Classification problems in Higher Dimensional Birational Geometry

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Mori and Fujino have generalized Kodaira's canonical bundle formula and proved, as an application, that algebraic varieties with Kodaira dimension at most three have finitely generated canonical rings. Mori, Miyaoka and Takagi, together with Kollar, have proved the boundedness of Fano 3-folds with only canonical singularities. Mori also gave an explicit description of every irreducible semistable extremal neighborhood with two non-Gorenstein points, in terms of coordinates with equations and patching.Mukai proved that every canonical curve of genus 9 with maximal Clifford index, is a linear space section of the symplectic Grassmannian variety of dimension six.Masahiko Saito have introduced the Okamoto-Painleve pair of an algebraic surface and its anti-canonical divisor which algebro-geometrically characterizes the initial value space of the Painleve equation. He also reconstructed the Painleve equation from the pair using its deformation theory.Nakayama described certain elliptic fiber structures over a given analytic space upto bimeromorphic equivalence using the a-etale cohomology group.Namikawa have studied birational maps between complex symplectic varieties and proved that the analogue of "Reid's dream" does not hold in the category of complex symplectic varieties. He also constructed a counterexample to birational Torelli problem for complex symplectic varieties.Oguiso has generalized the characterization of the Klein curve to the case of K3 surfaces, which states that the Klein-Mukai surface is the only K3 surface which admits a faithful action of the quartic extension of the simple group of order 168.Takagi has generalized Takeuchi's method, obtained a classification list of Q-Fano 3-folds with index two and confirmed the existence in several cases.Fujino has proved that every sequence of log flips terminates for 4 dimensional canonical pairs.