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

TL;DR

This paper proves that compact complex manifolds with big anti-canonical bundles and Gibbs stability admit no non-trivial holomorphic vector fields, linking measure preservation, anti-canonical divisors, and stability conditions.

Contribution

It establishes a vanishing result for measure-preserving holomorphic vector fields on such manifolds, extending to varieties with singularities and connecting to stability and functional coercivity.

Findings

No non-trivial holomorphic vector fields on Gibbs stable manifolds with big anti-canonical bundle.

Holomorphic vector fields tangent to anti-canonical divisors do not exist under these conditions.

Relations to Hamiltonians and Ding functional coercivity are discussed.

Abstract

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.