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

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

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

Veröffentlicht/Copyright: 16. Mai 2018

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Aus der Zeitschrift Journal für die reine und angewandte Mathematik (Crelles Journal) Band 2020 Heft 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-120022434Suche 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. 10.2307/3062130Suche 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/1145475446Suche 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.2002.103Suche 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/s00222-011-0347-2Suche 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/rnu165Suche 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. 10.1007/978-1-4757-5323-3Suche in Google Scholar

[8] A. Beauville, Prym varieties and the Schottky problem, Invent. Math. 41 (1977), no. 2, 149–196. 10.1007/BF01418373Suche 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/1214430642Suche 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/BF02565599Suche 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/S1056-3911-04-00379-0Suche 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. 10.2307/1970801Suche in Google Scholar

[13] I. V. Dolgachev, Classical algebraic geometry. A modern view, Cambridge University Press, Cambridge 2012. 10.1017/CBO9781139084437Suche in Google Scholar

[14] R. Donagi, The unirationality of 𝒜5, Ann. of Math. (2) 119 (1984), no. 2, 269–307. 10.2307/2007041Suche 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/136/1188194Suche 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/BF02392458Suche 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/S0002-9947-2011-05298-0Suche in Google Scholar

[18] E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Mat. Sbornik N.S. 30(72) (1952), 349–462. 10.1090/trans2/006/02Suche 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.1736Suche 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.0.0053Suche 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/S0002-9947-02-03165-3Suche 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/476Suche 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/214Suche 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.2289Suche 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/S0002-9939-01-05878-6Suche 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.a10Suche 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_8Suche 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/S1056-3911-2012-00587-6Suche 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.2009.015Suche 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/679Suche 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. Suche in Google Scholar

[32] V. Kanev, Principal polarizations of Prym–Tjurin varieties, Compos. Math. 64 (1987), no. 3, 243–270. Suche 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/BF01790341Suche 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/049.1/1013158Suche 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.2006.01.008Suche 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. Suche 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.2011.02.005Suche in Google Scholar

[38] J. Kollár, Projectivity of complete moduli, J. Differential Geom. 32 (1990), no. 1, 235–268. 10.4310/jdg/1214445046Suche 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-xSuche 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/BFb0099970Suche 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/S0894-0347-98-00261-6Suche 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/B978-0-12-044850-0.50032-0Suche 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/BFb0061652Suche 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/S002776300001597XSuche 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/BF01596390Suche in Google Scholar

[46] T. Oshima, A classification of subsystems of a root system, preprint (2006), https://arxiv.org/abs/math/0611904. Suche 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)90033-7Suche in Google Scholar

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

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