# DMM seminar

Speaker: Keiji Oguiso (U Tokyo) SOME EXAMPLES OF PROJECTIVE 4-FOLDS WITH PRIMITIVE AUTOMORPHISMS OF POSITIVE ENTROPY Mon, Dec 21, 2015, 14:00 - 17:00 Seminar Room A We work over C. The main result of this talk is: Theorem 1.1. There are 4-dimensional smooth projective abelian varieties, hyperkahler manifolds, Calabi-Yau manifolds, rational manifolds with primitive biregular automorphisms of positive entropy. A birational automorphism f ∈ Bir (X) of a projective variety X is called primitive if there is no dominant rational map ψ : X --→ B with 0 < dim B < dim X preserved by f. A biregular automorphism of f ∈ Aut (X) of a smooth projective variety X is of positive entropy, via Gromov-Yomdin theorem, iff the spectral radius of f*| H*(X, Z) is strictly greater than 1. Proof of Theorem 1.1 ovbiously consists of the two parts: (1) How to find candidates (X, f), especially, those which f ∈ Aut (X) of positive entropy. (2) How to check the primitivity of candidates (X, f). For (2), we use the notion of (relative) dynamical degrees and their basic properties due to Dinh-Sibony, Dinh-Nguyen-Troung, which are useful renements of the notion and properties of entropy of the automorphism due to Gromov-Yomdin. In the rst half of my talk, After giving a few examples and counterexamples, I would like to explain back ground materials: (i) reason why I consider particular classes of manifolds, smooth projective abelian vari- eties, hyperkahler manifolds, Calabi-Yau manifolds, rational manifolds; (ii) the notion of (relative) dynamical degrees and their basic properties. In the second half of my talk, after brief review of relevant results known in lower di- mension, I would like to give a proof of Theorem 1.1. Break; 15:00 - 15:30