On the Non-Uniform Hyperbolicity of the Kontsevich-Zorich Cocycle for Quadratic Differentials

We prove the non-uniform hyperbolicity of the Kontsevich-Zorich cocycle for a measure supported on abelian differentials which come from non-orientable quadratic differentials through a standard orientating, double cover construction. The proof uses Forni’s criterion for non-uniform hyperbolicity of the cocycle for SL(2,R)-invariant measures. We apply these results to the study of deviations in homology of typical leaves of the vertical and horizontal (non-orientable) foliations and deviations of ergodic averages.

💡 Research Summary

The paper addresses a long‑standing gap in the theory of the Kontsevich‑Zorich (KZ) cocycle: while the non‑uniform hyperbolicity of the cocycle is well‑understood for SL(2,ℝ)‑invariant measures supported on abelian (i.e. orientable) differentials, the case of non‑orientable quadratic differentials has remained elusive. The authors bridge this gap by exploiting the classical orientating double‑cover construction. Given a non‑orientable quadratic differential (q) on a Riemann surface (M), one builds the canonical double cover (\hat{M}) together with a holomorphic 1‑form (\hat{\omega}) that pulls back to (q). The pair ((\hat{M},\hat{\omega})) lies in the stratum of abelian differentials and inherits an SL(2,ℝ)‑invariant probability measure (\mu) that projects to the original measure on the space of quadratic differentials. This lift allows the authors to work entirely within the well‑developed framework of abelian differentials while keeping track of the involution (\sigma) that exchanges the two sheets of the cover.

The core technical contribution is a refined version of Forni’s criterion for non‑uniform hyperbolicity. Forni’s original condition requires that the Hodge bundle over the support of (\mu) contain a subbundle (the “Forni subspace”) on which the KZ cocycle has strictly positive Lyapunov exponents. In the double‑cover setting the Hodge bundle splits into (\sigma)‑invariant and (\sigma)‑anti‑invariant parts. The authors prove that the invariant part, which corresponds precisely to the dynamics of the original non‑orientable foliation, already carries a positive exponent. Their argument proceeds by (i) establishing that (\mu) has full support and is ergodic for the Teichmüller flow, (ii) showing that the symplectic form on cohomology respects the involution, and (iii) constructing a symmetrized Forni subspace that survives the projection back to the base surface. Consequently, the Lyapunov spectrum of the KZ cocycle over the lifted measure contains at least one strictly positive value, which by projection yields non‑uniform hyperbolicity for the original quadratic differential measure.

Having secured hyperbolicity, the authors turn to dynamical applications. For a typical leaf of the vertical (or horizontal) foliation defined by (q), they study the homology class (