Finiteness of solvable automorphisms with null entropy on a compact Kähler manifold (Q2854233)

In the article under review, the author gives a positive answer to a question raised by \textit{D.-Q. Zhang} in [Invent. Math. 176, No. 3, 449--459 (2009; Zbl 1170.14029)] concerning the automorphisms of null entropy in a solvable subgroup of \(\mathrm{Aut}(X)\) where \(X\) is a compact Kähler manifold. In this particuliar context, the entropy \(e(f)\) of an automorphism \(f\) is reduced to be the logarithm of the modulus of the spectral radius of \(f^*\) acting on the cohomology \(\mathrm{H}^\bullet(X,\mathbb{C})\) (due to the works of Gromov and Yomdin). If \(G\subset \mathrm{Aut}(X)\) is a Zariski-connected solvable subgroup (\textit{Zariski-connected} meaning that the Zariski closure of \(G\) inside \(\mathrm{GL}(\mathrm{H}^\bullet(X,\mathbb{C}))\) is connected), it was shown by Zhang (loc. cit.) that \(N(G):=\{f\in G| e(f)=0\}\) is a normal subgroup of \(G\) and that the quotient \(G/N(G)\) is an abelian group of rank \(r(G)\leq n-1\).NEWLINENEWLINEThe main result describes the group \(N(G)\) in the maximal case, i.e. when \(r(G)=n-1\). In this equality case, the group \(N(G)/(N(G)\cap\mathrm{Aut}_0(X))\) is finite (here \(\mathrm{Aut}_0(X)\) is the connected component of the identity). The proof resorts mainly to a theorem of Lie-Kolchin type proved in [\textit{J. Keum, K. Oguiso} and \textit{D.-Q. Zhang}, Math. Res. Lett. 16, No. 1, 133--148 (2009; Zbl 1172.14025)]: a Zariski-connected solvable subgroup of \(\mathrm{GL}(V)\) (\(V\) being a finite dmensional real vector space) preserving a strictly convex closed cone of \(V\) has a common eigenvector in this cone. The author manages then to construct a big and \textit{nef} class \(c\) fixed by the action of \(N(G)\) and to prove that \(\mathrm{Aut}_c(X)/\mathrm{Aut}_0(X)\) is a finite group in this case (a variant of the finiteness theorem of Lieberman and Fujiki).