Classification of connected components for k≥3 strata of k-differentials

Classify the connected components of the strata Ω^k M_g(μ) of primitive k-differentials on Riemann surfaces for all integers k ≥ 3 and all signatures μ with sum m_1+⋯+m_n = k(2g−2).

Background

The paper reviews known classifications of connected components for special cases: holomorphic differentials (k=1), meromorphic differentials (k=1 with some poles), and quadratic differentials (k=2) under various area constraints. These cases are settled through results of Kontsevich–Zorich, Boissy, Lanneau, and Chen–Möller/Chen–Gendron.

For higher odd powers k≥3, the situation is far less understood. The authors emphasize that beyond hyperelliptic and low-genus ad hoc structures, spin parity is one of the few invariants available, yet a complete classification of connected components of Ωk M_g(μ) is still not available in general.

References

However, for $k\geq 3$, the classification of connected components of $\Omegak\mathcal M_g(\mu)$ remains largely unknown.

— Parity of $k$-differentials in genus zero and one  (2602.03722 - Chen et al., 3 Feb 2026) in Section 1 (Introduction and Statement of Results)