The uniformization of the moduli space of principally polarized abelian 6-folds

Valery Alexeev; Ron Donagi; Gavril Farkas; Elham Izadi; Angela Ortega

doi:10.1515/crelle-2018-0005

Article

The uniformization of the moduli space of principally polarized abelian 6-folds

Valery Alexeev , Ron Donagi , Gavril Farkas , Elham Izadi and Angela Ortega

Published/Copyright: May 16, 2018

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2020 Issue 761

Abstract

Starting from a beautiful idea of Kanev, we construct a uniformization of the moduli space 𝒜6 of principally polarized abelian 6-folds in terms of curves and monodromy data. We show that the general principally polarized abelian variety of dimension 6 is a Prym–Tyurin variety corresponding to a degree 27 cover of the projective line having monodromy the Weyl group of the E6 lattice. Along the way, we establish numerous facts concerning the geometry of the Hurwitz space of such E6-covers, including: (1) a proof that the canonical class of the Hurwitz space is big, (2) a concrete geometric description of the Hodge–Hurwitz eigenbundles with respect to the Kanev correspondence and (3) a description of the ramification divisor of the Prym–Tyurin map from the Hurwitz space to 𝒜6 in the terms of syzygies of the Abel–Prym–Tyurin curve.

Funding source: Deutsche Forschungsgemeinschaft

Award Identifier / Grant number: Sonderforschungsbereich 647 “Raum-Zeit-Materie”

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-1200726

Award Identifier / Grant number: DMS-1603526

Award Identifier / Grant number: DMS-1103938/1430600

Funding statement: The authors acknowledge partial support by the NSF: Valery Alexeev under grant DMS-1200726, Ron Donagi under grant DMS-1603526, Elham Izadi under grant DMS-1103938/1430600. The work of Gavril Farkas and Angela Ortega has been partially supported by the DFG Sonderforschungsbereich 647 “Raum-Zeit-Materie”.

Acknowledgements

We owe a great debt to the work of Vassil Kanev, who first constructed the Prym–Tyurin map PT and raised the possibility of uniformizing 𝒜6 in this way.

References

[1] D. Abramovich, A. Corti and A. Vistoli, Twisted bundles and admissible covers, Comm. Algebra 31 (2003), no. 8, 3547–3618. 10.1081/AGB-120022434Search in Google Scholar

[2] V. Alexeev, Complete moduli in the presence of semiabelian group action, Ann. of Math. (2) 155 (2002), no. 3, 611–708. 10.2307/3062130Search in Google Scholar

[3] V. Alexeev, Compactified Jacobians and Torelli map, Publ. Res. Inst. Math. Sci. 40 (2004), no. 4, 1241–1265. 10.2977/prims/1145475446Search in Google Scholar

[4] V. Alexeev, C. Birkenhake and K. Hulek, Degenerations of Prym varieties, J. reine angew. Math. 553 (2002), 73–116. 10.1515/crll.2002.103Search in Google Scholar

[5] V. Alexeev and A. Brunyate, Extending the Torelli map to toroidal compactifications of Siegel space, Invent. Math. 188 (2012), no. 1, 175–196. 10.1007/s00222-011-0347-2Search in Google Scholar

[6] M. Arap and R. Varley, On algebraic equivalences among the twenty-seven Abel–Prym curves on a generic abelian five-fold, Int. Math. Res. Not. IMRN (2015), no. 16, 7323–7335. 10.1093/imrn/rnu165Search in Google Scholar

[7] E. Arbarello, M. Cornalba, P. A. Griffiths and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren Math. Wiss. 267, Springer, New York 1985. 10.1007/978-1-4757-5323-3Search in Google Scholar

[8] A. Beauville, Prym varieties and the Schottky problem, Invent. Math. 41 (1977), no. 2, 149–196. 10.1007/BF01418373Search in Google Scholar

[9] A. Borel, Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem, J. Differential Geometry 6 (1972), 543–560. 10.4310/jdg/1214430642Search in Google Scholar

[10] A. Borel and J. De Siebenthal, Les sous-groupes fermés de rang maximum des groupes de Lie clos, Comment. Math. Helv. 23 (1949), 200–221. 10.1007/BF02565599Search in Google Scholar

[11] S. Casalaina-Martin and R. Friedman, Cubic threefolds and abelian varieties of dimension five, J. Algebraic Geom. 14 (2005), no. 2, 295–326. 10.1090/S1056-3911-04-00379-0Search in Google Scholar

[12] C. H. Clemens and P. A. Griffiths, The intermediate Jacobian of the cubic threefold, Ann. of Math. (2) 95 (1972), 281–356. 10.2307/1970801Search in Google Scholar

[13] I. V. Dolgachev, Classical algebraic geometry. A modern view, Cambridge University Press, Cambridge 2012. 10.1017/CBO9781139084437Search in Google Scholar

[14] R. Donagi, The unirationality of 𝒜5, Ann. of Math. (2) 119 (1984), no. 2, 269–307. 10.2307/2007041Search in Google Scholar

[15] R. Donagi, The fibers of the Prym map, Curves, Jacobians, and abelian varieties (Amherst 1990), Contemp. Math. 136, American Mathematical Society, Providence (1992), 55–125. 10.1090/conm/136/1188194Search in Google Scholar

[16] R. Donagi and R. C. Smith, The structure of the Prym map, Acta Math. 146 (1981), no. 1–2, 25–102. 10.1007/BF02392458Search in Google Scholar

[17] M. J. Dyer and G. I. Lehrer, Reflection subgroups of finite and affine Weyl groups, Trans. Amer. Math. Soc. 363 (2011), no. 11, 5971–6005. 10.1090/S0002-9947-2011-05298-0Search in Google Scholar

[18] E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Mat. Sbornik N.S. 30(72) (1952), 349–462. 10.1090/trans2/006/02Search in Google Scholar

[19] E. Esteves, Wronski algebra systems on families of singular curves, Ann. Sci. Éc. Norm. Supér. (4) 29 (1996), no. 1, 107–134. 10.24033/asens.1736Search in Google Scholar

[20] G. Farkas, Koszul divisors on moduli spaces of curves, Amer. J. Math. 131 (2009), no. 3, 819–867. 10.1353/ajm.0.0053Search in Google Scholar

[21] G. Farkas and A. Gibney, The Mori cones of moduli spaces of pointed curves of small genus, Trans. Amer. Math. Soc. 355 (2003), no. 3, 1183–1199. 10.1090/S0002-9947-02-03165-3Search in Google Scholar

[22] G. Farkas, S. Grushevsky, R. Salvati Manni and A. Verra, Singularities of theta divisors and the geometry of 𝒜5, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 9, 1817–1848. 10.4171/JEMS/476Search in Google Scholar

[23] G. Farkas and K. Ludwig, The Kodaira dimension of the moduli space of Prym varieties, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 3, 755–795. 10.4171/JEMS/214Search in Google Scholar

[24] G. Farkas and A. Verra, The universal abelian variety over 𝒜5, Ann. Sci. Éc. Norm. Supér. (4) 49 (2016), no. 3, 521–542. 10.24033/asens.2289Search in Google Scholar

[25] W. N. Franzsen, Automorphisms of Coxeter groups, PhD thesis, University of Sydney, Sydney 2001, http://www.maths.usyd.edu.au/u/PG/Theses/franzsen.pdf. 10.1090/S0002-9939-01-05878-6Search in Google Scholar

[26] G. van der Geer and A. Kouvidakis, The Hodge bundle on Hurwitz spaces, Pure Appl. Math. Q. 7 (2011), no. 4, 1297–1307. 10.4310/PAMQ.2011.v7.n4.a10Search in Google Scholar

[27] J. Harris and D. Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982), no. 1, 23–88. 10.1007/978-1-4757-4265-7_8Search in Google Scholar

[28] E. Izadi and H. Lange, Counter-examples of high Clifford index to Prym–Torelli, J. Algebraic Geom. 21 (2012), no. 4, 769–787. 10.1090/S1056-3911-2012-00587-6Search in Google Scholar

[29] E. Izadi, H. Lange and V. Strehl, Correspondences with split polynomial equations, J. reine angew. Math. 627 (2009), 183–212. 10.1515/CRELLE.2009.015Search in Google Scholar

[30] E. Izadi, C. Tamás and J. Wang, The primitive cohomology of the theta divisor of an Abelian fivefold, J. Algebraic Geom. 26 (2017), no. 1, 107–175. 10.1090/jag/679Search in Google Scholar

[31] E. Izadi and D. van Straten, The intermediate Jacobians of the theta divisors of four-dimensional principally polarized abelian varieties, J. Algebraic Geom. 4 (1995), no. 3, 557–590. Search in Google Scholar

[32] V. Kanev, Principal polarizations of Prym–Tjurin varieties, Compos. Math. 64 (1987), no. 3, 243–270. Search in Google Scholar

[33] V. Kanev, Intermediate Jacobians and Chow groups of three-folds with a pencil of del Pezzo surfaces, Ann. Mat. Pura Appl. (4) 154 (1989), 13–48. 10.1007/BF01790341Search in Google Scholar

[34] V. Kanev, Spectral curves, simple Lie algebras, and Prym–Tjurin varieties, Theta functions – Bowdoin 1987. Part 1 (Brunswick 1987), Proc. Sympos. Pure Math. 49, American Mathematical Society, Providence (1989), 627–645. 10.1090/pspum/049.1/1013158Search in Google Scholar

[35] V. Kanev, Hurwitz spaces of Galois coverings of ℙ1, whose Galois groups are Weyl groups, J. Algebra 305 (2006), no. 1, 442–456. 10.1016/j.jalgebra.2006.01.008Search in Google Scholar

[36] S. Keel and J. McKernan, Contractible extremal rays on M¯0,n, Handbook of moduli. Vol. II, Adv. Lect. Math. (ALM) 25, International Press, Somerville (2013), 115–130. Search in Google Scholar

[37] A. Kokotov, D. Korotkin and P. Zograf, Isomonodromic tau function on the space of admissible covers, Adv. Math. 227 (2011), no. 1, 586–600. 10.1016/j.aim.2011.02.005Search in Google Scholar

[38] J. Kollár, Projectivity of complete moduli, J. Differential Geom. 32 (1990), no. 1, 235–268. 10.4310/jdg/1214445046Search in Google Scholar

[39] H. Lange and A. M. Rojas, A Galois-theoretic approach to Kanev’s correspondence, Manuscripta Math. 125 (2008), no. 2, 225–240. 10.1007/s00229-007-0143-xSearch in Google Scholar

[40] S. Mori and S. Mukai, The uniruledness of the moduli space of curves of genus 11, Algebraic geometry (Tokyo/Kyoto 1982), Lecture Notes in Math. 1016, Springer, Berlin (1983), 334–353. 10.1007/BFb0099970Search in Google Scholar

[41] A. Moriwaki, Relative Bogomolov’s inequality and the cone of positive divisors on the moduli space of stable curves, J. Amer. Math. Soc. 11 (1998), no. 3, 569–600. 10.1090/S0894-0347-98-00261-6Search in Google Scholar

[42] D. Mumford, Prym varieties. I, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York (1974), 325–350. 10.1016/B978-0-12-044850-0.50032-0Search in Google Scholar

[43] D. Mumford, On the Kodaira dimension of the Siegel modular variety, Algebraic geometry – Open problems (Ravello 1982), Lecture Notes in Math. 997, Springer, Berlin (1983), 348–375. 10.1007/BFb0061652Search in Google Scholar

[44] Y. Namikawa, On the canonical holomorphic map from the moduli space of stable curves to the Igusa monoidal transform, Nagoya Math. J. 52 (1973), 197–259. 10.1017/S002776300001597XSearch in Google Scholar

[45] Y. Namikawa, A new compactification of the Siegel space and degeneration of Abelian varieties. II, Math. Ann. 221 (1976), no. 3, 201–241. 10.1007/BF01596390Search in Google Scholar

[46] T. Oshima, A classification of subsystems of a root system, preprint (2006), https://arxiv.org/abs/math/0611904. Search in Google Scholar

[47] A. Verra, A short proof of the unirationality of 𝒜5, Nederl. Akad. Wetensch. Indag. Math. 46 (1984), no. 3, 339–355. 10.1016/1385-7258(84)90033-7Search in Google Scholar

[48] W. Wirtinger, Untersuchungen über Thetafunktionen, Teubner, Leipzig 1895. Search in Google Scholar

[49] Website with supporting computer computations, http://alpha.math.uga.edu/~valery/a6. Search in Google Scholar

Received: 2018-03-03

Published Online: 2018-05-16

Published in Print: 2020-04-01

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https://doi.org/10.1515/crelle-2018-0005