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