Iku Nakamura

Coinvariant Algebras of Finite Subgroups of SL(3, C)

FormostofthefinitesubgroupsofSL(3, C)wegiveexplicit formulaefortheMolienseriesof the coinvariant algebras, generalizing McKays formulae (McKay99) for subgroups of SU(2). We also study the G-orbit Hilbert scheme Hilb G (C 3 ) for any finite subgroup G of SO(3), which is known to be a minimal (crepant) resolution of the orbit space C 3 /G. In this case the fiber over the origin of the Hilbert-Chow morphism from Hilb G (C 3 ) to C 3 /G consists of finitely many smooth rational curves, whose planar dual graph is identified with a certain subgraph of the representation graph of G. This is an SO(3) version of the McKay correspondence in the SU(2) case.

Threefolds Homeomorphic to a Hyperquadric in P4

This chapter highlights the proofs of mainly three theorems and their collaries. A compact complex threefold homeomorphic to a nonsingular hyperquadric Q 3 in P 4 is isomorphic to Q 3 if H 1 ( X , O x ) = 0 and if there is a positive integer m such that dim H 0 { X , − m K x ) > 1. A Moishezon threefold homeomorphic to Q 3 is isomorphic to Q 3 if its Kodaira dimension is less than three. An arbitrary complex analytic (global) deformation of Q 3 is isomorphic to Q 3 . A compact complex threefold is called a Moishezon threefold if it has three algebraically independent meromorphic functions on it. The chapter also recalls elementary facts about algebraic two cycles on singular hyperquadrics in P 4 .

Compactification de l'espace de modules de variétés abéliennes sur Z[ζN,1/N]

Abstract We compactify canonically the moduli scheme Ag,K of abelian schemes over Z[ζN, 1/N] via geometric invariant theory by introducing certain degenerate abelian varieties with noncommutative level structures. Any degenerate abelian scheme on the boundary of the Compactification SQg,K of Ag,K is a singular one among our models — projectively stable quasi-abelian schemes. For a degenerate abelian variety we prove that its Hilbert points are Kempf-stable if and only if it is a projectively stable quasi-abelian scheme.