  Let X be a compact complex manifold whose anti-canonical line bundle is big.
We show that X admits no non-trivial holomorphic vector fields if it is Gibbs
stable (at any level). The proof is based on a vanishing result for measure
preserving holomorphic vector fields on X of independent interest. As an
application it shown that, in general, if the anti-canonical line bundle is
big, there are no holomorphic vector fields on X that are tangent to a
non-singular irreducible anti-canonical divisor S on X. More generally, the
result holds for varieties with log terminal singularities and log pairs.
Relations to a result of Berndtsson about generalized Hamiltonians and
coercivity of the quantized Ding functional are also pointed out.
