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A theorem of Tits type for compact Kähler manifolds. (English) Zbl 1170.14029
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.)

##### MSC:
 14J50 Automorphisms of surfaces and higher-dimensional varieties 14E07 Birational automorphisms, Cremona group and generalizations 32M05 Complex Lie groups, group actions on complex spaces 32Q15 Kähler manifolds
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