Intersection theory and volumes of moduli spaces of flat metrics on the sphere

Abstract

Let $\mathbb {P}\Omega ^d\mathcal {M}_{0,n}(\kappa )$, where $\kappa =(k_1,\dots ,k_n)$, be a stratum of (projectivized) d-differentials in genus 0. We prove a recursive formula which relates the volume of $\mathbb {P}\Omega ^d\mathcal {M}_{0,n}(\kappa )$ to the volumes of other strata of lower dimensions in the case where none of the $k_i$ is divisible by d. As an application, we give a new proof of the Kontsevich’s formula for the volumes of strata of quadratic differentials with simple poles and zeros of odd order, which was originally proved by Athreya–Eskin–Zorich. In another application, we show that up to some power of $\pi$, the volume of the moduli spaces of flat metrics on the sphere with prescribed cone angles is a continuous piecewise polynomial with rational coefficients function of the angles, provided none of the angles is an integral multiple of $2\pi$. This generalizes the results of Koziarz and Nguyen (Ann Sci l’Éc Normale Supér 51(6):1549–1597, 2018) and McMullen (Am J Math 139(1):261–291, 2017).

Appendices

Appendix A: A result on symmetric polynomials

Theorem A.1

Let a be a real number. For all $k\in \mathbb {Z}_{>0}$ define

$$\begin{aligned} P_{k,a}(X):=\prod _{i=1}^k(X+a+i). \end{aligned}$$

By convention, we set $P_{0,a}(X):=1$ for all $a\in \mathbb {R}$. Given $a,b\in \mathbb {R}$, and $n \in \mathbb {Z}, n \ge 2$, define

$$\begin{aligned}{} & {} F_{n,a,b}(X_1,\dots ,X_n):= \sum _{\begin{array}{c} I \subset \{1,\dots ,n\},\\ I\ne \varnothing , I \ne \{1,\dots ,n\} \end{array}} P_{|I|-1,a}\left( \sum _{i\in I}X_i\right) \cdot P_{|I^c|-1,b}\left( \sum _{i\in I^c}X_i\right) \\{} & {} \quad = \sum _{\begin{array}{c} I \subset \{1,\dots ,n\},\\ I\ne \varnothing , I \ne \{1,\dots ,n\} \end{array}} \left( \left( \sum _{i\in I}X_i+a+1\right) \dots \left( \sum _{i\in I}X_i+a+|I|-1\right) \cdot \right. \\{} & {} \quad \left. \times \, \left( \sum _{i\in I^c}X_i+b+1\right) \dots \left( \sum _{i\in I^c}X_i+b+|I^c|-1\right) \right) . \end{aligned}$$

Then $F_{n,a,b}(X_1,\dots ,X_n)$ depends only on the sum $X_1+\dots +X_n$, that is $F_{n,a,b}(X_1,\dots ,X_n)\in \mathbb {R}[\sum _{i=1}^nX_i]$.

Proof

Note that $F_{n,a,b}(X_1,\dots ,X_n)$ is a symmetric polynomial of degree $n-2$ in $(X_1,\dots ,X_n)$. We have

$$\begin{aligned} F_{2,a,b}(X_1,X_2)= 2, \end{aligned}$$

and

$$\begin{aligned} F_{3,a,b}(X_1+X_2+X_3)=4(X_1+X_2+X_3)+3(a+b+2). \end{aligned}$$

Assume that the conclusion holds for some $n\ge 3$. For all $i\in \{1,\dots ,n+1\}$, define

$$\begin{aligned} G^{(i)}_{n+1,a,b}(X_1,\dots ,\widehat{X}_i,\dots ,X_{n+1}):=F_{n+1,a,b}(X_1,\dots ,\underset{i}{\underbrace{0}},\dots ,X_{n+1}). \end{aligned}$$

Consider

$$\begin{aligned} G^{(n+1)}_{n+1,a,b}(X_1,\dots ,X_n):=F_{n+1,a,b}(X_1,\dots ,X_n,0). \end{aligned}$$

We will show that $G^{(n+1)}_{n+1,a,b}\in \mathbb {R}[\sum _{i=1}^n X_i]$. Let J be a subset of $\{1,\dots ,n\}$ such that $1< |J| < n$. Denote by $J^c$ the complement of J in $\{1,\dots ,n\}$. Consider J as a subset of $\{1,\dots ,n+1\}$. The term in $G^{(n+1)}_{n+1,a,b}$ associated to J is

$$\begin{aligned}{} & {} P_{|J|-1,a}\left( \sum _{i\in J}X_i\right) \cdot P_{|J^c|,b}\left( \sum _{i\in J^c}X_i\right) \\{} & {} \quad = \left( \sum _{i\in J^c}X_i+b+|J^c|\right) \cdot P_{|J|-1,a}\left( \sum _{i\in J}X_i\right) \cdot P_{|J^c|-1,b}\left( \sum _{i\in J^c}X_i\right) . \end{aligned}$$

Let $\hat{J}=J\sqcup \{n+1\}$. Then the term in $G^{(n+1)}_{n+1,a,b}$ corresponding to $\hat{J}$ is equal to

$$\begin{aligned}{} & {} P_{|J|,a}\left( \sum _{i\in J}X_i\right) \cdot P_{|J^c|-1,b}\left( \sum _{i\in J^c}X_i\right) \\{} & {} \quad = \left( \sum _{i\in J^c}X_i+a+|J|\right) \cdot P_{|J|-1,a}\left( \sum _{i\in J}X_i\right) \cdot P_{|J^c|-1,b}\left( \sum _{i\in J^c}X_i\right) . \end{aligned}$$

Thus the sum of the terms in $G^{(n+1)}_{n+1,a,b}$ corresponding to J and $\hat{J}$ is equal to

$$\begin{aligned} \left( \sum _{i=1}^nX_i+a+b+n\right) \cdot P_{|J|-1,a}\left( \sum _{i\in J}X_i\right) \cdot P_{|J^c|-1,b}\left( \sum _{i\in J^c}X_i\right) . \end{aligned}$$

Consider now a subset I of $\{1,\dots ,n+1\}$ such that $I\ne \varnothing$ and $I \ne \{1,\dots ,n+1\}$. If $I=\{n+1\}$, then the term corresponding to I in $G^{(n+1)}_{n+1,a,b}$ is

$$\begin{aligned} P_{n-1,b}\left( \sum _{i=1}^n X_i\right) \in \mathbb {R}[\sum _{i=1}^nX_i], \end{aligned}$$

and if $I=\{1,\dots ,n\}$, then the term corresponding to I is

$$\begin{aligned} P_{n-1,a}(X_1+\dots +X_n) \in \mathbb {R}[X_1+\dots +X_n]. \end{aligned}$$

Otherwise, there is a unique subset J of $\{1,\dots ,n\}$ such that $1< |J| < n$ and either $I=J$ of $I=J\sqcup \{n+1\}$. Therefore,

$$\begin{aligned}{} & {} G^{(n+1)}_{n+1,a,b}(X_1,\dots ,X_n)=P_{n-1,a}\left( \sum _{i=1}^nX_i\right) \\{} & {} \quad + P_{n-1,b}\left( \sum _{i=1}^nX_i\right) + \left( \sum _{i=1}^n X_i+a+b+n\right) \cdot F_{n,a,b}(X_1,\dots ,X_n). \end{aligned}$$

By assumption $F_{n,a,b}(X_1,\dots ,X_n)\in \mathbb {R}[\sum _{i=1}^nX_i]$, thus we have $G^{(n+1)}_{n+1,a,b}(X_1,\dots ,X_n)\in \mathbb {R}[\sum _{i=1}^nX_i]$. By the same argument, we also have $G^{(j)}_{n+1,a,b} \in \mathbb {R}[\sum _{\begin{array}{c} 1\le i\le n+1, i\ne j \end{array}}X_i]$ for all $j\in \{1,\dots ,n\}$.

For $d\in \{1,\dots ,n-1\}$, let $F^{(d)}_{n+1,a,b}$ be the degree d component of $F_{n+1,a,b}$. We can write

$$\begin{aligned} F^{(d)}_{n+1,a,b}=a_d(X_1+\dots +X_{n+1})^d+R_d(X_1,\dots ,X_{n+1}), \end{aligned}$$

where $R_d$ is a homogeneous symmetric polynomial of degree d that does not contain any term of the form $\lambda \cdot X_i^d$. We now observe that $F^{(d)}_{n+1,a,b}(X_1,\dots ,X_n,0)$ is the degree d component of $G^{(n+1)}_{n+1,a,b}$. Since $G^{(n+1)}_{n+1,a,b} \in \mathbb {R}[\sum _{i=1}^nX_i]$, we must have

$$\begin{aligned} F^{(d)}_{n+1,a,b}(X_1,\dots ,X_n,0)=a_d(X_1+\dots +X_{n})^d \end{aligned}$$

and

$$\begin{aligned} R_d(X_1,\dots ,X_n,0)=0. \end{aligned}$$

This means that every term of $R_d$ contains $X_{n+1}$. The same argument applied to $G^{(i)}_{n+1,a,b}$ implies that all the terms of $R_d$ contain $X_1\cdots X_{n+1}$. But the degree of $R_d$ is $d \le n-1$. Thus we must have $R_d\equiv 0$. It follows that

$$\begin{aligned} F^{(d)}_{n+1,a,b}(X_1,\dots ,X_n,X_{n+1})=a_d(X_1+\dots +X_{n+1})^d \end{aligned}$$

and hence $F_{n+1,a,b} \in \mathbb {R}[\sum _{i=1}^{n+1}X_i]$. $\square$

Appendix B: Computations of Masur–Veech volumes in low dimension by Vincent Koziarz and Duc-Manh Nguyen

In this section, we compute the self-intersection number $\hat{\mathcal {D}}_\mu ^{n-3}$, where $\hat{\mathcal {D}}_\mu$ is defined in (13), and the corresponding Masur–Veech volume of $\mathbb {P}\Omega ^d\mathcal {M}_{0,n}(\kappa )$, for $n=4,5$, $d\in \{3,4,6\}$. It is well known that the Masur–Veech volume of $\mathbb {P}\Omega ^d\mathcal {M}_{0,n}(\kappa )$ gives the asymptotics of the number of tilings of the sphere by triangles and squares with prescribed constraints at some vertices (see for instance [5, 6, 16, 28]). It is worth noticing that our calculations also show that $\hat{\mathcal {D}}_\mu ^{n-3}=0$ if there exists $i\in \{1,\dots ,n\}$ such that $\mu _i\in \mathbb {Z}_{\le 0}$, in accordance with Theorem 5.1.

To compute the ratio $\frac{d\textrm{vol}^{MV}}{d\textrm{vol}}$, our strategy goes as follows: we first observe that it is enough to compute this ratio locally near any point in $\Omega ^d\mathcal {M}_{0,n}(\kappa )$. Recall that an element $(\mathbb {P}^1,x_1,\dots ,x_n,q)\in \Omega ^d\mathcal {M}_{0,n}(\kappa )$ defines a flat metric with conical singularities on the sphere, where the cone angle at $x_i$ is $\alpha _i:=2\pi (1+\frac{k_i}{d})$. To simplify the discussion, let us assume that there is at most one angle in $\{\alpha _1,\dots ,\alpha _n\}$ that is greater than $2\pi$. Thus, we can assume that $0< \alpha _i< 2\pi$ for $i=1,\dots ,n-1$. We construct a flat surface in $\Omega ^d\mathcal {M}_{0,n}(\kappa )$ as follows: let P be a $2(n-1)$-gon in the plane whose vertices are denoted by $p_1,\dots ,p_{2(n-1)}$ in the counterclockwise ordering. We suppose that P satisfies the following

• the sides $\overline{p_{2i-1}p_{2i}}$ and $\overline{p_{2i}p_{2i+1}}$ have the same length, and

• the interior angle at $p_{2i}$ is $\alpha _i$,

for $i=1,\dots ,n-1$ (with the convention $p_{2n-1}=p_1$).

For each $i \in \{1,\dots ,n-1\}$, we endow the side $\overline{p_{2i-1}p_{2i}}$ with the orientation from $p_{2i}$ to $p_{2i-1}$, and the side $\overline{p_{2i}p_{2i+1}}$ with the orientation from $p_{2i}$ to $p_{2i+1}$. Gluing $\overline{p_{2i-1}p_{2i}}$ and $\overline{p_{2i}p_{2i+1}}$ together in respecting their orientation, we obtain a flat surface M homeomorphic to the sphere with n conical singularities $x_1,\dots ,x_n$, where $x_i$ corresponds to the vertex $p_{2i}$ of P for $i=1,\dots ,n-1$, and $x_n$ is the identification of $p_1,p_3,\dots ,p_{2n-3}$. The cone angle at $x_i$ is clearly $\alpha _i$ for $i=1,\dots ,n-1$. Thus the cone angle at $x_n$ must be $\alpha _n$. Hence M is an element of $\Omega ^d\mathcal {M}_{0,n}(\kappa )$.

Let $z_i$ be the complex number associated to the segment $\overline{p_{2i-1}p_{2i}}$ with the orientation from $p_{2i}$ to $p_{2i-1}$ for $i=1,\dots ,n-1$. The complex number associated to the segment $\overline{p_{2i}p_{2i+1}}$ with the orientation from $p_{2i}$ to $p_{2i+1}$ is then $e^{\imath \alpha _i}z_i$. Since the sum of the complex numbers associated to the sides of P must be 0, we get

$$\begin{aligned} (1-e^{\imath \alpha _1})z_1+\dots +(1-e^{\imath \alpha _{n-1}})z_{n-1}=0. \end{aligned}$$

(40)

A neighborhood of M in $\Omega ^d\mathcal {M}_{0,n}(\kappa )$ can be identified with an open subset of the hyperplane V defined by (40) in $\mathbb {C}^{n-1}$. Since $e^{\imath \alpha _i} \ne 1$ for all $i=1,\dots ,n-1$, we can identify this hyperplane with $\mathbb {C}^{n-2}$ via the natural projection $\mathbb {C}^{n-1} \rightarrow \mathbb {C}^{n-2}, \; (z_1,\dots ,z_{n-1}) \mapsto (z_1,\dots ,z_{n-2})$. In the coordinates $(z_1,\dots ,z_{n-2})$, the area of M is given by a Hermitian matrix H, that is

$$\begin{aligned} \text {Area}(M)=(\bar{z}_1,\dots ,\bar{z}_{n-2})\cdot H \cdot {}^t(z_1,\dots ,z_{n-2}). \end{aligned}$$

Hence $d\textrm{vol}=\det H \cdot d\lambda _{2(n-2)}$, where $\lambda _{2(n-2)}$ is the Lebesgue measure on $\mathbb {C}^{n-2}$.

In this setting, the Masur–Veech volume on $\Omega ^d\mathcal {M}_{0,n}(\kappa )$ is the unique (real) $(n-2,n-2)$-form on V such that the lattice $\Lambda :=V\cap (\mathbb {Z}\oplus \zeta \mathbb {Z})^{n-1}$ has covolume 1, where $\zeta =\imath$ if $d=4$, and $\zeta =e^{\frac{2\pi \imath }{3}}$ if $d\in \{3,6\}$. Let $\Lambda '$ be the projection of $\Lambda$ in $\mathbb {C}^{n-2}$. Then $\Lambda '$ is a sublattice of $(\mathbb {Z}\oplus \zeta \mathbb {Z})^{n-2}$. Let m be the index of $\Lambda '$ in $(\mathbb {Z}\oplus \zeta \mathbb {Z})^{n-2}$. Since $\Lambda '$ has covolume $m\cdot \textrm{Im}(\zeta )^{n-2}$ with respect to the Lebesgue measure, we must have

$$\begin{aligned} \frac{d\textrm{vol}^{MV}}{d\lambda _{2(n-2)}}=\frac{1}{m\cdot \textrm{Im}(\zeta )^{n-2}}. \end{aligned}$$

As a consequence, we get

$$\begin{aligned} \frac{d\textrm{vol}^{MV}}{d\textrm{vol}}=\frac{1}{m\cdot \textrm{Im}(\zeta )^{n-2}\cdot \det H}. \end{aligned}$$

(41)

In the cases where there are more than one angle in $\{\alpha _1,\dots ,\alpha _n\}$ that are greater than $2\pi$, we construct a surface in $\Omega ^d\mathcal {M}_{0,n}(\kappa )$ from several polygons, and compute the ratio $d\textrm{vol}^{MV}/d\textrm{vol}$ by the same method.

1.1 B.1. Case $n=4$

In this case, $\widehat{\mathcal {M}}_{0,n}(\kappa )$ is isomorphic to $\overline{\mathcal {M}}_{0,4}\simeq \mathbb {P}^1$, and $\hat{\mathcal {D}}_\mu =\mathcal {D}_\mu$. Thus, the self-intersection number is simply the degree of $\mathcal {D}_\mu$. The values of the ratio $d\textrm{vol}^{MV}/d\textrm{vol}$ and the Masur–Veech volume of $\mathbb {P}\Omega ^d\mathcal {M}_{0,n}(\kappa )$ are recorded in Table 1 (the values on the last column are the product of the ones on the third and the fourth columns with $-\frac{\pi ^2}{2\cdot d}$).

Table 1 Volume of $\mathbb {P}\Omega ^d\mathcal {M}_{0,n}(\kappa )$ when $n=4$

1.2 B.2. Case $n=5$

In this case the exceptional divisor $\mathcal {E}$ is non-trivial if and only if there exists a partition $\{I_0,I_1,I_2\}$ of $\{1,\dots ,5\}$ such that $\sum _{i\in I_1} \mu _i>1$ and $\sum _{i\in I_2}\mu _i>1$ (cf. Theorem 3.3(a)). Since all the $\mu _i$’s satisfy $\mu _i <1$, we must have $|I_0|=1$ and $|I_1|=|I_2|=2$. Thus the stratum of $\overline{\mathcal {M}}_{0,5}$ corresponding to this partition is just one point. Let E be the Weil divisor associated to the Cartier divisor $\mathcal {E}$. Then each component of E maps to a single point in $\overline{\mathcal {M}}_{0,5}$. Therefore, we have

$$\begin{aligned} \hat{p}^*\mathcal {D}_\mu \cdot E= \mathcal {D}_\mu \cdot \hat{p}_* E=0. \end{aligned}$$

Hence

$$\begin{aligned} ({\hat{p}}^*\mathcal {D}_\mu +\mathcal {E})^2=\mathcal {D}^2_\mu +\mathcal {E}^2. \end{aligned}$$

(42)

Since we have an explicit expression of $\mathcal {D}_\mu$ and the intersection of boundary divisors in $\overline{\mathcal {M}}_{0,n}$ is well understood, the computation of $\mathcal {D}_\mu ^2$ is straightforward. It remains to compute $\mathcal {E}^2$.

To simplify the discussion, let us assume that E has a unique irreducible component associated to a partition $\{I_0,I_1,I_2\}$ as above. Define $r_i:=d(\sum _{k\in I_i} \mu _k-1), \; i=1,2$. By assumption, we have $r_i \in \mathbb {Z}_{>0}$.

By construction, a neighborhood of E in $\widehat{\mathcal {M}}_{0,5}(\kappa )$ can be described as $\widehat{\mathcal {U}}=\{(t_1,t_2,[u:v]) \in \mathcal {U}\times \mathbb {P}^1, \; vt^{r_1}_1=ut_2^{r_2}\}$, where $\mathcal {U}$ is a neighborhood of (0, 0) in $\mathbb {C}^2$. Define $U=\{(t_1,t_2,[u:v]) \in \widehat{\mathcal {U}}, \; v \ne 0\}$ and $V=\{(t_1,t_2,[u:v])\in \widehat{\mathcal {U}}, \; u\ne 0\}$. We have

$$\begin{aligned} U\simeq \{(t_1,t_2,u) \in \mathcal {U}\times \mathbb {C}, \; t_1^{r_1}=ut_2^{r_2}\} \text { and } V\simeq \{(t_1,t_2,v) \in \mathcal {U}\times \mathbb {C}, \; vt_1^{r_1}=t_2^{r_2}\}. \end{aligned}$$

Note that $E\cap U$ and $E\cap V$ are both defined by $t_1=t_2=0$.

Lemma B.1

We have $\mathcal {E}\cdot E=-1$.

Proof

Since $E\simeq \mathbb {P}^1$, it is enough to check that the restriction of the line bundle associated to $\mathcal {E}$ to E is precisely the tautological line bundle $\mathcal {O}_{\mathbb {P}^1}(-1)$. This is straightforward from the construction. $\square$

We now show the following formula, which generalizes the usual self-intersection number of the exceptional divisor of the blow-up of a surface at one point.

Lemma B.2

We have

$$\begin{aligned} \mathcal {E}^2=-r_1r_2. \end{aligned}$$

(43)

By Lemma B.1, it is enough to show that the order of $\mathcal {E}$ along E is $r_1r_2$. This is in fact a consequence of [24, Th.1.3 (a)]. However we will give here below an alternative proof using complex analytic arguments.

Table 2 Volume of $\mathbb {P}\Omega ^d\mathcal {M}_{0,n}(\kappa )$ when $n=5$

Proof of Lemma B.2 in the analytic setting

Since the order of a function along a Weil divisor is only well defined for normal spaces, we first need to pass to the normalization of $\widehat{\mathcal {U}}$. Let $\tilde{U}$ (resp. $\tilde{V}$) be the normalization of U (resp. V). Then the normalization $\tilde{\mathcal {U}}$ of $\widehat{\mathcal {U}}$ is obtained by gluing $\tilde{U}$ and $\tilde{V}$ over $U\cap V$. Let $\tilde{\varphi }: \tilde{\mathcal {U}} \rightarrow \widehat{\mathcal {U}}$ denote the normalizing map.

Let $c:=\gcd (r_1,r_2)$ and $m_i:=r_i/c, \; i=1,2$. Denote by $G_{m_i}$ the cyclic group $\mathbb {Z}/m_i\mathbb {Z}$. It is a well known fact that $\tilde{U}$ is isomorphic to the quotient $\mathbb {C}^2/G_{m_1}$, where the action of $G_{m_1}$ on $\mathbb {C}^2$ is generated by $(t,x) \mapsto (\zeta _1t,\zeta _1^{-m_2}x)$ (here $\zeta _1$ is a primitive $m_1$-th root of unity). The normalizing projection $\tilde{\varphi }_{|\tilde{U}}: \tilde{U} \rightarrow U$ is induced by the map $\varphi _U: \mathbb {C}^2 \rightarrow U, \; (t,x) \mapsto (t_1,t_2,u)=(xt^{m_2},t^{m_1},x^{r_1})$.

Let $\tilde{E}$ be the pre-image of E in $\tilde{\mathcal {U}}$. The intersection $\tilde{E}\cap \tilde{U}$ is identified with $(\{0\}\times \mathbb {C})/G_{m_1}$. Observe that the restriction of $\tilde{\varphi }$ to $\tilde{E}$ has degree c. This is because $\widehat{\mathcal {U}}$ has c local branches through every point $\{(0,0)\}\times \mathbb {C}^*$. Therefore, we have

$$\begin{aligned} \tilde{E}\cdot \tilde{\varphi }^*\mathcal {E}= c\cdot (E\cdot \mathcal {E})=-c. \end{aligned}$$

A neighborhood of a regular point in $\tilde{E}$ can be identified with a neighborhood in $\mathbb {C}^2$ of a point (0, x) with $x\ne 0$. In this neighborhood $\tilde{E}$ is defined by the equation $t=0$, while the Cartier divisor $\tilde{\varphi }^*\mathcal {E}$ is defined by $t_2^{r_2}=t^{r_2m_1}=t^{cm_1m_2}$. Thus the order of $\tilde{\varphi }^*\mathcal {E}$ along $\tilde{E}$ is $cm_1m_2$. It follows that

$$\begin{aligned} \tilde{\varphi }^*\mathcal {E}\cdot \tilde{\varphi }^*\mathcal {E}=cm_1m_2\cdot (\tilde{E}\cdot \tilde{\varphi }^*\mathcal {E})=-c^2m_1m_2=-r_1r_2. \end{aligned}$$

Since $\deg \tilde{\varphi }=1$, we get $\mathcal {E}^2=-r_1r_2$. $\square$

In Table 2, we record the values of $\hat{\mathcal {D}}_\mu ^2$ in the case where $n=5$ and $d\in \{3,4,6\}$, together with the values of the Masur–Veech volume of the corresponding stratum of d-differentials. The values on the last column are the product of the ones in the third and the fourth columns with $\frac{\pi ^3}{3!d}$.