Perfect forms and the moduli space of abelian varieties

By N. Shepherd-barron. 2005

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Toroidal compactifications of the moduli space Ag, or the stack Ag, of principally polarized abelian g-folds have been constructed over C in [AMRT] and over any base in [FC]. Roughly speaking, each such compactification corresponds to choosing a way of decomposing the cone of real positive quadratic forms in g variables. The choice made here is the perfect cone decomposition, also called the first Voronoi decomposition, which leads to the compactification Ag . This carries divisor classes M , the line bundle of weight 1 modular forms, and D, the reduced boundary Ag \ Ag. (Sometimes, but not in this paper, M is denoted instead by ω.) It is easy to see that D is geometrically irreducible, and it follows ([M] or [F2]) that the classes M and D generate NS(Ag ) ⊗ Q. Here is the first main result of this paper, where a divisor class E on a projective variety X is nef if E.C ≥ 0 for all curves C on X :

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

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Citations in the corpus

Log minimal model program for the moduli space of stable curves: The first flip (2008, methodology): “ding canonical models of moduli spaces. For moduli spaces Ag of principally polarized abelian varieties of dimension g≥ 12, the canonical model exists and is equal to the ﬁrst Voronoi compactiﬁcation [SB06]. Unfortunately, no analogous results are known for Mg, even for g≫ 0. Our approach is to appr”
Classification of eight dimensional perfect forms (2006, background): “ot lattices (see [CoSl98, Chapter 6] or [Ma03, Chapter 4]) since perfect domains form a tessellation of Sd >0, their interest is larger than only sphere packing theory (see, for example, [AMRT75], [Sh05] and [So05]). Since the group GLd(Z) acts on the tessellation with perfect domains, this provi”
On the tropical Torelli map (2009, background): “Each of them plays a significant (and different) role in the theory of the toroidal compactifications of the moduli space of principally polarized abelian varieties (see [31,1,54]).”
The Kodaira dimension of the moduli of K3 surfaces (2007, background): “. Now the theorem follows from Proposition 2.18 and Proposition 2.19, below. ✷ Note that we only needed to choose Σ smooth: no further subdivision is necessary. A version of Theorem 2.17 is stated in [S-B] and proved in [Sn]. There the variety X Σ is itself allowed to have canonical singularities, b”
Vanishing theorems for torsion automorphic sheaves on general PEL-type Shimura varieties (2013, methodology): “In [63], Stroh obtained a liftability theorem based on a result of Shepherd-Barron [61] (for projective toroidal compactifications of Siegel modular schemes), and deduced from it certain results on classicality in Hida theory.”