Mathematics > Algebraic Geometry
[Submitted on 4 Dec 2007 (v1), last revised 20 Jun 2010 (this version, v9)]
Title:Solvable automorphism groups of a compact Kaehler manifold
View PDFAbstract:Let X be a compact Kaehler manifold of complex dimension n. Let G be a connected solvable subgroup of the automorphism group Aut(X), and let N(G) be the normal subgroup of G of elements of null entropy. One of the goals of this paper is to show that G/N(G) is a free abelian group of rank r(G) less than or equal to n-1 and that the rank estimate is optimal. This gives an alternative proof of the conjecture of Tits type. In addition, we show some non-obvious implications on the structure of solvable automorphism groups of compact Kaehler manifolds. Furthermore, we also show that if the rank r(G) of the quotient group G/N(G) is equal to n-1 and the identity component Aut_0(X) of Aut(X) is trivial, then N(G) is a finite set. The main strategy of this paper is to combine the method of Dinh and Sibony and the theorem of Birkhoff-Perron-Frobenius (or Lie-Kolchin type), and one argument of D.-Q. Zhang originated from the paper of Dinh and Sibony plays an important role.
Submission history
From: Jin Hong Kim [view email][v1] Tue, 4 Dec 2007 06:21:36 UTC (9 KB)
[v2] Sun, 2 Mar 2008 10:40:03 UTC (9 KB)
[v3] Tue, 11 Mar 2008 08:50:43 UTC (9 KB)
[v4] Wed, 12 Mar 2008 12:28:13 UTC (9 KB)
[v5] Thu, 20 Mar 2008 11:08:12 UTC (10 KB)
[v6] Fri, 21 Mar 2008 09:19:18 UTC (10 KB)
[v7] Thu, 29 May 2008 08:08:15 UTC (9 KB)
[v8] Wed, 21 Apr 2010 12:32:31 UTC (10 KB)
[v9] Sun, 20 Jun 2010 11:47:57 UTC (11 KB)
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