  Karshon constructed the first counterexample to the log-concavity conjecture
for the Duistermaat-Heckman measure: a Hamiltonian six manifold whose fixed
points set is the disjoint union of two copies of $T^4$. In this article, for
any closed symplectic four manifold $N$ with $b+$ greater than 1, we show that
there is a Hamiltonian six manifold $M$ such that its fixed points set is the
disjoint union of two copies of $N$ and such that its Duistermaat-Heckman
function is not log-concave.
  On the other hand, we prove that if there is a torus action of complexity two
such that all the symplectic reduced spaces taken at regular values satisfy the
condition $b+=1$, then its Duistermaat-Heckman function has to be log-concave.
As a consequence, we prove the log-concavity conjecture for Hamiltonian circle
actions on six manifolds such that the fixed points sets have no four
dimensional components, or only have four dimensional pieces with $b+=1$.
