Stable reductive varieties. II: Projective case
This is the second part of the authors work on stable reductive varieties where they extend the results of the affine part [Invent. Math. 157, No.~2, 227--274 (2004; Zbl 1068.14003)] to the projective setting. The authors construct a moduli space of stable reductive pairs which are the higher dimensional analog of stable \(n\)-pointed curves, and generalize to the noncommutative case a functorial compactification of moduli of abelian varieties, and of toric pairs. So, the main aim of the article, is to construct a compactified moduli of higher dimensional projective varieties in the special case when there is a nontrivial reductive group action. The compactification of the moduli of stable \(n\)-pointed curves \(\overline{M}_{g,n}\) was extended to pairs \((X,D)\) of log general type by Kollár, Shepherd-Barron, Alexeev, and others, using the methods of the log minimal program. Finally it was extended by Alexeev to the case of pairs \((X,D)\) in which the variety has arbitrary dimension but comes with a nontrivial semiabelian group action, for example, an abelian or toric variety. In the abelian case, this gives a moduli compactification of the moduli space of principally polarized abelian varieties. The toric case is closely related to \(A\)-hypergeometric functions. Each stable toric pair \((X,D)\) comes with a canonical map to \(\mathbb{P}^n\), and the image is its ``shadow cycle''. Now there is a toric Chow variety parameterizing these cycles. The authors study pairs of varieties with divisors rather than just polarized projective varieties. The most important reasons for this is the following: Without the additional structure there are only finitely many isomorphism classes of varieties of a fixed numerical type, the automorphism group is (possibly) infinite, stable limits does not necessarily exist uniquely. These things are repaired by adding the additional structure of the divisor. Also, the log minimal program states that varieties and pairs that have good moduli spaces must have an ample log canonical class \(K_X+D\), and semi log canonical singularities. Thus, because reductive varieties are rational, a nonzero divisor is necessary. In section 1, the authors define polarized stable varieties and pairs. This clearly depends on, and is a generalization of the affine case. The next two sections give their complete combinatorial description in terms of discrete data -- complexes of polytopes invariant under the action of the Weyl group, and continuous data -- certain cohomology groups. Here it becomes clear that the objects under consideration are generalizations of the affine case. It is proved that a linearized stable reductive variety for \(G\) is \((X,L)=({\text{Proj}} R,\mathcal{O}(1))\) for a graded algebra \(R\) such that \(\tilde{X}=\text{Spec} R\) is an affine stable reductive variety for the action of the group \(\tilde{G}=\mathbb{G}_m\times G\). Then the affine classification is the reason for the discrete data explained in these sections. In section 4, the moduli space of stable reductive pairs is constructed, and it is proved to be a disjoint union of projective schemes. Also, in the case of conjugation-invariant pairs, the authors prove that the moduli space is a union of projective toric varieties corresponding to special fiber polytopes. In section 5 it is proved that the constructed varieties have log canonical singularities as predicted by the log minimal program. The last section is devoted to generalizations.
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