A theorem of Tits type for compact Kähler manifolds
In this paper, the author proves a theorem of Tits type about the automorphism groups of compact Kähler manifolds. Let \(X\) be a compact Kähler manifold of dimension \(n\geq 2\) and \(G\) a subgroup of \(\text{Aut}(X)\), then either \(G\) contains a subgroup isomorphic to the non-abelian free group \(\mathbb Z * \mathbb Z\) or there is a finite index subgroup \(G_1\subset G\) such that its induced action \(G_1|H^{1,1}(X)\) is solvable and \(Z\)-connected (i.e. its Zariski closure in \(\text{GL}(H^{1,1}(X))\) is connected with respect to the Zariski topology). Moreover, the subset \[ N(G_1):=\{ g\in G_1| g\;\text{is \;of\;null\;entropy}\} \] is a normal subgroup of \(G_1\) and the quotient \(G_1/N(G_1)\) is a free abelian group of rank \(r\leq n-1\). The case in which \(r = n-1\) is then analyzed in detail. It is shown that: If \(G\) is a subgroup of \(\text{Aut}(X)\) such that the induced action \(G|H^{1,1}(X)\) is solvable and \(Z\)-connected, then the subgroup \(N(G)\subset G\) is normal and \(G/N(G)\cong \mathbb Z ^r\) where \(r\leq n-1\). If \(r=n-1\) then the algebraic dimension \(a(X)\in \{0,n\}\), the anti-Kodaira dimension is \(\kappa (-K_X)\leq 0\) and \(X\) is either 1) bimeromorphic to a torus, or 2) a weak Calabi-Yau (i.e. \(q(X)=0\) and \(\kappa (X)=0\)), or 3) \(\text{Aut}_0(X)=(1)\), \(q(X)=0\) and \(\kappa (X)=-\infty\) (if \(X\) is projective and uniruled, then it is rationally connected), or 4) \(\text{Aut}_0(X)\) is a non-trivial linear algebraic group, \(X\) is a almost homogeneous projective manifold dominated by every positive dimensional characteristic closed subgroup of \(\text{Aut}_0(X)\). (So \(X\) is unirational and ruled and therefore rational unless \(\dim X \geq 4\) and \(\text{Aut}_0(X)\) is semi-simple.)
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