Oscillations and the Kontsevich-Zorich cocycle

Abstract: We present a mechanism for producing oscillations along the lift of the Teichm\"uller geodesic flow to the (real) Hodge bundle, as the basepoint surface is deformed by a unipotent element of $\text{SL}_2(\mathbb{R})$. Invoking Chen-M\"oller, we apply our methods to all but finitely many strata in genus $4$, those exhibiting a varying Lyapunov-exponents phenomenon.