Compact K\"ahler manifolds with automorphism groups of maximal rank

For an automorphism group G on an n-dimensional (n > 2) normal projective variety or a compact K\"ahler manifold X so that G modulo its subgroup N(G) of null entropy elements is an abelian group of maximal rank n-1, we show that N(G) is virtually contained in Aut_0(X), the X is a quotient of a complex torus T and G is mostly descended from the symmetries on the torus T, provided that both X and the pair (X, G) are minimal.Comment: Added Hypothesis (C) to Theorem 1.2. No change of the proof

Automorphism groups and anti-pluricanonical curves

We show the existence of an anti-pluricanonical curve on every smooth projective rational surface X which has an infinite group G of automorphisms of either null entropy or of type Z . Z (semi-direct product), provided that the pair (X, G) is minimal. This was conjectured by Curtis T. McMullen (2005) and further traced back to Marat Gizatullin and Brian Harbourne (1987). We also prove (perhaps) the strongest form of the famous Tits alternative theorem.Comment: Mathematical Research Letters (to appear); 20 page

Jordan property for non-linear algebraic groups and projective varieties

A century ago, Camille Jordan proved that the complex general linear group $GL_n(C)$ has the Jordan property: there is a Jordan constant $C_n$ such that every finite subgroup $H \le GL_n(C)$ has an abelian subgroup $H_1$ of index $[H : H_1] \le C_n$. We show that every connected algebraic group $G$ (which is not necessarily linear) has the Jordan property with the Jordan constant depending only on $\dim \, G$, and that the full automorphism group $Aut(X)$ of every projective variety $X$ has the Jordan propertyComment: American Journal of Mathematics (to appear); minor change