Projectivized Strata of Abelian Differentials

• Projectivized strata of Abelian differentials are moduli spaces of Riemann surfaces with scaled holomorphic 1-forms, stratified by prescribed zero orders and geometric invariants.
• They model flat surfaces with conical singularities using Jenkins–Strebel differentials and polygonal representations to capture dynamic properties.
• The framework integrates algebraic, combinatorial, and dynamical techniques to classify connected components, enable compactification, and facilitate cohomological studies.

A projectivized stratum of Abelian differentials is a quasi-projective variety parametrizing (equivalence classes of) pairs (C, [ω]), where C is a compact Riemann surface of genus g and ω is a nonzero holomorphic 1-form on C with prescribed multiplicities of zeros, up to scaling ω by nonzero complex numbers. The corresponding moduli space is stratified by zero orders, and after projectivization, geometric and dynamical invariants of translation surfaces, spin structures, and Teichmüller orbits are encoded in its structure. The theory of projectivized strata provides a rigorous framework for the paper of flat metrics with conical singularities, for the classification of connected components via algebraic and dynamical means, and for the analysis of their compactifications, cohomology, and birational geometry.

1. Stratification and Connected Components

The moduli space of Abelian differentials $\mathcal{H}_g$ on genus $g$ curves decomposes into strata $\mathcal{H}(d_1, \ldots, d_n)$, with $d_1+\cdots+d_n=2g-2$, each stratum consisting of pairs $(C,\omega)$ where the zero divisor $(\omega)_0=\sum d_ip_i$ and $p_i$ are distinct points on $C$. Projectivization identifies $(C, \lambda\omega)\sim(C,\omega)$ for $\lambda\in\mathbb{C}^*$, yielding the projectivized stratum $\mathbb{P}\mathcal{H}(d_1,\ldots,d_n)$.

Connected components of a stratum are determined by:

• Hyperellipticity (whether $C$ is hyperelliptic with anti-invariant differential),
• Parity of spin structure for strata with all even zero orders (even/odd, computable from the Gauss map indices, see formula $(1)$: $\sum_{i=1}^g (ind(a_i)+1)(ind(b_i)+1) \pmod{2}$).

These invariants give rise to nontrivial splitting of strata, e.g., $\mathcal{H}(2g-2)$ splits into three components in high enough genus: hyperelliptic, even spin, and odd spin. Under the $GL(2,\mathbb{R})$-action, ergodic components of the Teichmüller flow match these connected components (hence stratification governs dynamic and geometric invariants) (Zorich, 2010).

2. Jenkins–Strebel Differentials and Polygonal Models

Within each stratum, explicit representatives can be constructed as Jenkins–Strebel differentials—1-forms whose horizontal trajectories are periodic, filling a single cylinder bounded by saddle connections meeting at the zeros. This construction provides:

• A "cylinder representation" for each connected component,
• A translation surface structure giving a flat metric with conical singularities where the zeros of order $d$ have total angle $2\pi(d+1)$,
• Flat surfaces modeled as polygons in the plane with parallel edge identifications: cutting and unfolding the cylinder yields polygons whose gluing data encodes the Abelian differential (Zorich, 2010).
• From a representative in the principal stratum with only simple zeros, merging zeros via deformation yields models for more degenerate strata.

3. Combinatorics: Generalized Permutations and Rauzy Classes

The identification pattern of polygon edges is captured by a generalized permutation, an ordered pair of two rows where each label appears exactly twice (once per row in the Abelian case). This generalizes the interval exchange permutation: $$(\alpha_1, \alpha_2, ..., \alpha_r \ ; \ \beta_1, \beta_2, ..., \beta_s).$$ Saddle connections along the cylinder boundaries are labeled by these symbols, and the cyclic order and pairing suffice to reconstruct the surface.

The dynamics of these permutations under Rauzy induction (operations $a$, $b$; permutation reversal $c$) generate extended Rauzy classes, minimal sets closed under these moves. There is a canonical correspondence:

• Connected component of the stratum $\leftrightarrow$ extended Rauzy class $\leftrightarrow$ ergodic component of the Teichmüller flow.

Constructing a Jenkins–Strebel differential in a connected component gives an explicit element in its extended Rauzy class, thus transferring the geometric classification to a combinatorial one (Zorich, 2010).

4. Stratification, Geometric Structures, and Dynamics

Strata parametrize flat surfaces with singularities, and after projectivization (quotient by scaling), each is an orbifold locally modeled on period coordinates modulo automorphisms. For abelian (and quadratic) differentials, the structure supports:

• The action of $GL(2,\mathbb{R})$ (including the Teichmüller geodesic flow),
• Isoperiodic submanifolds indexed by period lattices and equivalence classes,
• Incidence with algebraic subloci such as spin/hyperelliptic loci, and
• Fibrations onto moduli of curves with additional structures, e.g., mapping $[(C,\omega)]$ to $(C, D=(\omega)_0)$.

Furthermore, each projectivized stratum can be visualized in terms of families of translation surfaces, and can be encoded by the associated polygonal data, generalized permutation, and the collection of interval exchange or linear involution models.

5. Compactification and Degenerations

Degenerations of Abelian differentials require the analysis of stable, possibly nodal curves. The closure of a stratum in the projectivized Hodge bundle is described via:

• Twisted differentials on each irreducible component, with prescribed vanishing and pole orders,
• Matching conditions at nodes: for a node $q$ joining components $X_{v_1}$ and $X_{v_2}$,

$$\operatorname{ord}_q(\eta_{v_1}) + \operatorname{ord}_q(\eta_{v_2}) = -2,$$

and for simple poles, residue balancing

$$\operatorname{Res}_q(\eta_{v_1}) + \operatorname{Res}_q(\eta_{v_2}) = 0.$$

• Level graph structure encoding the scaling limits along hierarchies of components,
• Global residue condition: a sum of residues over separating nodes must vanish, ensuring plumbability.

The complex analytic (plumbing) and flat geometric approaches both yield criteria for when a pointed stable differential lies in the compactified stratum (Bainbridge et al., 2016). Explicit coordinates near the boundary are given via period coordinates plus scaling parameters.

6. Cohomological and Birational Aspects

Projectivized strata are natural subvarieties (often of codimension $n-1$ for $n$ zeros) in the projectivized Hodge bundle over $\mathcal{M}_g$. Cohomological studies show:

• The (co)homology is generated by the pullback of the first Mumford–Morita–Miller class $K_1$ and the tautological class $n$ of the projective bundle; e.g.,

$\eta = 2P^*K_1 - (6g-6)n,$

where $\eta$ is the class dual to the hypersurface of abelian differentials vanishing at a point (Hamenstädt, 2020).

• Effective divisor theory and intersection computations yield further constraint on the birational geometry, such as extremal divisors arising from boundary stratum formed by merging zeros (Chen, 2012).
• The stratification and associated divisor classes control the Kodaira dimension of compactified strata for large genus and for even/odd spin components (Chen et al., 2022, Bud et al., 24 Oct 2024).
• The structure of extended Rauzy classes and corresponding monodromy representations encode the topology of the strata, with orbifold fundamental groups identified with stabilizers of the induced framing (Salter, 29 Sep 2025).

7. Geometric, Combinatorial, and Dynamical Interplay

The geometry of projectivized strata intertwines:

• Flat geometry (polygonal representations and conical singularity structure),
• Combinatorics (generalized permutations, Rauzy classes, and linear involutions),
• Algebraic geometry (tautological classes, effective divisors, stratification),
• Teichmüller dynamics (ergodicity, $GL(2,\mathbb{R})$-action correspondence to connected components),
• Moduli of curves (fibration structure, relationships to spin and hyperelliptic loci),
• Cohomological/topological invariants (fundamental groups, stratifications, and monodromy),
• Bridging complex analysis (plumbing, degeneration), algebraic geometry (divisor theory), and dynamical systems.

This framework allows for explicit geometric and combinatorial models (Jenkins–Strebel polygons, cylinder/ribbon graphs), cohomological and intersection-theoretic computations, and rigorous classification of components, compactifications, and boundary structure. The approach is foundational for the modern analysis of dynamics on moduli of abelian and quadratic differentials, as well as for applications in interval exchange transformations and translation surface geometry.