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日時: 2016年1月8日(金) 16:50〜17:50
会場: 数理科学研究科棟(駒場) 123号室
小木曽 啓示 氏 (東京大学大学院数理科学研究科)
Birational geometry through complex dymanics (ENGLISH)
Birational geometry and complex dymanics are rich subjects having interactions with many branches of mathematics. On the other hand, though these two subjects share many common interests hidden especially when one considers group symmetry of manifolds, it seems rather recent that their rich interations are really notified, perhaps after breaking through works for surface automorphisms in the view of topological entropy by Cantat and McMullen early in this century, by which I was so mpressed. The notion of entropy of automorphism is a fundamental invariant which measures how fast two general points spread out fast under iteration. So, the exisitence of surface automorphism of positive entropy with Siegel disk due to McMullen was quite surprizing. The entropy also measures, by the fundamenal theorem of Gromov-Yomdin, the logarithmic growth of the degree of polarization under iteration. For instance, the Mordell-Weil group of an elliptic fibration is a very intersting rich subject in algebraic geometry and number theory, but the group preserves the fibration so that it might not be so interesting from dynamical view point. However, if the surface admits two different elliptic fibrations, which often happens in K3 surfaces of higher Picard number, then highly non-commutative dynamically rich phenomena can be observed. In this talk, I would like to explain the above mentioned phenomena with a few unexpected applications that I noticed in these years: (1) Kodaira problem on small deformation of compact Kaehler manifolds in higher dimension via K3 surface automorphism with Siegel disk; (2) Geometric liftability problem of automorphisms in positive characteristic to chacateristic 0 via Mordell-Weil groups and number theoretic aspect of entropy; (3) Existence problem on primitive automorphisms of projective manifolds, through (relative) dynamical degrees due to Dinh-Sibony, Dinh-Nguyen- Troung, a powerful refinement of the notion of entropy, with by-product for Ueno-Campana's problem on (uni)rationality of manifolds of torus quotient.