Various Problems on the Classification in Higher Dimensional Birational Geometry

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Mori, jointly with Prokhorov, proved Iskovskikh's conjecture on the singular points of the base surface of a 3-dimensional terminal Q-conic bundle, and classified the fibers over the points. Mukai explicitly described the K3 surfaces with primitive polarization of degree 24, and proved the unirationality of the moduli and universal family. Namikawa explicitly described the equivalence of the derived categories of coherent sheaves for Mukai flops. He also moved the equivalence of deformation smoothability and existence of a crepent resolution for projective complex symplectic varieties. Nakayama published the numerical study on divisors of algebraic varieties. Jointly with Fujimoto, he determined the structure of a nonsingular projective 3-fold with non-negaive Kodaira dimension and with a surjective self morphism of degree >1. Kawakita published the classification of 3-dimensional divisorial contractions contracting a divisor to a non-Gorenstein point. He proved the inverse adjunction … More for log canonicity. Oguiso, jointly with Hosono, Lian and Yau, gave an explicit formula for the number of the Fourier Mukai pairs for a complex projective K3 surface. He else determined the maximal finite solvable group acting on some complex K3 surface. Takagi classified the primary singular Fano threefolds with only quotient terminal singularities satisfying General Elephant Conjecture on anti-canonical systems. Saito, jointly with Budur and Mustata, gave a combinatorial formula on the b-function of a principal ideal, defined the b-function for an arbitrary ideal, and proved its relation with multiplier ideals. Abe studied how the moduli of vector bundles with fixed determinant bundle degenerates when the base curve degenerates to a nodal curve. Hayakawa revised and proved Reid's conjecture on the existence of an economical blowup of a 3-dimensional terminal singularity. The overseas cooperative researcher Matsuki successfully revised the invariant and bipassed the termination conjecture in his project with Kawanoue toward desingularization in positive characteristics. Less