AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Moduli of double EPW-sextics
About this Title
Kieran G. O’Grady, Sapienza Università di Roma
Publication: Memoirs of the American Mathematical Society
Publication Year:
2016; Volume 240, Number 1136
ISBNs: 978-1-4704-1696-6 (print); 978-1-4704-2824-2 (online)
DOI: https://doi.org/10.1090/memo/1136
Published electronically: October 9, 2015
Keywords: GIT quotient,
period map,
hyperkähler varieties
MSC: Primary 14J10, 14L24, 14C30
Table of Contents
Chapters
- 1. Introduction
- 2. Preliminaries
- 3. One-parameter subgroups and stability
- 4. Plane sextics and stability of lagrangians
- 5. Lagrangians with large stabilizers
- 6. Description of the GIT-boundary
- 7. Boundary components meeting $\mathfrak {I}$ in a subset of $\mathfrak {X}_{\mathcal {W}}\cup \{\mathfrak {x},\mathfrak {x}^{\vee }\}$
- 8. The remaining boundary components
- A. Elementary auxiliary results
- B. Tables
Abstract
We will study the GIT quotient of the symplectic grassmannian parametrizing lagrangian subspaces of $\bigwedge ^3{\mathbb C}^6$ modulo the natural action of SL$_6$, call it $\mathfrak {M}$. This is a compactification of the moduli space of smooth double EPW-sextics and hence birational to the moduli space of HK $4$-folds of Type $K3^{[2]}$ polarized by a divisor of square $2$ for the Beauville-Bogomolov quadratic form. We will determine the stable points. Our work bears a strong analogy with the work of Voisin, Laza and Looijenga on moduli and periods of cubic $4$-folds. We will prove a result which is analogous to a theorem of Laza asserting that cubic $4$-folds with simple singularities are stable. We will also describe the irreducible components of the GIT boundary of $\mathfrak {M}$. Our final goal (not achieved in this work) is to understand completely the period map from $\mathfrak {M}$ to the Baily-Borel compactification of the relevant period domain modulo an arithmetic group. We will analyze the locus in the GIT-boundary of $\mathfrak {M}$ where the period map is not regular. Our results suggest that $\mathfrak {M}$ is isomorphic to Looijenga’s compactification associated to $3$ specific hyperplanes in the period domain.- Arnaud Beauville, Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom. 18 (1983), no. 4, 755–782 (1984) (French). MR 730926
- Arnaud Beauville and Ron Donagi, La variété des droites d’une hypersurface cubique de dimension $4$, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 14, 703–706 (French, with English summary). MR 818549
- Corrado De Concini and Claudio Procesi, Topics in hyperplane arrangements, polytopes and box-splines, Universitext, Springer, New York, 2011. MR 2722776
- Olivier Debarre and Claire Voisin, Hyper-Kähler fourfolds and Grassmann geometry, J. Reine Angew. Math. 649 (2010), 63–87. MR 2746467, DOI 10.1515/CRELLE.2010.089
- David Eisenbud, Sorin Popescu, and Charles Walter, Lagrangian subbundles and codimension 3 subcanonical subschemes, Duke Math. J. 107 (2001), no. 3, 427–467. MR 1828297, DOI 10.1215/S0012-7094-01-10731-X
- Andrea Ferretti, Special subvarieties of EPW sextics, Math. Z. 272 (2012), no. 3-4, 1137–1164. MR 2995162, DOI 10.1007/s00209-012-0980-5
- Robert Friedman, A new proof of the global Torelli theorem for $K3$ surfaces, Ann. of Math. (2) 120 (1984), no. 2, 237–269. MR 763907, DOI 10.2307/2006942
- V. Gritsenko, K. Hulek, and G. K. Sankaran, Moduli spaces of irreducible symplectic manifolds, Compos. Math. 146 (2010), no. 2, 404–434. MR 2601632, DOI 10.1112/S0010437X0900445X
- Daniel Huybrechts, A global Torelli theorem for hyperkähler manifolds [after M. Verbitsky], Astérisque 348 (2012), Exp. No. 1040, x, 375–403. Séminaire Bourbaki: Vol. 2010/2011. Exposés 1027–1042. MR 3051203
- Daniel Huybrechts and Manfred Lehn, The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig, 1997. MR 1450870
- Atanas Iliev and Laurent Manivel, Fano manifolds of degree ten and EPW sextics, Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), no. 3, 393–426 (English, with English and French summaries). MR 2839455, DOI 10.24033/asens.2146
- Atanas Iliev and Kristian Ranestad, $K3$ surfaces of genus 8 and varieties of sums of powers of cubic fourfolds, Trans. Amer. Math. Soc. 353 (2001), no. 4, 1455–1468. MR 1806733, DOI 10.1090/S0002-9947-00-02629-5
- Atanas Iliev and Kristian Ranestad, Addendum to “$K3$ surfaces of genus 8 and varieties of sums of powers of cubic fourfolds” [Trans. Amer. Math. Soc. 353 (2001), no. 4, 1455–1468; MR1806733], C. R. Acad. Bulgare Sci. 60 (2007), no. 12, 1265–1270. MR 2391437
- Radu Laza, The moduli space of cubic fourfolds, J. Algebraic Geom. 18 (2009), no. 3, 511–545. MR 2496456, DOI 10.1090/S1056-3911-08-00506-7
- Radu Laza, The moduli space of cubic fourfolds via the period map, Ann. of Math. (2) 172 (2010), no. 1, 673–711. MR 2680429, DOI 10.4007/annals.2010.172.673
- M. Lehn - C Lehn - C. Sorger - D. van Straten, Twisted cubics on cubic fourfolds, Mathematics arXiv: 1305.0178
- Eduard Looijenga, Compactifications defined by arrangements. II. Locally symmetric varieties of type IV, Duke Math. J. 119 (2003), no. 3, 527–588. MR 2003125, DOI 10.1215/S0012-7094-03-11933-X
- Eduard Looijenga, The period map for cubic fourfolds, Invent. Math. 177 (2009), no. 1, 213–233. MR 2507640, DOI 10.1007/s00222-009-0178-6
- D. Luna, Adhérences d’orbite et invariants, Invent. Math. 29 (1975), no. 3, 231–238 (French). MR 376704, DOI 10.1007/BF01389851
- Eyal Markman, Integral constraints on the monodromy group of the hyperKähler resolution of a symmetric product of a $K3$ surface, Internat. J. Math. 21 (2010), no. 2, 169–223. MR 2650367, DOI 10.1142/S0129167X10005957
- Eyal Markman, A survey of Torelli and monodromy results for holomorphic-symplectic varieties, Complex and differential geometry, Springer Proc. Math., vol. 8, Springer, Heidelberg, 2011, pp. 257–322. MR 2964480, DOI 10.1007/978-3-642-20300-8_{1}5
- U. Morin, Sui sistemi di piani a due a due incidenti, Atti del Reale Istituto Veneto di Scienze, Lettere ed Arti LXXXIX, 1930, pp. 907-926.
- D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906
- Kieran G. O’Grady, Desingularized moduli spaces of sheaves on a $K3$, J. Reine Angew. Math. 512 (1999), 49–117. MR 1703077, DOI 10.1515/crll.1999.056
- Kieran G. O’Grady, A new six-dimensional irreducible symplectic variety, J. Algebraic Geom. 12 (2003), no. 3, 435–505. MR 1966024, DOI 10.1090/S1056-3911-03-00323-0
- Kieran G. O’Grady, Irreducible symplectic 4-folds and Eisenbud-Popescu-Walter sextics, Duke Math. J. 134 (2006), no. 1, 99–137. MR 2239344, DOI 10.1215/S0012-7094-06-13413-0
- Kieran G. O’Grady, Dual double EPW-sextics and their periods, Pure Appl. Math. Q. 4 (2008), no. 2, Special Issue: In honor of Fedor Bogomolov., 427–468. MR 2400882, DOI 10.4310/PAMQ.2008.v4.n2.a6
- Kieran G. O’Grady, EPW-sextics: taxonomy, Manuscripta Math. 138 (2012), no. 1-2, 221–272. MR 2898755, DOI 10.1007/s00229-011-0472-7
- Kieran O’Grady, Double covers of EPW-sextics, Michigan Math. J. 62 (2013), no. 1, 143–184. MR 3049300, DOI 10.1307/mmj/1363958245
- Kieran G. O’Grady, Periods of double EPW-sextics, Math. Z. 280 (2015), no. 1-2, 485–524. MR 3343917, DOI 10.1007/s00209-015-1434-7
- Antonio Rapagnetta, On the Beauville form of the known irreducible symplectic varieties, Math. Ann. 340 (2008), no. 1, 77–95. MR 2349768, DOI 10.1007/s00208-007-0139-6
- Jayant Shah, A complete moduli space for $K3$ surfaces of degree $2$, Ann. of Math. (2) 112 (1980), no. 3, 485–510. MR 595204, DOI 10.2307/1971089
- Misha Verbitsky, Mapping class group and a global Torelli theorem for hyperkähler manifolds, Duke Math. J. 162 (2013), no. 15, 2929–2986. Appendix A by Eyal Markman. MR 3161308, DOI 10.1215/00127094-2382680
- C. Voisin, Théorème de Torelli pour les cubiques de $P^ 5$, Invent. Math. 86, 1986, pp. 577-601. Erratum Invent. Math. 172, 2008, pp. 455-458.