Contractions and flips for varieties with group action of small complexity

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We consider projective, normal algebraic varieties $X$ equipped with the action of a reductive algebraic group $G$. We assume that a Borel subgroup of $G$ has an orbit of codimension at most one in $X$ (i.e.\ the complexity of the $G$-variety $X$ is at most one) and that $X$ is unirational. Then we prove that the cone of effective one-cycles $NE(X)$ is finitely generated, and that each face of $NE(X)$ can be contracted. Moreover, flips exist when $X$ is $\bold Q$-factorial, and any sequence of directed flips terminates. Finally, we prove that any homogeneous space of complexity at most one admits an equivariant completion whose anticanonical divisor is ample.
東京大学の論文