By Martin C. Olsson

This quantity provides the development of canonical modular compactifications of moduli areas for polarized Abelian forms (possibly with point structure), development at the prior paintings of Alexeev, Nakamura, and Namikawa. this offers a special method of compactifying those areas than the extra classical strategy utilizing toroical embeddings, which aren't canonical. There are major new contributions during this monograph: (1) The creation of logarithmic geometry as understood by way of Fontaine, Illusie, and Kato to the examine of degenerating Abelian types; and (2) the development of canonical compactifications for moduli areas with larger measure polarizations according to stack-theoretic concepts and a examine of the theta group.

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Let f : (X, MX ) → (Y, MY ) be a morphism of ﬁne log schemes locally of ﬁnite presentation. For a scheme T and a quasi–coherent sheaf I on T , let T [I] denote the scheme with same underlying topological space as that of T , but with structure sheaf the OT –algebra OT ⊕ I with algebra structure given by (a + i)(c + j) = ac + (aj + ci). The ideal I deﬁnes a closed immersion j : T → T [I] for which the natural map T [I] → T induced by OT → OT [I] sending a to a is a retraction. If MT is a ﬁne log structure on T , let MT [I] denote the log structure on T [I] obtained by pullback along OT → OT [I] so that we have a diagram of ﬁne log schemes j π (T, MT ) −−−−→ (T [I], MT [I] ) −−−−→ (T, MT ).

Iv) θ ∈ f∗ L is a section such that for every geometric point s¯ → S the zero locus of the section θs¯ ∈ H 0 (Ps¯, Ls¯) does not contain any T -orbit. 10]) The stack T P fr [Q] is a proper algebraic stack over Z with ﬁnite diagonal. 7. 2], a stable semiabelic variety over an algebraically closed ﬁeld k is a scheme P/k together with an action of a semiabelian scheme G/k such that the following condition holds: 1. The dimension of G is equal to the dimension of each irreducible component of P . 2.

There are only ﬁnitely many orbits for the G–action. 4. The stabilizer group scheme of every point of P is connected, reduced, and lies in the toric part T of G. A stable semiabelic pair is a projective stable semiabelic variety P (with action of the semiabelian group scheme G) together with an ample line bundle L on P and a section θ ∈ H 0 (P, L ) whose zero locus does not contain any G–orbits. 8. If S is a scheme, then a stable semiabelic pair over S is a collection of data (G, f : P → S, L , θ ∈ f∗ L ) where: 1.