The Kodaira dimension of Siegel modular varieties of genus 3 or higher

Schellhammer, Eric

Bollettino dell'Unione Matematica Italiana, Tome 9-A (2006), p. 749-776 / Harvested from Biblioteca Digitale Italiana di Matematica

Abstract

We consider the moduli space $A_{\text{pol}}(n)$ of (non-principally) polarised abelian varieties of genus $g \geq 3$ with coprime polarisation and full level-n structure. Based upon the analysis of the Tits building in [S], we give an explicit lower bound on n that is sufficient for the compactified moduli space to be of general type if one further explicit condition is satisfied.

Résumé

Consideriamo lo spazio dei moduli $A_{\text{pol}}(n)$ delle varietà abeliane (non principalmente) polarizzate di genere $g \geq 3$ con polarizzazione coprima e struttura di livello n completa. Basandoci sull'analisi dei building di Tits di [S], diamo un limite inferiore esplicito per n che è sufficiente affinché lo spazio dei moduli compattificato sia di tipo generale, se un'ulteriore condizione esplicita viene soddisfatta.

Détail

Publié le : 2006-10-01

MR 2274125 | Zbl 1182.14017

Bibliographie

[AMRT] Ash, A. - Mumford, D. - Rapoport, M. - Tai, Y., Smooth Compactification of Locally Symmetric Varieties, Brookline: Math. Sci Press 1975. | Zbl 0334.14007

[B] Brasch, H.-J., Lifting level D-structures of abelian varieties, Arch. Math., Vol. 60 (1993), 553-562. | Zbl 0787.14025

[BC] Barnes, E. S. - Cohn, M. J., On the inner product of positive quadratic forms, J. London Math. Soc. (2), 12 (1975), 32-36. | Zbl 0312.10013

[BL] Birkenhake, C. - Lange, H., An isomorphism between moduli spaces of abelian varieties, Math. Nach., Vol. 253, Iss. 1 (2003), 3-7. | Zbl 1034.14019

[C] Cassels, J. W. S., An introduction to the geometry of numbers, Springer-Verlag, 1959. | Zbl 0086.26203

[F] Freitag, E., Siegelsche Modulfunktionen, Springer-Verlag, 1983.

[H] Hulek, K., Igusa's Modular Form and the Classification of Siegel Modular Threefolds, Moduli of Abelian Varieties. Progress in Mathematics195, Birkhauser Verlag (2001), 217-229. | Zbl 1054.14027

[HKW] Hulek, K. - Kahn, C. - Weintraub, S. H., Moduli Spaces of Abelian Surfaces: Compactification, Degenerations and Theta Functions, De Gruyter Expositions in Mathematics 12, 1993. | Zbl 0809.14035

[HS] Hulek, K. - Sankaran, G. K., The Geometry of Siegel Modular Varieties, Advanced Studies in Pure Mathematics 35 (2002), Higher Dimensional Birational Geometry, 89-156. | Zbl 1074.14021

[M] Mumford, D., Hirzebruch's Proportionality Theorem in the Non-Compact Case, Inv. Math., 42 (1977), 239-272. | Zbl 0365.14012

[M2] Mumford, D., On the Kodaira Dimension of the Siegel Modular Variety, Lec. Notes in Math. Vol. 997, Springer-Verlag, 1983, 348-376. | Zbl 0527.14036

[Nami] Namikawa, Y., Toroidal Compactification of Siegel Spaces, Lec. Notes in Math. Vol. 812, Springer-Verlag, 1980. | Zbl 0466.14011

[Nath] Nathanson, M. B., Elementary Methods in Number Theory, Springer-Verlag, 1991.

[O] Oda, T., Convex Bodies and Algebraic Geometry. An Introduction to the Theory of Toric Varieties, Ergeb. Math. Grenzgeb., 3 Folge, 15 (1988), Berlin, Springer-Verlag. | Zbl 0628.52002

[S] Schellhammer, E., On the Tits building of paramodular groupshttp://arXiv.org/abs/math/0405321 | Zbl 1177.51011

[T] Tai, Y.-S., On the Kodaira Dimension of the Moduli Space of Abelian Varieties, Invent. Math., 68 (1982), 425-439. | Zbl 0508.14038

[T2] Tai, Y.-S., On the Kodaira Dimension of the Moduli Spaces of Abelian Varieties with non-principal polarizations, In: Abelian Varietes, Egloffstein 1993, 293-302, de Gruyter, 1995. | Zbl 0848.14021