Measure preserving holomorphic vector fields, invariant anti-canonical divisors and Gibbs stability

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.

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