Kummer Rigidity for Hyperkähler Automorphisms

The uploaded paper (Jang, 2021) proves: if X is a projective hyperkähler manifold and a holomorphic automorphism f has positive entropy and the volume form is a measure of maximal entropy, then X is the normalization of a torus quotient and f is induced by a hyperbolic affine automorphism on that quotient, i.e., a hyperkähler Kummer example. This is stated explicitly as Theorem 1 and proved via a detailed argument using Green currents, a “simplicity” criterion yielding uniqueness of the measure of maximal entropy, Jensen’s inequality to force constant expansion/contraction, and a flatness-to-torus-quotient step . The candidate solution correctly concludes the same result and cites Jang’s theorem, so it reaches the right answer. However, it glosses over a key technical point by asserting uniqueness of the maximal entropy measure essentially from the hyperkähler dynamical-degree pattern d_p = d_1^p, whereas the paper verifies the needed “simple” spectral property and wedge-positivity of Green currents to deduce uniqueness (via Dinh–De Thélin and Dinh–Sibony) . Aside from this under-justified step, the candidate’s reduction to Jang’s theorem is correct.