On a compact Kähler manifold we introduce a cohomological
obstruction to the solvability of the constant scalar curvature
(cscK) equation twisted by a semipositive form, appearing
in works of Fine and Song-Tian.
¶ As a special case we find an obstruction for a manifold to be
the base of a holomorphic submersion carrying a cscK metric in
certain “adiabatic” classes. We apply this to find 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 Kähler manifold.
Thus we find examples of non-projective slope unstable manifolds.