Moduli Space of Abelian Differentials

Updated 24 November 2025

The moduli space of Abelian differentials is a parameter space for pairs (X, ω) on Riemann surfaces, stratified by distinct zero and pole orders.
Analytical and combinatorial methods, including period coordinates and Jenkins–Strebel differentials, uncover the geometric and topological structure of these strata.
Applications span dynamics, intersection theory, and arithmetic geometry, aiding in computations of volumes, Lyapunov exponents, and mapping class group actions.

The moduli space of Abelian differentials parameterizes isomorphism classes of pairs $(X,\omega)$ , where $X$ is a smooth compact Riemann surface of genus $g$ , and $\omega$ is a holomorphic (possibly meromorphic) 1-form with prescribed zero and pole orders. These moduli spaces decompose into "strata" classified by the multiplicities of zeros (and prescribed poles) of $\omega$ , forming a rich geometric and topological structure. The subject brings together algebraic geometry, flat surface theory, Teichmüller dynamics, and arithmetic geometry, and underpins the paper of translation surfaces, interval-exchange transformations, and Lyapunov exponents. Strata of Abelian differentials can be compactified, stratified further by invariants such as spin structure and hyperellipticity, and analyzed via a blend of combinatorial, algebro-geometric, and analytic techniques.

1. Strata, Local Coordinates, and Connected Components

Given a genus $g$ , fix a partition $\kappa=(k_1,\ldots,k_n)$ of $2g-2$, with $k_i\geq1$ . The corresponding stratum $\mathcal{H}(\kappa)$ consists of pairs $(X,\omega)$ where the divisor of zeros of $\omega$ is $\sum k_i P_i$ , with the $P_i$ distinct points. The dimension of $\mathcal{H}(\kappa)$ is $2g + n - 1$. Period coordinates are given by integrating $\omega$ over a basis of relative homology $H_1(X, \{P_i\}; \mathbb{Z})$ , providing affine charts whose transition maps lie in $SL(n,\mathbb{Z})$ (Zorich, 2010, Wright, 2012).

The connected components of each stratum were classified by Kontsevich–Zorich and Lanneau:

Each stratum is either connected, or splits by spin parity (when all zero orders are even), or into up to three components if hyperellipticity arises (notably in $\mathcal{H}(2g-2)$ and $\mathcal{H}(g-1,g-1)$ ) (Zorich, 2010, Calderon et al., 2019).
The non-hyperelliptic components are indexed by $r$ -spin structures, where $r = \gcd(k_1,\dots,k_n)$ , and for even $r$ further split by the Arf invariant (Calderon et al., 2019).

A table summarizing Konstevich–Zorich component structures:

Stratum	Number of Components	Invariant
Principal: $(1,\ldots,1)$	1	–
All $k_i$ even	2 ( $g\geq 4$ )	spin
$(2g-2)$ , $(g-1,g-1)$	3 ( $g\geq 4$ )	spin, hyp
Other cases	1	–

The cylinder configuration for each component can be explicitly modeled by Jenkins–Strebel representatives and generalized permutations, leading to polygonal decompositions and connections with extended Rauzy classes and interval-exchange transformations (Zorich, 2010).

2. Compactifications and Boundary: Multiscale and Incidence-Variety Approaches

Compactifying the strata is essential for intersection-theoretic and algebro-geometric applications. Two principal frameworks are:

Incidence-Variety Compactification (IVC): As described by Bainbridge–Chen–Gendron–Grushevsky–Möller, this is the closure of the stratum in the projectivized Hodge bundle over the Deligne–Mumford compactification, parameterizing pointed stable differentials on nodal curves. Boundary points are described in terms of "twisted differentials" satisfying local order and residue matching conditions at nodes (order sum $-2$ , residues match for simple poles) and a global residue condition (GRC) induced by a level graph on the dual graph of the stable curve (Bainbridge et al., 2016, Chen et al., 2016).
Multiscale Differential (LMS) Compactification: The moduli space of multiscale differentials (LMS) is an orbifold with normal crossing boundary, constructed as a normalization of a toroidal blowup of the IVC. Boundary divisors are indexed by enhanced level graphs with prong-matching data at nodes. This stacky perspective is necessary for precise intersection theory and Kodaira dimension calculations (Bainbridge et al., 2019, Chen et al., 2022).

Smoothing constructions (complex-analytic via plumbing and flat-geometric via residue slits and basic domains) relate boundary twisted differentials with families degenerating in the stratum, ensuring the boundary has a modular interpretation matching analytic and algebro-geometric expectations (Bainbridge et al., 2016, Chen et al., 2016).

3. Intersection Theory, Cycle Classes, and Volumes

Intersection theory on the strata is central for computations of cycle classes, Chern invariants, and for quantitative geometry. The cycle class of a stratum in the Hodge bundle is determined by the Porteous formula, which, after explicit expansion of Chern classes, allows calculation of classes such as the non-simple zero divisor and extremal divisors (Chen, 2012).

The compactified Euler characteristic and Chern classes of strata are calculated via intersection theory on the boundary, with the key formula: $\chi(B)=(-1)^d \int_{\overline B} c_d(\Omega^1_{\overline{B}}(\log D))$ where the boundary divisor structure is encoded by the multiscale compactification (Costantini et al., 2020, Costantini et al., 2020).

Masur–Veech volumes (areas of unit area loci in strata) and Siegel–Veech constants are realized as explicit intersection numbers involving the tautological $\xi$ , $\lambda$ , $\psi$ and boundary classes on incidence variety compactifications. This framework enables large genus asymptotics for volumes and comparisons of even/odd spin volumes, confirming conjectures of Eskin–Zorich (Chen et al., 2019).

4. Monodromy, Mapping Class Groups, and the Fundamental Group

Monodromy representations of the orbifold fundamental group of each stratum into the mapping class group are described in terms of $r$ -spin stabilizers, with explicit finite generating sets. For low codimension strata (with many simple zeros), the Salter-Calderón theorem asserts that the inclusion of the stratum into the moduli of pointed curves induces an injection at the orbifold $\pi_1$ level: the orbifold fundamental group is isomorphic to a "framed" mapping class group, reflecting the extra structure provided by the framing given by the 1-form on the punctured surface (Salter, 29 Sep 2025, Calderon et al., 2019). For nonhyperelliptic strata with at least one simple zero, the monodromy is isomorphic to the stabilizer of the corresponding framing.

Applications to homological monodromy are explicit: if $r$ (the gcd of zero orders) is odd, the monodromy group is the full symplectic group; if $r$ is even, it is the stabilizer of the quadratic refinement of the intersection pairing (given by the Arf invariant) (Calderon et al., 2019).

5. Affine Invariant Submanifolds and Arithmetic Properties

Affine invariant submanifolds—immersed submanifolds locally defined by real linear period equations—form the orbit closures for the $\mathrm{SL}(2, \mathbb{R})$ action. The field of definition of such a submanifold is the intersection of holonomy fields of its translation surfaces, always a degree at most $g$ real number field. The absolute cohomology bundle over such a submanifold admits a semisimple, Galois-compatible decomposition, key for understanding orbit closures, typical and generic behavior, and arithmetic constraints, with finiteness results for algebraically primitive Teichmüller curves (Wright, 2012).

The local period coordinates encode deep arithmetic and dynamical information, and bi-algebraic subvarieties in these coordinates are strictly richer than in the case of Shimura varieties: genuinely nonlinear bi-algebraic loci exist, as exhibited by families admitting finite monodromy in Picard–Fuchs theory, shattering naive analogies to the classification of special subvarieties in homogeneous cases (Deroin et al., 2023).

6. Flat Geometry, Dynamics, and Spectral Theory

Each stratum admits models via Jenkins–Strebel differentials yielding explicit combinatorial and polygonal representations, categorizing connected components and supporting the structure of Rauzy classes and interval exchange transformations (Zorich, 2010). The SL(2, $\mathbb{R}$ )-action preserves Lebesgue (Masur–Veech) measure on each stratum, with unique ergodicity and dynamic invariants (including Teichmüller flow, Lyapunov exponents, and the spectrum of the foliated Laplacian).

Extensions of Selberg’s Eigenvalue Conjecture to congruence covers of strata (in the sense of Yoccoz) yield positive uniform lower bounds on the smallest positive eigenvalue of the Laplacian, generalizing spectral gap phenomena to high-dimensional moduli spaces (Magee, 2016).

Teichmüller dynamics has implications for extremal rays of the pseudo-effective cone: for instance, if certain divisor classes are cut out by Zariski-dense collections of Teichmüller curves, they are extremal (Chen, 2012). Masur–Veech volumes and Siegel–Veech constants also admit recursions via intersection theory, connecting volumes, counting problems, and flat geometry (Chen et al., 2019).

7. Kodaira Dimension, Projectivity, and Effective Classes

Recent work demonstrates that the projectivized strata, compactified by multi-scale differentials, are projective varieties, and establishes that with sufficiently many zeros or in suitable large genus situations, these strata are of general type. The computation relies on explicitly tracking the canonical class, effective boundary divisors associated to noncanonical singularities, and constructing pluri-canonical forms extending over resolutions via vanishing orders determined by combinatorial boundary data. Minimal, "few zeros," and equidistributed zero signature cases for large genus are addressed through a mixture of algebraic, combinatorial, and volume-theoretic tools (Chen et al., 2022).

References

(Zorich, 2010) Jenkins–Strebel, explicit representatives and combinatorics
(Wright, 2012) Field of definition, affine invariant submanifolds
(Chen, 2012) Intersection theory, divisor classes
(Bainbridge et al., 2016, Chen et al., 2016) Compactification, boundary, twisted differentials
(Bertola et al., 2018) Period coordinates, cone structures, Hitchin systems
(Chen et al., 2019) Masur–Veech volumes, intersection theory
(Calderon et al., 2019) Connected components, r-spin, mapping class groups
(Bainbridge et al., 2019) Multi-scale differentials, orbifold compactifications
(Costantini et al., 2020, Costantini et al., 2020) Chern classes, Euler characteristic, algorithmic calculation
(Lanneau et al., 2022) Trace field degrees, pseudo-Anosov stretch factors
(Chen et al., 2022) Kodaira dimension, general type
(Deroin et al., 2023) Non-linear bi-algebraic loci
(Salter, 29 Sep 2025) Fundamental groups, framed mapping class groups