  We introduce a cohomological obstruction to solving the constant scalar
curvature K\"ahler (cscK) equation twisted by a semipositive form, appearing in
works of Fine and Song-Tian. Geometrically this gives an obstruction for a
manifold to be the base of a holomorphic submersion carrying a cscK metric in
certain ``adiabatic'' classes. In turn this produces many new examples of
general type threefolds with classes which do not admit a cscK representative.
When the twist vanishes our obstruction extends the slope stability of
Ross-Thomas to effective divisors on a K\"ahler manifold. Thus we find examples
of non-projective slope unstable manifolds.
