カラビ・ヤウ多様体の錐予想について

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Calabi-Yau varieties play a prominent role in many branches of mathematics including algebraic geometry. The mirror symmetry of Calabi-Yau varieties enlightened D. Morrison to formulate the cone conjecture. Y. Kawamata proposed a refinement whose motivation originated from birational geometry. In this direction, besides two published papers, I completed one preprint.The study of the birational automorphism groups is not only interrelated with the cone conjecture but also of independent interest. One attractive area is the dynamics of birational automorphisms from both geometric and arithmetic viewpoints. Recently, S. Kawaguchi and J. H. Silverman proposed a conjecture which connects dynamical and arithmetic degrees of dominant rational self-maps. This conjecture is known for some cases of morphisms, but is much less known for arbitrary rational maps due to the lack of functoriality of height functions. In this direction, besides two published papers, I completed two preprints.