𝑛-dimensional projective varieties with the action of an abelian group of rank 𝑛-1
Copyright policy )𝑛-dimensional projective varieties with the action of an abelian group of rank 𝑛-1
Let X X be a normal projective variety of dimension n ≥ 3 n \ge 3 admitting the action of the group G := Z ⊕ n − 1 G := \mathbb {Z}^{\oplus n-1} such that every non-trivial element of G G is of positive entropy. We show: ‘ X X is not rationally connected’ ⇒ \Rightarrow ‘ X X is G G -equivariant birational to the quotient of a complex torus’ ⇐⇒ \Leftarrow \Rightarrow ‘ K X + D K_X + D is pseudo-effective for some G G -periodic effective fractional divisor D D ’. To apply, one uses the above and the fact: ‘the Kodaira dimension κ ( X ) ≥ 0 \kappa (X) \ge 0 ’ ⇒ \Rightarrow ‘ X X is not uniruled’ ⇒ \Rightarrow ‘ X X is not rationally connected’. We may generalize the result to the case of solvable G G .
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automorphism, Topological entropy, COMPACT KAHLER-MANIFOLDS, positive entropy, Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables, Dynamical Systems (math.DS), iteration, Complex Lie groups, group actions on complex spaces, 530, 510, Abelian varieties of dimension \(> 1\), Mathematics - Algebraic Geometry, holomorphic dynamics, topological entropy, FOS: Mathematics, AUTOMORPHISM-GROUPS, Mathematics - Dynamical Systems, complex dynamics, Algebraic Geometry (math.AG), Science & Technology, 32H50, 14J50, 32M05, 11G10, 37B40, Automorphism, abelian varieties, tori, Physical Sciences, Automorphisms of surfaces and higher-dimensional varieties, Mathematics
automorphism, Topological entropy, COMPACT KAHLER-MANIFOLDS, positive entropy, Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables, Dynamical Systems (math.DS), iteration, Complex Lie groups, group actions on complex spaces, 530, 510, Abelian varieties of dimension \(> 1\), Mathematics - Algebraic Geometry, holomorphic dynamics, topological entropy, FOS: Mathematics, AUTOMORPHISM-GROUPS, Mathematics - Dynamical Systems, complex dynamics, Algebraic Geometry (math.AG), Science & Technology, 32H50, 14J50, 32M05, 11G10, 37B40, Automorphism, abelian varieties, tori, Physical Sciences, Automorphisms of surfaces and higher-dimensional varieties, Mathematics
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