and 8], which at the same time exhibits this space as the quotient stack of the projectivised simple Dehn space

(29) $$ \begin{align} \mathbb{P}\Xi\mathcal{D}_\Delta^s = \Bigl(\coprod_{\Gamma {\rightsquigarrow} \Pi {\rightsquigarrow} {\Delta}} \big(\mathfrak{W}_{\operatorname{pm}}(\Pi)/T_{\Pi}^s\big)/\mathrm{Tw}_{\Delta}^s \Bigr) /\mathbb{C}^*. \end{align} $$

We define $U_\Delta ^s \to U_\Delta $ to be the cover induced locally by the covering $\mathbb {P}\Xi \mathcal {D}_\Delta ^s \to \mathbb {P}\Xi \mathcal {D}_\Delta $ and moreover by labelling the edges of the ‘ambient’ graph $\Gamma $ , killing automorphisms that permute these. The first step is well-defined since we work on strata undegenerating $\Delta $ , and the passage to the simple twist group defines a cover independently of the chosen auxiliary Teichmüller marking.

Since the edges of $\Gamma $ are labelled for points in $U^s_\Delta $ , and since the equivalence relation in equation (29) is defined level by level, we may decompose the differentials parametrised by $U^s_\Delta $ according to the levels of $\Gamma $ . In this way, we obtain maps $p_\Gamma ^{\Delta ,[i]}: U_\Delta ^s \to B_\Gamma ^{[i]}$ such that the product map $p_\Gamma ^\Delta = \prod _i p_\Gamma ^{\Delta ,[i]}$ is a finite cover of an open subset $B_{\Gamma ,\Delta }$ of the product of level strata $B_\Gamma = \prod _i B_\Gamma ^{[i]}$ .

The last step is to define a covering dominating all the $c_\Gamma ^\Delta $ . For technical reasons, we first define the ‘generically simple’ intermediate space $D_\Gamma ^{\mathrm {gs}}$ that removes the stack structure over the open subset $U_\Gamma $ (if there is) just as above, by passage to the simple covering and marking edges. The maps $U_\Delta ^s \to D_\Gamma $ factor through this $D_\Gamma ^{\mathrm {gs}}$ , defining an ‘intermediate’ open substack $U_\Delta ^{\mathrm {int}}$ . Finally, we take $D_\Gamma ^s$ to be the normalization of $D_\Gamma ^{\mathrm {gs}}$ in a Galois field extension of the function field of $D_\Gamma ^{\mathrm {gs}}$ that contains all the extensions defined by $U_\Delta ^s \to D_\Gamma ^{\mathrm {gs}}$ . (If $D_\Gamma $ happens to be reducible, we perform the construction on each connected component. Actually, the $U_\Delta ^s$ still have a stack structure due to automorphisms of the underlying stable curves. The details for how to construct the covering with this caveat are in [Reference MumfordMum83, Section 2b].) This space comes with a forgetful map $c_\Gamma : D_\Gamma ^s \to D_\Gamma $ that factors as $c_\Gamma = c_\Gamma ^\Delta \circ q_\Delta : \widetilde {U}_\Delta \to U_\Delta $ over the preimages of $U_\Delta $ . We may now define $p_\Gamma ^{[i]} = p_\Gamma ^{\Delta ,[i] } \circ q_\Delta $ , since the $\widetilde {U}_\Delta $ for all degenerations $\Gamma {\rightsquigarrow } \Delta $ cover $D_\Gamma ^s$ . This completes the proof of Proposition 4.4.

4.3 Push-pull comparison

Let $\Gamma \in \mathrm {LG}_L(\overline {B})$ be a level graph. Several recursive computations in the sequel are performed on the level strata $B_\Gamma ^{[i]}$ , and we want to transfer the result via $p^{[i]}$ -pullback and $c_\Gamma $ -pushforward to $D_\Gamma $ . This section provides the basic relations in this push-pull procedure. The degree of $c_\Gamma $ seems difficult to compute. In applications, we only need the following relative statement.

Lemma 4.5. The ratio of the degrees of the projections in Proposition 4.4 is

(30) $$ \begin{align} \frac{\deg(p_\Gamma)}{\deg(c_\Gamma)} = \frac{K_\Gamma}{|\mathrm{Aut}(\Gamma)|\,\ell_\Gamma}\,. \end{align} $$

Proof. The degrees can be computed at the generic point, where both maps factor through $q_\Gamma $ . Using the notation introduced in Section 3.4, we find that the degree of $p_\Gamma $ is the number of equivalence classes of prong-matchings, which is $K_\Gamma / [R_\Gamma : \mathrm {Tw}_{\Gamma }]$ by (21). The degree of $c_\Gamma $ is the index $[\mathrm {Tw}_{\Gamma }: \mathrm {Tw}_{\Gamma }^s] \cdot |\mathrm {Aut}(\Gamma )|$ . The claimed equality

(31) $$ \begin{align} \frac{\deg(p_\Gamma)}{\deg(c_\Gamma)} = \frac{1}{|\mathrm{Aut}(\Gamma)} \frac{K_\Gamma} {[R_\Gamma: \mathrm{Tw}_{\Gamma}^s]} = \frac{K_\Gamma}{|\mathrm{Aut}(\Gamma)|\,\ell_\Gamma} \end{align} $$

follows from the definition of the simple twist group.

Next we compare codimension 1 boundary classes on the strata $D_\Gamma \in \mathrm {LG}_L(B)$ and on their level strata $B_\Gamma ^{[i]}$ to pull back tautological relations. We use the symbol $[D_\Gamma ]$ to denote the fundamental class of the substack of $\overline {B}$ parametrising multi-scale differentials compatible with a degeneration of $\Gamma $ . Let $i \in \mathbb {Z}_{\leq 0}$ .

Consider a graph $\Delta \in \mathrm {LG}_1(B_\Gamma ^{[i]})$ defining a divisor in $B_\Gamma ^{[i]}$ . We aim to compute its pullback to $D_\Gamma ^s$ and the push forward to $D_\Gamma $ and to $\overline {B}$ . Recall that in $D_\Gamma ^s$ the edges of $\Gamma $ have been labeled once and for all (we write $\Gamma ^\dagger $ for this labeled graph) and that the level strata $B_\Gamma ^{[i]}$ inherit these labels. Consequently, there is a unique graph $\widehat {\Delta }^\dagger $ that is a degeneration of $\Gamma ^\dagger $ and such that extracting the levels i and $i-1$ of $\widehat {\Delta }^\dagger $ equals $\Delta $ . The resulting unlabeled graph will simply be denoted by $\widehat {\Delta }$ . (Recall from Remark 3.2 that $\delta _{(-i+1)}^\complement (\widehat {\Delta })=\Gamma $ .) On the other hand, the procedure of glueing in and forgetting labels is not injective. For a fixed labeled graph $\Gamma ^\dagger $ , we denote by $J(\Gamma ^\dagger ,\widehat {\Delta })$ the set of $\Delta \in \mathrm {LG}_1(B_\Gamma ^{[i]})$ such that $\widehat {\Delta }$ is the result of that procedure. Obviously the graphs in $J(\Gamma ^\dagger ,\widehat {\Delta })$ differ only by the labeling of their half-edges.

Lemma 4.6. The cardinality of $J(\Gamma ^\dagger ,\widehat {\Delta })$ is determined by

$$ \begin{align*} |J(\Gamma^\dagger,\widehat{\Delta})|\cdot |\mathrm{Aut}(\widehat{\Delta})| = |\mathrm{Aut}(\Delta)|\cdot |\mathrm{Aut}(\Gamma)|\,. \end{align*} $$

We now determine the multiplicities of the push-pull procedure. Recall from equation (22) the definition of $\ell _{\Gamma ,j}$ for $j\in \mathbb {Z}_{\geq 1}$ .

Proposition 4.7. For a fixed $\Delta \in \mathrm {LG}_1(B_\Gamma ^{[i]})$ , the divisor classes of $D_{\widehat {\Delta }}$ and the clutching of $D_\Delta $ are related by

(32) $$ \begin{align} \frac{|\mathrm{Aut}(\widehat{\Delta})|} {|\mathrm{Aut}(\Delta)| |\mathrm{Aut}(\Gamma)|} \cdot c_\Gamma^* [D_{\widehat{\Delta}}] = \frac{\ell_{{\Delta}}}{\ell_{\widehat{\Delta},-i+1}} \cdot p_\Gamma^{[i],*} [D_\Delta] \end{align} $$

in $\mathrm {CH}^1(D_\Gamma ^s)$ and consequently by

(33) $$ \begin{align} \frac{|\mathrm{Aut}(\widehat{\Delta})|}{|\mathrm{Aut}(\Gamma)|} \cdot \ell_{\widehat{\Delta},-i+1} \cdot [D_{\widehat{\Delta}}] = \frac{|\mathrm{Aut}(\Delta)|}{\deg(c_\Gamma)} \cdot \ell_\Delta \cdot \,c_{\Gamma,*} \big(p_\Gamma^{[i],*} [D_\Delta]\big) \end{align} $$

in $\mathrm {CH}^1(D_\Gamma )$ .

Next we compare various versions of the $\xi $ -class on boundary strata. A first definition is by a local description. Consider a level $i\in \{0,\dots ,-L\}$ of a boundary stratum $D_\Gamma $ , and recall that it is a moduli space of multi-scale differentials compatible with a degeneration of $\Gamma $ . We define the line bundle $\mathcal {O}_\Gamma ^{[i]}(-1)$ on $D_\Gamma $ as follows. On open sets where $\Gamma $ does not degenerate further, it is generated by the ith component $\eta _{(i)}$ of the multi-scale differential. If $\Gamma $ degenerates to $\Gamma _1$ , the level i splits up into an interval i to $i-k$ of levels, and then the local generator of $\mathcal {O}_\Gamma ^{[i]}(-1)$ is the multi-scale component $\eta _{(i)}$ for the top of these levels. We let $\xi _\Gamma ^{[i]} = c_1(\mathcal {O}_\Gamma ^{[i]}(-1))$ and write $\xi ^\top _\Gamma $ for the top-level contribution.

Proposition 4.9. The first Chern classes of the tautological bundles on the levels of a boundary divisor are related by

(34) $$ \begin{align} c_\Gamma^* \, \xi^{[i]}_\Gamma = p_\Gamma^{[i],*} \xi_{B_\Gamma^{[i]}} \qquad \text{in} \quad \mathrm{CH}^1(D_\Gamma^s)\,. \end{align} $$

Proof. Comparing local generators, we obtain a collection of isomorphisms

$$ \begin{align*} c_\Gamma^{\Delta,*} \mathcal{O}_{B_\Gamma^{[i]}(-1)} \,\cong \, (p_\Gamma^{\Delta,[i]})^*\,\mathcal{O}_\Gamma^{[i]}(-1) \end{align*} $$

compatible with restrictions to undegenerations. The $q_\Delta $ -pullback of this collection of maps gives the isomorphism on $D_\Gamma ^s$ , and then we take the first Chern class.

We will continue the study of the tautological ring in Sections 7 and 8, using local descriptions near the boundary introduced along with Section 6.

6.1 Over the open stratum

Recall that moduli spaces of Abelian differentials have an affine structure given by period coordinates. Concretely, for a pointed flat surface $(X,\omega ,\mathbf {z})$ , we denote by $Z = \{z_1,\ldots , z_{r+s}\}$ the zeros and by $P = \{z_{r+s+1},\ldots , z_n\}$ the poles among the marked points, thus including marked ordinary points in Z. By [Reference Hubbard and MasurHM79] or [Reference VeechVee86] (see also [Reference Bainbridge, Chen, Gendron, Grushevsky and MöllerBCGGM2]), integration of the one-form along relative homology classes is a local biholomorphism and thus provides local charts of ${\Omega \mathcal M}_{g,n}(\mu )$ in the vector space

The changes in charts are linear with $\mathbb {Z}$ -coefficients. This makes the projectivisation B into a $(\mathrm {PGL}_{N}, \mathbb {P}^{N-1})$ -manifold.

We denote by $\mathcal {H}_{\text {rel}}^1$ the local system on B with fibre the relative cohomology $V = \mathrm {H}^1(X\setminus P, Z;\mathbb {C})$ and recall that $N = \dim (V) = \dim (B) +1$ . Recall that the fibre of $\mathcal {O}_B(-1)$ at the point $(X,\omega ,\mathbf {z})$ is the complex one-dimensional vector space generated by $\omega $ . We thus obtain the evaluation map

$$\begin{align*}\mathrm{ev}\colon(\mathcal{H}_{\text{rel}}^1)^\vee\otimes \mathcal{O}_B(-1)\to \mathcal{O}_B,\quad \gamma\otimes \omega\mapsto \int_{\gamma} \omega\end{align*}$$

by integrating the one-form.

Proposition 6.2. There is a short exact sequence of vector bundles on B

$$\begin{align*}0\longrightarrow \Omega^1_B \longrightarrow (\mathcal{H}_{\text{rel}}^1)^\vee\otimes \mathcal{O}_B(-1) \overset{\mathrm{ev}}{\longrightarrow} \mathcal{O}_B\longrightarrow 0\end{align*}$$

that locally on a chart $\mathbb {P} V$ is given by the standard Euler sequence.

Proof. Let $\pi \colon \tilde {B}\to B$ be the universal cover of B. Consider the developing map $\mathrm {dev}\colon \tilde {B}\longrightarrow \mathbb {P}(V)$ , which is a $\pi _1(B)$ -equivariant local isomorphism. We use the sequence on the standard charts of $\mathbb {P}(V)$ , and we claim that its $\mathrm {dev}$ -pullback descends to an exact sequence B.

To justify this, consider paths $\{\alpha _i\}_{i=1}^N$ that form a local frame of $(\mathcal {H}_{\text {rel}}^1)^\vee $ . Let $\{a_i\}_{i=1}^N$ be the corresponding local coordinates and $\{\mathrm {d} a_i\}$ the local frame of $\Omega ^1_{\mathbb {P}(V)}$ . On the open subset $U_k=\{a_k\not =0\}\subseteq \mathbb {P}(V)$ , the monomorphism of the Euler sequence in equation (36) is given by

(38) $$ \begin{align} \mathrm{d} a_i\mapsto \Bigl(\alpha_i-\frac{a_i}{a_k}\alpha_k\Bigr)\otimes \omega,\qquad i=1,\dots,\hat{k},\dots,N\,, \end{align} $$

where $\omega $ is the representative of the line bundle with $\int _{\alpha _k} \omega =1$ . The pullback sequence gives rise to an isomorphism of short exact sequences

Each vector bundle appearing is provided with a canonical $\pi _1(B)$ action, and the vertical maps are isomorphisms of $\pi _1(B)$ -vector bundles. The first vertical map is an isomorphism since the developing map is a local isomorphism and $ \pi ^*(\Omega ^i_B)\cong \Omega ^i_{\tilde {B}}$ for every i. Since the evaluation map is $\pi _1(B)$ -equivariant, so is the kernel. Hence the short exact sequence passes to the quotient by the action of $\pi _1(B)$ and yields the claim.

6.2 Coordinates near the boundary

Coordinates near the boundary of the moduli space are perturbed period coordinates ([Reference Bainbridge, Chen, Gendron, Grushevsky and MöllerBCGGM3, Section 11] or [Reference Costantini, Möller and ZachhuberCMZ19, Section 3]) that we now illustrate in typical cases that exhibit all the relevant features. The reader is encouraged to read this subsection in parallel with the subsequent one, where the Euler sequence is extended step by step to these boundary strata.

Case 2: two levels, only vertical nodes

For concreteness, we suppose that in the $2$ -level graph $\Gamma \in \mathrm {LG}_1(B)$ there is only one vertex on each level and for concreteness, say, with three edges $e_1,e_2,e_3$ joining the two vertices. Suppose moreover that there is no marked zero on the lower level. (If there is such a marked point on each level, the loops $\beta _i$ below have to be replaced by relative periods across the level, leading to similar constructions.) At a point close to $D_\Gamma $ , the relative homology can be grouped into

• ○ loops $\beta _1$ through $e_1$ and $e_3$ and $\beta _2$ through $e_2$ and $e_3$ ,

• ○ loops $\alpha _1$ and $\alpha _2$ , the vanishing cycles corresponding to $e_1$ and $e_2$ ,

• ○ paths $\gamma _1^{[0]}, \ldots , \gamma _{d_0}^{[0]}$ forming a basis of the relative homology on the top level,

• ○ loops $\gamma _1^{[-1]}, \ldots , \gamma _{d_1}^{[-1]}$ forming a basis of the homology on the bottom level,

for some $d_0,d_1 \in \mathbb {Z}$ ; see also Figure 3.

On the other hand, the surfaces on the boundary stratum $D_\Gamma $ have a basis of relative homology that can be grouped into

• ○ relative periods $\widetilde {\beta }_i$ joining the marked points at the upper ends of the edge $e_i$ to the upper end of the edge $e_3$ for $i=1,2$ ,

• ○ loops $\widetilde {\alpha }_i$ around the poles at the lower ends of $e_i$ for $i=1,2$ ,

• ○ paths $\widetilde {\gamma }_1^{[0]}, \ldots , \widetilde {\gamma }_{d_0}^{[0]}$ forming a basis of the relative homology on the top level,

• ○ loops $\widetilde {\gamma }_1^{[-1]}, \ldots , \widetilde {\gamma }_{d_1}^{[-1]}$ forming a basis of the homology on the bottom level.

From this description it is apparent that $d_i$ is related to the projectivised and unprojectivised dimensions of the level strata previously introduced by $d_i = N_\Gamma ^{[i]} -2 = d_\Gamma ^{[i]}-1$ .

The main statement about perturbed period coordinates [Reference Bainbridge, Chen, Gendron, Grushevsky and MöllerBCGGM3, Section 11] is that on the one hand, coordinates near the boundary are given by the periods on the boundary surfaces and on the other hand, periods with and without tildes are nearly the same after appropriate rescaling. To make this statement concrete, let $\kappa _i$ be the enhancements corresponding to the edges $e_i$ , and let $\ell = \mathrm {lcm}(\kappa _1,\kappa _2,\kappa _3)$ . Near our current boundary divisor $D_\Gamma $ , the universal family of curves has a (universal) family of differentials $\omega $ and $\ell $ is chosen so that rescaling $\eta _{(-1)} = t^{-\ell } \omega _{(-1)}$ is holomorphic and generically nonzero for a coordinate with $D_\Gamma = \{t = 0\}$ locally ([Reference Bainbridge, Chen, Gendron, Grushevsky and MöllerBCGGM3, Section 12]). At each point $p \in D_\Gamma $ , we find a nonzero $\eta $ -period on the lower level, say the period along $\widetilde {\gamma }_1^{\,[-1]}$ , and choose t and thus $\eta $ so that $\int _{\widetilde {\gamma }_1^{\,[-1]}} \eta = 1$ .

A chart of near p is then nearly the product of a neighbourhood of the irreducible components $(X_0,\omega )$ and $(X_1,\eta )$ of the fibre over p in their respective strata of meromorphic differentials. Here, ‘nearly’ refers to the fact that, because of prong-matchings; it is an $\ell $ -fold cover fully ramified over $t=0$ , and moreover, because of enhanced level graph automorphisms, it is a quotient stack by the subgroup G of $S_3$ that exchanges edges with the same enhancement.

Coordinates on this chart are then given by t and the periods

$$ \begin{align*}\begin{aligned} \widetilde{b_i} & = \int_{\widetilde{\beta}_i} \eta_{(0)} \quad (i=1,2), & r_i & = \int_{\widetilde{\alpha}_i} \eta_{(-1)} \quad (i=1,2), \\ \widetilde{c_i}^{\,[0]} & = \int_{\widetilde{\gamma}_i^{\,[0]}} \eta_{(0)} \quad (i=1,\ldots,d_0), \quad & \widetilde{c_i}^{\,[-1]} & = \int_{\widetilde{\gamma}_i^{\,[-1]}} \eta_{(-1)} \quad (i=2,\ldots,d_1)\,. \end{aligned}\end{align*} $$

To provide charts of the projectivisation $\overline {B}$ , we simply fix one of the periods on the top level, say $\widetilde {c}_1^{\,[0]}$ , to be identically one. (If $d_0=0$ , we take $\widetilde {b}_1 \equiv 1$ instead.)

In each sector near the boundary, the perturbed period coordinates are related to the $\omega $ -periods by

(39) $$ \begin{align}\begin{aligned} b_i &:= \int_{{\beta}_i} \omega \, \sim \, \widetilde{b_i} \qquad & a_i &:= \int_{\alpha_i} \omega = t^\ell r_i \ \\ c_i^{[0]} &:= \int_{\gamma_i^{[0]}} \omega \, \sim \, \widetilde{c_i}^{\,[0]}, \qquad & c_i^{[-1]}&:= \int_{\gamma_i^{[-1]}} \omega = t^\ell \widetilde{c_i}^{\,[-1]}, \end{aligned}\end{align} $$

where $\sim $ indicates that the difference is $O(t^\ell )$ . The difference stems (for $c_i^{[0]}$ ) from the fact that the $\omega $ in the universal family is not just the deformation of the twisted differential $(\eta _{(0)},\eta _{(-1)})$ in the fibre over p in its product moduli space, but blurred by some modification differentials. For the $b_i$ , there is an additional error term in the same order of magnitude due to a choice of a nearby base point in the plumbing construction.

6.3 The Euler sequence on the Deligne extension

Recall that the Deligne extension of a local system on B is a canonical extension to a vector bundle on $\overline {B}$ admitting an extension of the Gauss-Manin connection to a connection with regular singular points ([Reference DeligneDel70]). In this section, we want to extend the Euler sequence across the boundary to construct equation (37). For this purpose, we exhibit local generators of the Deligne extension $\overline {\mathcal {H}}^1_{\text {rel}}$ of $\mathcal {H}_{\text {rel}}^1$ , extend the map $\mathrm {ev}$ and determine its kernel in each of the cases as we discussed perturbed period coordinates in Section 6.2, adopting notation from there.

Case 1: only horizontal nodes

A basis of $(\overline {\mathcal {H}}^1_{\text {rel}})^\vee $ consists of the cycles $\alpha _1,\ldots ,\alpha _k$ and $\gamma _1,\ldots ,\gamma _{N-2k}$ that extend across $D_\Gamma $ , together with the linear combinations

$$ \begin{align*} \widehat{\beta}_i = \beta_i - \frac1{2\pi i}\log(q_i)\alpha_i \end{align*} $$

designed to be monodromy invariant. Since the family of one-forms $\omega $ extends across $D_\Gamma $ to a family of stable differentials, the definition

$$ \begin{align*} \mathrm{ev}(\widehat{\beta}_i \otimes \omega) = \int_{\beta_i} \omega - \frac1{2\pi i}\log(q_i) \int_{\alpha_i} \omega = b_i - \frac1{2\pi i}\log(q_i) a_i = 0 \end{align*} $$

extends the definition of $\mathrm {ev}$ in the interior and gives a well-defined holomorphic function. To check the surjectivity of $\mathrm {ev}$ , we can use any of the periods that extend across $D_\Gamma $ . We claim that the kernel of $\mathrm {ev}$ is on the chart U with $a_1 \equiv 1$

(40) $$ \begin{align} \mathcal{K} = \big\langle dq_1/q_1, da_2, dq_2/q_2, \ldots, da_k, dq_k/q_k, dc_1,\ldots, dc_{N-2k} \big\rangle \, \end{align} $$

as $\mathcal {O}_U$ -module. In fact, using the definition in equation (38) in the interior, one checks that

(41) $$ \begin{align} dq_i/q_i = d\log(q_i) = d\left(2\pi i \frac{b_i}{a_i}\right)\, \mapsto \, \frac{2\pi i}{a_i} \Bigl(\beta_i - \frac{b_i}{a_i} \alpha_i\Bigr) \otimes \omega \end{align} $$

is mapped to a local generator of $(\overline {\mathcal {H}}^1_{\text {rel}})^\vee \otimes \mathcal {O}_{\overline {B}}(-1)$ since the functions $a_i$ do not vanish near such a boundary point. Moreover, $dq_i/q_i$ is mapped to the kernel of $\mathrm {ev}$ by the preceding calculation. For the other elements, these claims follow as in the interior.

Case 2: two levels, only vertical nodes

We first work in the special case near a boundary divisor $D_\Gamma $ , where $\Gamma $ has three edges as in the case discussed in Section 6.2. A basis of $(\overline {\mathcal {H}}^1_{\text {rel}})^\vee $ consists of the cycles $\alpha _1,\alpha _2, \gamma _1^{[0]}, \ldots , \gamma _{d_0}^{[0]}, \gamma _1^{[-1]}, \ldots , \gamma _{d_1}^{[-1]}$ that extend across $D_\Gamma $ , together with the linear combinations

(42) $$ \begin{align} \widehat{\beta}_i = \beta_i - \log(t)(\alpha_i + (\alpha_1+\alpha_2)) \qquad(i=1,2) \end{align} $$

that are monodromy invariant since turning once around the divisor acts by simultaneous Dehn-twists around the core curves of the three plumbing cylinders. Sending cycles that extend across $D_\Gamma $ to their $\omega $ -integrals and letting

$$ \begin{align*} \mathrm{ev}(\widehat{\beta}_i \otimes \omega) = \int_{\beta_i} \omega - \log(t)\Bigl(\int_{\alpha_i} \omega + \int_{\alpha_1+\alpha_2} \omega \Bigr) \qquad(i=1,2) \end{align*} $$

extends the definition of $\mathrm {ev}$ in the interior and is well-defined since the function

$$ \begin{align*} \log (t)\Bigl (2\int _{\alpha _i} \omega + \int _{\alpha _1+\alpha _2} \omega \Bigr ) = O(t^\ell \log (t)) \end{align*} $$

is bounded near $D_\Gamma $ and $\int _{\beta _i} \omega \to \int _{\widetilde {\beta }_i} \omega $ is bounded as well.

Obviously, on the chart with $c_1^{[0]} = 1$ , the kernel of $\mathrm {ev}$ is

(43) $$ \begin{align}\begin{aligned} \mathrm{Ker}(\mathrm{ev}) & = \Big\langle \gamma_i^{[0]} - {c_i^{[0]}} \gamma_1^{[0]}\,\, (i=2,\ldots,d_0);\,\,\, \alpha_i - {a_i} \gamma_1^{[0]}\,\, (i=1,2);\,\, \\ & \qquad \quad \gamma_i^{[-1]} - {c_i^{[-1]}} \gamma_1^{[0]}\,\, (i=1,\ldots,d_1);\, \widehat{\beta_i} - \widehat{b}_i \gamma_1^{[0]}\,\, (i=1,2) \Big\rangle, \end{aligned}\end{align} $$

where $\widehat {b}_i$ is the integral of $\widehat {\beta _i}$ . We claim that via the identification of periods in equation (39), this kernel is precisely the image of

$$ \begin{align*} \mathcal{K} = \big\langle d\widetilde{c}_2^{\,[0]}, \ldots, d\widetilde{c}_{d_0}^{\,[0]},\, d\widetilde{b}_1, d\widetilde{b}_2, \, t^\ell dt/t, \, t^\ell d\widetilde{c}_2^{\,[-1]}, \ldots, t^\ell d\widetilde{c}_{d_1}^{\,[-1]}, \, t^\ell dr_1, \, t^\ell dr_2, \big \rangle \, \end{align*} $$

under the map in equation (38). First, since we used the coordinate $\widetilde {c}_1^{\,[1]}$ to fix the scaling on the bottom level, the differential form $\ell t^\ell dt/t = d c_1^{[-1]}$ is mapped to $\gamma _1^{[-1]} - {c_1^{[-1]}} \gamma _1^{[0]}$ . Then from equation (39), we see that $t^\ell d\widetilde {c}_i^{\,[-1]}$ is mapped to a linear combination of $\gamma _i^{[-1]} - {c_i^{[-1]}} \gamma _1^{[0]}$ and the previous generator for any $i \geq 2$ . Similarly, $t^\ell dr_i$ maps to $\alpha _i - {a_i} \gamma _1^{[0]}$ and a linear combination of the previous generators. In the second step, we consider the generators that correspond to top level. The form $d\widetilde {c}_i^{\,[0]}$ does not quite map to $\gamma _i^{[0]} - {c_i^{[0]}} \gamma _1^{[0]}$ because of the presence of modification differentials, but the difference is a linear combination of the differential of some $\eta $ -periods that we have shown already in the first step to belong to $\mathrm {Ker}(\mathrm {ev})$ . Similarly, the images of $d\widetilde {b_i}$ and $\widehat {\beta _i} - \widehat {b}_i \gamma _1^{[0]}$ are differentials of periods supported on the lower level (from equation (39) to compare with $db_i$ and from equation (42)).

We now rename and regroup the generators of $\mathcal {K}$ in a form that generalises to other level graphs. Since the $\beta $ -periods become relative periods, and since the $\alpha $ -periods for the edges joining the levels are simply residues appearing on a lower level, we may name the set of all periods on the top level by $\widetilde {c}_i^{\,[0]}$ for $1 \leq i \leq N_0$ and those on the bottom level by $\widetilde {c}_i^{\,[-1]}$ for $1 \leq i \leq N_1$ . Then the above argument gives that

(44) $$ \begin{align} \mathcal{K} = \big\langle d\widetilde{c}_2^{\,[0]}, \ldots, d\widetilde{c}_{N_0}^{\,[0]},\, t^\ell dt/t, \, t^\ell d\widetilde{c}_2^{\,[-1]}, \ldots, t^\ell d\widetilde{c}_{N_1}^{\,[-1]} \big\rangle \,. \end{align} $$

Case 3: two levels, additional horizontal nodes

We mix the conclusion of the two previous cases. If the horizontal node is at the top level, then

(45) $$ \begin{align} \mathcal{K} = \langle d\widetilde{c}_2^{\,[0]}, \ldots, d\widetilde{c}_{N_0}^{\,[0]},\, da,\, dq/q, \, t^\ell dt/t, \, t^\ell d\widetilde{c}_2^{\,[1]}, \ldots, t^\ell d\widetilde{c}_{N_1}^{\,[1]} \rangle \,, \end{align} $$

while in the case of a horizontal node at the bottom level

(46) $$ \begin{align} \mathcal{K} = \langle d\widetilde{c}_2^{\,[0]}, \ldots, d\widetilde{c}_{N_0}^{\,[0]},\, t^\ell dt/t, \, t^\ell d\widetilde{c}_2^{\,[-1]}, \ldots, t^\ell d\widetilde{c}_{N_1}^{\,[-1]}, \, t^\ell da,\, t^\ell dq/q \rangle \,. \end{align} $$

Case 4: three levels, three nodes

Using the same arguments as in Case 2, we can show that the $\alpha _j$ -periods corresponding to the edges joining different levels become residues and that the monodromy-invariant modifications $\widehat {\beta _j}$ of the dual $\beta _j$ -periods have $\mathrm {ev}$ -images that tend to the $\beta _j$ -integrals. We claim that thus $\mathrm {Ker}(\mathrm {ev})$ is the image of

(47) $$ \begin{align}\begin{aligned} \mathcal{K} = \langle d\widetilde{c}_2^{\,[0]}, \ldots, d\widetilde{c}_{N_0}^{\,[0]},\,\, & t_1^{\ell_1} dt_1/t_1,\, \, t_1^{\ell_1} d\widetilde{c}_2^{\,[-1]}, \ldots, \,t_1^{\ell_1} d\widetilde{c}_{N_1}^{\,[-1]} \\ & t_1^{\ell_1} t_2^{\ell_2} dt_2/t_2, \, t_1^{\ell_1}t_2^{\ell_2} d\widetilde{c}_2^{\,[-2]}, \ldots, t_1^{\ell_1}t_2^{\ell_2} d\widetilde{c}_{N_2}^{\,[-2]} \rangle \end{aligned}\end{align} $$

under the map in equation (38). We justify this, starting at the bottom level. The differential form

$$ \begin{align*} d(\widetilde{c}_1^{\,[-2]}) = d(t_1^{\ell_1}t_2^{\ell_2}) = \ell_2 t_1^{\ell_1}t_2^{\ell_2} dt_2/t_2 + \ell_1 t_1^{\ell_1}t_2^{\ell_2} dt_1/t_1 \quad \in \mathcal{K}\,, \end{align*} $$

since it is mapped to $\gamma _1^{[-2]} - {c_1^{[-2]}} \gamma _1^{[0]}$ , which in analogy with equation (43) belongs to the natural basis of $\mathrm {Ker}(\mathrm {ev})$ . Next, the form $dt_1^{\ell _1}t_2^{\ell _2} d\widetilde {c}_{i}^{\,[-2]}$ map to a linear combination of the elements $\gamma _i^{[-2]} - {c_i^{[-2]}} \gamma _1^{[0]}$ in the natural basis of $\mathrm {Ker}(\mathrm {ev})$ and the previous generator.

We next proceed to the middle level. There, the form $\ell _1t^{\ell _1} dt_1/t_1$ is not quite equal to $d(c_1^{[-1]})$ because of the presence of modification differentials. It thus does not quite map to the basis element $\gamma _i^{[-1]} - {c_i^{[-1]}} \gamma _1^{[0]}$ of $\mathrm {Ker}(\mathrm {ev})$ . But the difference is a combination of elements that we have already shown to belong to $\mathcal {K}$ . As a combination of this form and $d(\widetilde {c}_1^{\,[-2]})$ , we now have $\ell _2 t_1^{\ell _1}t_2^{\ell _2} dt_2/t_2 \in \mathcal {K}$ . Considering the remaining form $dc_i^{[-1]}$ from periods on the middle level and then all the form $d{c}_i^{[0]}$ for $i \geq 2$ on the top level identifies the remaining elements listed in $\mathcal {K}$ with elements of $\mathrm {Ker}(\mathrm {ev})$ , up to the effect of modification differentials, which produce differentials of periods already shown to belong to $\mathcal {K}$ .

The notation

(48) $$ \begin{align} t_{\lceil j \rceil} = \prod_{i=1}^j t_i^{\ell_i}, \quad j\in \mathbb{N} \end{align} $$

will be convenient here and in the sequel.

Proof of Theorem 6.1.

Continuing the argument as in the preceding cases, we see that near a point $p \in D_\Gamma $ , the elements

• ○ $ t_{\lceil j \rceil } dt_j/t_j,$ for every level $-j$ ,

• ○ the $ t_{\lceil j \rceil }$ -multiples of differential forms associated to periods on level $-j$ ,

• ○ $ t_{\lceil j \rceil } dq_k^{[-j]}/q_k^{[-j]}$ for every horizontal node with parameter $q_k$ on level $-j$

freely generate $\mathcal {K}$ .

8.1 Excess intersection formula

Suppose we are given two level graphs $\Lambda _1$ and $\Lambda _2$ without horizontal nodes and the corresponding inclusion maps into a compactified stratum. For a class $\alpha \in \mathrm {CH}^{\bullet }(D_{\Lambda _2})$ , we want to compute $\mathfrak {i}_{\Lambda _1}^* \mathfrak {i}_{\Lambda _2,*} \alpha $ as the pushforward from the maximal-dimensional boundary strata in the support of $D_{\Lambda _1}\cap D_{\Lambda _2}$ in terms of an $\alpha $ -pullback and normal bundle classes encoding the excess intersection of $D_{\Lambda _1}$ and $D_{\Lambda _2}$ . We say that a level graph $\Pi $ is a $({\Lambda _1,\Lambda _2})$ -graph if there are undegeneration morphisms $\rho _i \colon \Pi \to \Lambda _i$ : that is, edge contraction morphisms with the property that there are subsets $I_{\Lambda _1}$ and $I_{\Lambda _2}$ of level passages of $\Pi $ such that $\delta _{I_{\Lambda _1}}(\Pi ) = \Lambda _1$ and $\delta _{I_{\Lambda _2}}(\Pi ) = \Lambda _2$ . (Automorphisms of $\Lambda _i$ – that is, the stack structure of $D_{\Lambda _i}$ – stemming from permuting the edges require the distinction between $\delta $ ’s and the $\rho _i$ ’s.) We call $\Pi $ a generic $(\Lambda _1, \Lambda _2)$ -graph if $I_{\Lambda _1}^\complement \cap I_{\Lambda _2}^\complement = \emptyset $ . The intersection formula will use the inclusion maps as indicated in the diagram:

Proposition 8.1. For any $\alpha \in \mathrm {CH}^{\bullet }(D_{\Lambda _2})$ , we can express its pushforward pulled back to $\Lambda _1$ as

(60) $$ \begin{align} \mathfrak{i}_{\Lambda_1}^* \mathfrak{i}_{\Lambda_2,*} \alpha = \sum_{\Pi} \mathfrak{j}_{\Pi, \Lambda_1,*} \Bigl(\nu^\Pi_{\Lambda_1\cap \Lambda_2} \cdot \mathfrak{j}_{\Pi, \Lambda_2}^* \alpha \Bigr)\,, \end{align} $$

where the sum is over all generic $(\Lambda _1,\Lambda _2)$ -graphs $\Pi $ . In this expression,

$$\begin{align*}\nu^\Pi_{\Lambda_1\cap \Lambda_2} = \prod_{k \in I_{\Lambda_1}\cap I_{\Lambda_2}} \mathfrak{j}_{\Pi,\delta_k(\Pi)}^*\left(\nu_{\delta_k(\Pi)}\right)\end{align*}$$

is the product of the pullback to $D_\Pi $ of the first Chern classes of the normal bundles of the divisors containing both $D_{\Lambda _1}$ and $D_{\Lambda _2}$ .

An example of the excess intersection formula that illustrates both the appearance of a nontrivial normal bundle and of a sum over more than one graph $\Pi $ is given in Figure 4. In this example, $\Pi _1$ and $\Pi _2$ are the only generic $(\Lambda _1,\Lambda _2)$ -graphs. Since $\Gamma =\delta _{1}(\Lambda _1)=\delta _{1}(\Lambda _2)$ and $D_\Gamma $ is the only boundary divisor containing both $D_{\Lambda _1}$ and $D_{\Lambda _2}$ , we can compute $\mathfrak {i}_{\Lambda _1}^* \mathfrak {i}_{\Lambda _2,*} [1_{D_{\Lambda _2}}]=\sum _{i=1}^2\mathfrak {j}_{\Pi _i, \Lambda _1,*}\left (\mathfrak {j}_{\Pi _i,D_\Gamma }^*\left (\mathcal {N}_{D_\Gamma }\right )\right ) $ .

At the expense of introducing more notation, the excess intersection formula can be generalised in two ways. First, the ambient space might be a boundary stratum associated to a codimension L-level graph $\Gamma $ , as summarised in the diagram

of inclusions. In this situation, we define $\nu ^\Pi _{(\Lambda _1\cap \Lambda _2)/\Gamma }$ to be the product of the pullback to $\Pi $ of the Chern classes of the normal bundles $\mathcal {N}_{\Gamma '/\Gamma }$ , where $\Gamma '$ ranges over all codimension 1 nonhorizontal degenerations $\Gamma '$ of $\Gamma $ that are common to $\Lambda _1$ and $\Lambda _2$ . As above, we denote appropriate pullbacks of this product by the same letter. The excess intersection formula then reads

(61) $$ \begin{align} \mathfrak{j}_{\Lambda_1,\Gamma}^* \mathfrak{j}_{\Lambda_2,\Gamma*} \alpha = \sum_{\Pi} \mathfrak{j}_{\Pi, \Lambda_1,*} \Bigl(\nu^\Pi_{(\Lambda_1\cap \Lambda_2)/\Gamma} \cdot \mathfrak{j}_{\Pi, \Lambda_2}^* \alpha \Bigr)\,, \end{align} $$

where the sum ranges over all $(\Lambda _1,\Lambda _2)$ -graphs $\Pi $ .

In the more general case that the level graphs $\Lambda _i$ also have horizontal nodes, there is an obvious generalisation of this proposition. A general undegeneration of boundary graphs is given by a pair $\delta = (\delta _{\mathrm {ver}},\delta _{\mathrm {hor}})$ consisting of a level undegeneration $\delta _{\mathrm {ver}}$ as in Section 3.3 and an undegeneration of horizontal nodes $\delta _{\mathrm {hor}}$ . One defines $\Pi $ to be a $(\Lambda _1,\Lambda _2)$ -graph if there are undegenerations $\delta _i$ such that $\delta _i(\Pi ) = \Lambda _i$ , for $i=1,2$ . Such a graph is generic if the vertical undegenerations are generic as above and, moreover, if the horizontal contractions are generic in the usual sense of ${\overline {\mathcal M}}_{g}$ (see [Reference Graber and PandharipandeGP03] or [Reference Arbarello, Cornalba and GriffithsACG11, Chapter XVII]). We leave it to the reader to adapt the previous proposition and the subsequent argument to the general context.

8.2 Relations in the tautological ring and the proof of Theorem 1.5

Before concluding the proof of Theorem 1.5, we need some relations in the tautological ring. These relations are essentially known, but we restate them here for convenience and to justify a version for the spaces : that is, possibly disconnected, with residue conditions, and for multi-scale differentials rather than on the incidence variety compactification. Recall the notation of Section 4.1 for generalised strata, where the $(i,j)$ th marked point is the jth marked point of the ith surface and has order $m_{i,j}\in \mathbb {Z}$ .

Proposition 8.2 [Reference SauvagetSau19, Theorem 6(1)].

The class $\xi $ on can be expressed using the $\psi $ -class at the $(i,j)$ th marked point as

(62)

where are two-level graphs with the leg $(i,j)$ on the lower level.

The fact that our $D_\Gamma $ records prong-matching equivalence classes makes up for the difference between our formula and the one appearing in [Reference SauvagetSau19]. Indeed, the original formula of [Reference SauvagetSau19] can be retrieved using the substitution $\ell _\Gamma [D_\Gamma ]=\frac {K_\Gamma } {|\mathrm {Aut}(\Gamma )|}c_{\Gamma ,*} p_\Gamma ^*\left [ B_\Gamma ^\top \times B_\Gamma ^\bot \right ]$ , which follows from Lemma 4.5.

Proof. We expand the argument given in [Reference ChenChe19, Proposition 2.1] including the boundary terms. Let $\pi : \mathcal {X} \to \overline {B}$ be the universal family and $S_i$ be the image of the section given by the ith marked point. The evaluation map gives an isomorphism of $\pi ^* \mathcal {O}(-1)$ and $\omega _{\mathcal {X}/\overline {B}}$ outside the locus $S_i$ and the lower level components of the boundary divisors. Consider the construction of the universal differential over in [Reference Bainbridge, Chen, Gendron, Grushevsky and MöllerBCGGM3, Section 12], in particular in the plumbing fixture (12.6) of [Reference Bainbridge, Chen, Gendron, Grushevsky and MöllerBCGGM3, Section 12]. The difference of t-powers at the two branches is just $\ell _\Gamma $ in our notation, and all this is unchanged in the presence of a GRC $\mathfrak {R}$ . We deduce that

(63) $$ \begin{align} \pi^* \xi = c_1(\omega_{\mathcal{X}/\overline{B}}) - \sum_{i=1}^n m_i S_i - \sum_{\Gamma \in \mathrm{LG_1(B)}} \ell_\Gamma [\mathcal{X}_\Gamma^\bot]\,, \end{align} $$

where $\mathcal {X}_\Gamma ^\bot $ is the lower-level component in the universal family over the divisor $D_\Gamma $ . We intersect both sides with $S_i$ and apply $\pi _*$ . Using $\pi _*(S_i^2) = -\psi _i$ and $\pi _*( \omega _{\mathcal {X}/\overline {B}} \cdot S_i)=\psi _i$ , this gives the claim.

We need a similar generalisation of another relation of Sauvaget to our framework that will be needed for the final evaluation of top degree classes (see the end of Section 9 and [Reference Costantini, Möller and ZachhuberCMZ20]). Consider a generalised stratum defined by a residue condition $\mathfrak {R}$ as defined in Section 4.1. Suppose we remove one element from the set $\lambda _{\mathfrak {R}}$ constraining the residues in the definition of $\mathfrak {R}$ . We denote this new set by $\lambda _{\mathfrak {R}_0}$ and $\mathfrak {R}_0$ the new set of residue conditions. Two cases might occur. Either or is a divisor. We consider the second case here and note that this condition is equivalent to $S:=R \cap \mathfrak {R} \subset S_0:=R \cap \mathfrak {R}_0$ is codimension one (rather than the two being equal), where R is the space of residues defined in equation (25). Consider now a boundary stratum $D_\Gamma $ in . For each level i of $D_\Gamma $ and any GRC $\mathfrak {R}$ containing $\mathfrak {R}_0$ , we define the residue condition $\mathfrak {R}^{[i]}$ induced by $\mathfrak {R}$ to be the residue condition given at level i by the auxiliary level graph $\widetilde {\Gamma }_{\mathfrak {R}}$ as defined in Section 4.1, created with the help of the auxiliary vertices of $\mathfrak {R}$ . For the top level, we write $\mathfrak {R}^\top $ for the induced residue condition on the top level. It can be simply computed by discarding from the parts $\lambda _{\mathfrak {R}}$ all indices of edges that go to lower level in $D_\Gamma $ .