Download e-book for kindle: Compactifying Moduli Spaces for Abelian Varieties by Martin C. Olsson

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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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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Compactifying Moduli Spaces for Abelian Varieties by Martin C. Olsson

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