Finiteness of solvable automorphisms with null entropy on a compact Kähler manifold

Let G be a solvable subgroup of the automorphism group Aut( X ) of a compact Kähler manifold X of complex dimension n , and let N ( G ) be the normal subgroup of G consisting of elements with null entropy. Let us denote by G * the image of G under the natural map from Aut( X ) to GL( V , R ), where V is the Dolbeault cohomology group H 1, 1 ( X , R ). Assume that the Zariski closure of G * in GL( V C ) is connected. The main aim of this paper is to show that, when the rank r ( G ) of the quotient group G / N ( G ) is equal to n − 1 and the identity component of Aut( X ) is trivial, then the normal subgroup N ( G ) of G is finite. This affirmatively answers a question in Invent. Math. posed by D.‐Q. Zhang.