Codimension-one holomorphic Anosov diffeomorphisms

Jiesong Zhang proves that every codimension-one holomorphic Anosov diffeomorphism of a compact connected complex manifold is biholomorphically conjugate to a hyperbolic automorphism of a complex torus. This settles Ghys's conjecture in that case, and so for all compact complex threefolds, where Xu and Zhang (2024) recorded it as open. The proof removes the topological transitivity assumption from Ghys's Theorem B to get a topological conjugacy to a hyperbolic toral automorphism, proves that both invariant distributions are holomorphic (the stable one by an $L^2$ estimate), and shows a torus with holomorphic invariant distributions carries the standard complex structure. Cantat (2004) had the codimension-one case for projective manifolds. The paper also gives a nilmanifold example whose invariant distributions are smooth but not holomorphic, so the method stops at codimension one. The general conjecture remains open.

Jiesong Zhang proves that every codimension-one holomorphic Anosov diffeomorphism of a compact connected complex manifold is biholomorphically conjugate to a hyperbolic automorphism of a complex torus. This settles Ghys's conjecture in that case, and so for all compact complex threefolds, where Xu and Zhang (2024) recorded it as open. The proof removes the topological transitivity assumption from Ghys's Theorem B to get a topological conjugacy to a hyperbolic toral automorphism, proves that both invariant distributions are holomorphic (the stable one by an $L^2$ estimate), and shows a torus with holomorphic invariant distributions carries the standard complex structure. Cantat (2004) had the codimension-one case for projective manifolds. The paper also gives a nilmanifold example whose invariant distributions are smooth but not holomorphic, so the method stops at codimension one. The general conjecture remains open.