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.
