Compactification by GIT-stability of the moduli space of abelian varieties

摘要

The moduli space M_g of nonsingular projective curves of genus g is compactified into the moduli M_g of Deligne-Mumford stable curves of genus g. We compactify in a similar way the moduli space of abelian varieties by adding some mildly degenerating limits of abelian varieties. A typical case is the moduli space of Hesse cubics. Any Hesse cubic is GIT-stable in the sense that its SL(3)-orbit is closed in the semistable locus, and conversely any GIT-stable planar cubic is one of Hesse cubics. Similarly in arbitrary dimension, the moduli space of abelian varieties is compactified by adding only GIT-stable limits of abelian varieties (§ 14). Our moduli space is a projective “fine” moduli space of possibly degenerate abelian schemes with non-classical non-commutative level structure over Z[ζ_N,1/N] for some N > 3. The objects at the boundary are singular schemes, called PSQASes, projectively stable quasi-abelian schemes.