Invariant Subvarieties of Minimal Homological Dimension, Zero Lyapunov Exponents, and Monodromy

We classify the \(\mathrm{GL}(2,\mathbb{R})\)-invariant subvarieties \(\mathcal{M}\) in strata of Abelian differentials for which any two \(\mathcal{M}\)-parallel cylinders have homologous core curves. As a corollary we show that outside of an explicit list of exceptions, if \(\mathcal{M}\) is a \(\mathrm{GL}(2,\mathbb{R})\)-invariant subvariety, then the Kontsevich-Zorich cocycle has nonzero Lyapunov exponents in the symplectic orthogonal of the projection of the tangent bundle of \(\mathcal{M}\) to absolute cohomology.