By Martin C. Olsson

ISBN-10: 354070518X

ISBN-13: 9783540705185

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

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**Example text**

To see this let h : E → E be such an automorphism. Since h is a morphism over X × X and E is a Gm –torsor over X × X, for any local section e ∈ E there exists a unique element u ∈ Gm such that h(e) = u(e). Furthermore, since h is compatible with the Gm –action the element u depends only on π(e). We therefore obtain a set map b : X × X → Gm by associating to any pair (x, y) the section of Gm obtained by locally choosing a lifting e ∈ E of (x, y) and sending (x, y) to the corresponding unit u ∈ Gm .

The ﬁrst concerns polarized toric varieties and the second abelian varities. We summarize here the main results from both problems. 7. Broken Toric Varieties. 2. Let X be a ﬁnitely generated free abelian group, and let T = Spec(Z[X]) be the corresponding torus. If S is a scheme, we write TS for the base change of T to S. 3. Let B = Spec(k) be the spectrum of an algebraically closed ﬁeld k, and let P/B be an aﬃne integral scheme with action of the torus TB such that the action has only ﬁnitely many orbits.

1) i∈I ×n where A denotes the n–fold ﬁber product over S of A with itself. If I is the empty set then mI sends all points of An to the identity element of A. Then B → A × A is canonically isomorphic to m∗I M (−1) Λ(M ) := card(I) . 2) I⊂{1,2} In other words, for any two scheme–valued points a, b ∈ A, the ﬁber of B over (a, b) ∈ A × A is equal to M (a + b) ⊗ M (a)−1 ⊗ M (b)−1 ⊗ M (0). 3) Note also that the deﬁnition of Λ(M ) is symmetric in the two factors of A × A so there is a canonical isomorphism ι : B → B over the ﬂip map A × A → A × A.

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