Exact Masur–Veech volumes of strata of quadratic differentials: the general intersection-theoretic formula

Eskin and Okounkov (2001) computed the volumes of all strata of Abelian differentials by identifying them with the asymptotic enumeration of branched coverings of a torus, and in an explicit 'Some open problems' subsection posed two problems: separate the volumes of the connected components of a stratum, and compute the volumes of the analogously defined strata of quadratic differentials. The component problem for Abelian strata has since been resolved — Chen, Möller, Sauvaget and Zagier (2020) express the volume of every spin and hyperelliptic component as an intersection number — while the quadratic problem has been reduced to a sharp conjecture: Chen, Möller and Sauvaget conjecture an intersection-theoretic formula for the Masur–Veech volume of every stratum of quadratic differentials with at most simple poles, proven so far only for odd zero orders and for the principal strata, with exact genus-zero values known from the work of Athreya–Eskin–Zorich. Volumes of general quadratic strata are currently known only through lattice-point counting algorithms and numerical tables (exact values up to complex dimension 10, due to Goujard), exact hyperelliptic-component formulas, and recursions in the odd/principal sector. The open problem is to determine, by a proven exact formula valid for arbitrary singularity data (including even-order zeros, i.e. relative periods), the Masur–Veech volumes of all strata of quadratic differentials — equivalently, to prove the Chen–Möller–Sauvaget conjecture in full generality; the March 2026 survey of Chen and Mullane records the general formula as its Problem 3.9.

Background

A holomorphic (Abelian) differential $\omega$ on a Riemann surface $\Sigma$ defines a flat metric with conical singularities at the zeros of $\omega$; away from the zeros, integrating $\omega$ gives charts whose transition maps are translations, and the pair $(\Sigma,\omega)$ is called a translation surface. Fixing the multiplicities $\mu=(\mu_1,\dots,\mu_n)$ of the zeros, with $\sum\mu_i=2g-2$, cuts out a stratum $\mathcal{H}(\mu)$ inside the moduli space of pairs (curve, differential). A stratum carries period coordinates — the integrals of $\omega$ over a basis of the relative homology $H_1(\Sigma,\{\text{zeros}\})$ form a local coordinate system — so Lebesgue measure in these coordinates is well defined up to a lattice. After restricting to the area-one hypersurface $\mathcal{H}_1(\mu)$, where the area is $\tfrac{i}{2}\int\omega\wedge\bar\omega$, and passing to the induced cone measure, Masur and Veech proved independently that the total volume $\nu(\mathcal{H}_1(\mu))$ is finite. This Masur–Veech volume is the fundamental normalization constant of the $SL(2,\mathbb{R})$ action and the Teichmüller geodesic flow on the stratum: it enters the Siegel–Veech constants, i.e. the quadratic growth rates of counts of flat cylinders or saddle connections (geodesic segments for the flat metric joining two singularities with none in their interior) on a typical surface, which in turn govern renormalization statistics for billiards in rational polygons and interval exchange transformations (piecewise-translation maps of an interval defined by finitely many subinterval permutations).

Eskin and Okounkov (2001) computed these volumes for all strata of Abelian differentials. Their starting point is the interpretation, due to Kontsevich–Zorich and independently Masur–Eskin, of the volume as the leading asymptotics of a weighted count of connected branched coverings of a torus of fixed ramification type — equivalently, of square-tiled surfaces, i.e. translation surfaces tiled by finitely many unit squares. The relevant generating functions are quasimodular forms (polynomials in the classical Eisenstein series of modular-forms theory), and Eskin–Okounkov extracted the $q\to 1$ asymptotics through the asymptotic analysis of sums over partitions, obtaining an effective algorithm for every volume constant together with the rationality statement $\pi^{-2g}\nu(\mathcal{H}_1(\mu))\in\mathbb{Q}$, conjectured earlier by Kontsevich and Zorich.

Two issues were explicitly left open in the paper's subsection 'Some open problems'. First, strata can be disconnected: by the Kontsevich–Zorich classification, a stratum $\mathcal{H}(\mu)$ whose entries are all even has up to three components, distinguished by the parity of the spin structure that the differential induces on the curve — a theta characteristic, i.e. a square root of the canonical line bundle, whose parity is a $\mathbb{Z}/2$-valued invariant constant on connected components — and, in two exceptional families, by hyperellipticity (existence of a central involution whose quotient is a sphere); the Eskin–Okounkov formulas give only the total volume of $\mathcal{H}_1(\mu)$, not the volume of each component, although component volumes are the quantities relevant for ergodic-theoretic applications (small-genus component volumes had been determined by Zorich). Second, the paper treats only holomorphic 1-forms: 'Another problem important for applications is to compute the volumes of similarly defined moduli spaces of quadratic differentials.'

A meromorphic quadratic differential $q$ with at most simple poles (order $-1$) and prescribed zero/pole orders $d_i\geq -1$ with $\sum d_i=4g-4$ defines a half-translation surface: flat charts whose transition maps identify only up to sign, a simple pole appearing as a cone point of angle $\pi$. The strata $\mathcal{Q}(d_1,\dots,d_n)$ of such $q$ carry period coordinates on the canonical double cover (the translation surface obtained by taking a square root of $q$), hence a Masur–Veech measure and volume of their own; quadratic strata are the natural setting for billiards in arbitrary rational polygons. The counting counterpart of Eskin–Okounkov's torus covers is pillowcase covers — covers of the pillowcase orbifold, the quotient of a square torus by its elliptic involution $z\mapsto -z$ — whose counting functions Eskin–Okounkov (2006) proved to be quasimodular, making quadratic-stratum volumes computable in principle from lattice-point counting. Strata of quadratic differentials with at most simple poles are connected or split into hyperelliptic and non-hyperelliptic connected components (classification of Lanneau).

The decisive later development is the intersection-theoretic reinterpretation of volumes. Strata admit modular compactifications (Bainbridge–Chen–Gendron–Grushevsky–Möller), on which one can integrate tautological classes, i.e. the natural cohomology classes given as Chern classes of the standard bundles: the class $\xi=c_1(\mathcal{O}(1))$ of the Hodge bundle (whose fiber records the cohomology class of the differential), its quadratic analogue $\zeta$, and the $\psi$-classes, the first Chern classes of the cotangent line bundles at the marked points. Chen–Möller–Sauvaget–Zagier (2020) proved that for every stratum of Abelian differentials the Masur–Veech volume is, up to an explicit factor, one such intersection number, that the same formula and an effective recursion hold for every spin and hyperelliptic component (resolving Eskin–Okounkov's first open problem, up to a technical metric-extension assumption on minimal strata), and derived the large-genus limits conjectured by Eskin–Zorich. For quadratic differentials, Chen–Möller–Sauvaget proved the analogous intersection formula for strata with odd zero orders and for the principal strata (simple zeros), where it reduces to linear Hodge integrals, and conjectured it in general; Goujard had earlier computed explicit volume values for all quadratic strata of complex dimension at most 10 by pillowcase counting, together with exact closed formulas for hyperelliptic components; Athreya–Eskin–Zorich computed the genus-zero strata exactly. What remains open is the general case — strata containing even-order zeros, i.e. with relative periods — in closed, proven form.

Problem Statement

Let $g\geq 0$ and let $(\mu,\nu)$ be a partition of $4g-4$ into even parts $\mu=(2m_1,\dots,2m_r)$ and odd parts $\nu=(2n_1-1,\dots,2n_s)$, with $m_i,n_j\geq 0$ (an entry $n_j=0$ is a simple pole of order $-1$; an entry $m_i=0$ is an ordinary marked point). Denote by $\mathcal{Q}_{g,r+s}(\mu,\nu)$ the stratum of meromorphic quadratic differentials on genus-$g$ Riemann surfaces with these singularity orders, all zeros, poles, and marked points labeled (the convention of Chen–Möller–Sauvaget; only simple poles are allowed), and by $\operatorname{vol}(\mathcal{Q}_{g,r+s}(\mu,\nu))$ its Masur–Veech volume, i.e. the total cone measure of the area hypersurface of the stratum (in the Chen–Möller–Sauvaget convention, the hypersurface of half-translation surfaces of area $1/2$, so that the canonical double cover has area one), in any of the standard normalizations (Eskin–Okounkov, Athreya–Eskin–Zorich, Goujard, or Chen–Möller–Sauvaget), converted to one another explicitly.

Determine $\operatorname{vol}(\mathcal{Q}_{g,r+s}(\mu,\nu))$ for arbitrary data $(\mu,\nu)$ by a proven exact formula. The recognized precise form of this target is Conjecture 1.1 of Chen–Möller–Sauvaget:

$$\operatorname{vol}\bigl(\mathcal{Q}_{g,r+s}(\mu,\nu)\bigr)=\frac{2^{\,r-s+3}\,(2\pi i)^{\,2g-2+s}}{(2g-3+r+s)!}\int_{\overline{\mathbb{P}\mathcal{Q}}_{g,r+s}(\mu,\nu)}\zeta^{\,2g+s-3}\,\psi_1\cdots\psi_r$$

where the integral is over the closure of the projectivized stratum $\mathbb{P}\mathcal{Q}_{g,r+s}(\mu,\nu)=\mathcal{Q}_{g,r+s}(\mu,\nu)/\mathbb{C}^*$ in its incidence-variety compactification (the compactification of Bainbridge–Chen–Gendron–Grushevsky–Möller in which a stratum with its labeled singularities closes up), $\zeta=c_1(\mathcal{O}(1))$ is the first Chern class of the universal line bundle $\mathcal{O}(1)$ on the projectivized stratum (the quadratic analogue of the Hodge class), and $\psi_1,\dots,\psi_r$ are the cotangent line classes at the even-order zeros. The target is to prove this formula for all singularity data, in particular the case $r\geq 1$ of even-order zeros (relative periods), which contains every case still open; the formula is known for odd zero orders (Chen–Möller–Sauvaget, Theorem 1.2), for the principal strata, and, in genus-zero form, from the work of Athreya–Eskin–Zorich. Refuting the conjecture would resolve it negatively without determining the volumes. The expected principal application — not a required deliverable — is a proof of the conjectural large-genus asymptotics for arbitrary quadratic strata (Aggarwal–Delecroix–Goujard–Zograf–Zorich), currently proven only for principal strata.

References

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