A geometric criterion for the nonuniform hyperbolicity of the Kontsevich--Zorich cocycle

We establish a geometric criterion on a $SL(2, R)$-invariant ergodic probability measureon the moduli space of holomorphic abelian differentials on Riemann surfacesfor the nonuniform hyperbolicity of the Kontsevich--Zorich cocycle on the realHodge bundle. Applications include measures supported on the $SL(2, R)$-orbits ofall algebraically primitive Veech surfaces (see also [7]) andof all Prym eigenforms discovered in [34], as well as all canonicalabsolutely continuous measures on connected components of strata of the modulispace of abelian differentials (see also [4, 17]). The argumentsimplifies and generalizes our proof for the case of canonical measures[17]. In the Appendix, Carlos Matheus discusses several relevantexamples which further illustrate the power and the limitations of ourcriterion.