Ghys conjectured that every holomorphic Anosov diffeomorphism is biholomorphically conjugate to an affine automorphism of a complex infra-nilmanifold. The conjecture is open in higher dimensions. This work treats the case where the stable or unstable bundle has complex dimension one.

The unstable bundle is shown to be holomorphic, and a compatible invariant connection is built on the quotient line bundle. Franks–Manning conjugacy to a toral automorphism is combined with density of projected lattices, which shows that all translations act biholomorphically. A distributional holomorphicity criterion produces an invariant closed holomorphic 1-form. The main results, together with the Ghys results they use, were formalised in Lean with the SHEAF agent.

Every codimension-one holomorphic Anosov diffeomorphism of a compact complex manifold is biholomorphically conjugate to a hyperbolic torus automorphism. This verifies Ghys's conjecture in complex dimension three. An example on a six-dimensional complex nilmanifold has stable and unstable distributions that are both non-holomorphic.