The Reading Seminar on moduli of abelian varieties will take place Tuesdays 16:00-17:00 in g117.

Description

The standard Torelli map is the map M_g -> A_g from the moduli space of smooth curves to the moduli space of (principally polarized) abelian varieties defined by the rule that sends a curve to its Jacobian. Both M_g and A_g admit natural compactifications. The moduli space M_g is contained in the moduli space of stable curves, and the moduli space of abelian varieties admits many compactifications constructed using the theory of toroidal compactification. Given a toroidal compactification, a fundamental question is: When does the Torelli map extend to a regular map on compactifications? Using ideas of Mumford, Namikawa proved that the Torelli map extends to the second Voronoi compactification. Namikawa showed that the extension exists using toroidal methods, but Alexeev later proved that this construction can be constructed using moduli theory. Alexeev proved that the second Voronoi compactification can (almost) be described as a moduli space of degenerate abelian varieties, and the extended Torelli map can be described as the morphism that sends a stable curve to its compactified Jacobian. More recent work of Alexeev and Brunyate answers the question of when an extension of the Torelli map exists for the first Voronoi or the perfect cone compactification. In the seminar, we will try to learn more about this exciting body of work!

Schedule of upcoming talks

06 November	Jesse Leo Kass	Introductory talk
13 November	Matteo Tommasini	Polarizations
20 November	Nicola Tarasca	The algebraic Torelli map
27 November	Jesse Leo Kass	Degenerations of abelian varieties
04 December	Nicola Pagani	Degenerations of abelian varieties
11 December	Jesse Leo Kass	Definition of the slope semistable compactified Jacobian
8 January	Jesse Leo Kass	Combinatorics of semistability conditions
15 January	Jesse Leo Kass	Combinatorics of semistability conditions
22 January	Jesse Leo Kass	Combinatorics of semistability conditions
29 January	Jesse Leo Kass	Combinatorics of semistability conditions
5 February	Nicola Pagani	Examples of semistabilty conditions
12 February	Nicola Pagani	Examples of semistabilty conditions
19 February	Jesse Leo Kass	Variation of semistability conditions
26 February	No Meeting
5 March	Nicola Tarasca	The theta divsior

Notes

Download here the notes (preliminary version)

References

V. Alexeev, Ch. Birkenhake, and K. Hulek. Degenerations of Prym varieties. J. Reine Angew. Math., 553:73-116, 2002.

Valery Alexeev. Complete moduli in the presence of semiabelian group action. Ann. of Math. (2), 155(3):611-708, 2002.

Valery Alexeev. Compactified Jacobians and Torelli map. Publ. Res. Inst. Math. Sci., 40(4):1241-1265, 2004.

Valery Alexeev and Iku Nakamura. On Mumford's construction of degenerating abelian varieties. Tohoku Math. J. (2), 51(3):399-420, 1999.

Arnaud Beauville. Prym varieties and the Schottky problem. Invent. Math., 41(2):149-196, 1977.

Michel Brion. Compactification de l'espace des modules des variétés abéliennes principalement polarisées (d'après V. Alexeev). Astérisque, (311):Exp. No. 952, vii, 1-31, 2007. Séminaire Bourbaki. Vol. 2005/2006.

Lucia Caporaso. A compactification of the universal Picard variety over the moduli space of stable curves. J. Amer. Math. Soc., 7(3):589-660, 1994.

Sebastian Casalaina-Martin, Jesse Leo Kass, and Filippo Viviani. The local structure of compactified Jacobians: Deformation theory. arXiv:1107.4166v1, 2011.

P. Deligne and M. Rapoport. Les schémas de modules de courbes elliptiques. In Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pages 143-316. Lecture Notes in Math., Vol. 349. Springer, Berlin, 1973.

Eduardo Esteves. Very ampleness for theta on the compactified Jacobian. Math. Z., 226(2):181-191, 1997.

Gerd Faltings and Ching-Li Chai. Degeneration of abelian varieties, volume 22 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Springer-Verlag, Berlin, 1990. With an appendix by David Mumford.

Klaus Hulek, Constantin Kahn, and Steven H. Weintraub. Moduli spaces of abelian surfaces: compactification, degenerations, and theta functions, volume 12 of de Gruyter Expositions in Mathematics. Berlin, 1993.

Kai-Wen Lan. Arithmetic compactifications of PEL-type Shimura varieties. Pro-Quest LLC, Ann Arbor, MI, 2008. Thesis (Ph.D.)-Harvard University.

David Mumford. An analytic construction of degenerating abelian varieties over complete rings. Compositio Math., 24:239-272, 1972.

Tadao Oda and C. S. Seshadri. Compactifications of the generalized Jacobian variety. Trans. Amer. Math. Soc., 253:1-90, 1979.

Rahul Pandharipande. A compactification over M_g of the universal moduli space of slope-semistable vector bundles. J. Amer. Math. Soc., 9(2):425-471, 1996.

Carlos T. Simpson. Moduli of representations of the fundamental group of a smooth projective variety. I. Inst. Hautes Études Sci. Publ. Math., (79):47-129, 1994.

A. Soucaris. The ampleness of the theta divisor on the compactified Jacobian of a proper and integral curve. Compositio Math., 93(3):231-242, 1994.