Fano manifolds of degree ten and EPW sextics

Atanas Iliev; Laurent Manivel

Annales scientifiques de l'École Normale Supérieure (2011)

  • Volume: 44, Issue: 3, page 393-426
  • ISSN: 0012-9593

Abstract

top
O’Grady showed that certain special sextics in ℙ 5 called EPW sextics admit smooth double covers with a holomorphic symplectic structure. We propose another perspective on these symplectic manifolds, by showing that they can be constructed from the Hilbert schemes of conics on Fano fourfolds of degree ten. As applications, we construct families of Lagrangian surfaces in these symplectic fourfolds, and related integrable systems whose fibers are intermediate Jacobians.

How to cite

top

Iliev, Atanas, and Manivel, Laurent. "Fano manifolds of degree ten and EPW sextics." Annales scientifiques de l'École Normale Supérieure 44.3 (2011): 393-426. <http://eudml.org/doc/272141>.

@article{Iliev2011,
abstract = {O’Grady showed that certain special sextics in $\mathbb \{P\}^5$ called EPW sextics admit smooth double covers with a holomorphic symplectic structure. We propose another perspective on these symplectic manifolds, by showing that they can be constructed from the Hilbert schemes of conics on Fano fourfolds of degree ten. As applications, we construct families of Lagrangian surfaces in these symplectic fourfolds, and related integrable systems whose fibers are intermediate Jacobians.},
author = {Iliev, Atanas, Manivel, Laurent},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {holomorphic symplectic manifold; Fano manifold; grassmannian; Hilbert scheme; conic; double cover; lagrangian surface; integrable system},
language = {eng},
number = {3},
pages = {393-426},
publisher = {Société mathématique de France},
title = {Fano manifolds of degree ten and EPW sextics},
url = {http://eudml.org/doc/272141},
volume = {44},
year = {2011},
}

TY - JOUR
AU - Iliev, Atanas
AU - Manivel, Laurent
TI - Fano manifolds of degree ten and EPW sextics
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2011
PB - Société mathématique de France
VL - 44
IS - 3
SP - 393
EP - 426
AB - O’Grady showed that certain special sextics in $\mathbb {P}^5$ called EPW sextics admit smooth double covers with a holomorphic symplectic structure. We propose another perspective on these symplectic manifolds, by showing that they can be constructed from the Hilbert schemes of conics on Fano fourfolds of degree ten. As applications, we construct families of Lagrangian surfaces in these symplectic fourfolds, and related integrable systems whose fibers are intermediate Jacobians.
LA - eng
KW - holomorphic symplectic manifold; Fano manifold; grassmannian; Hilbert scheme; conic; double cover; lagrangian surface; integrable system
UR - http://eudml.org/doc/272141
ER -

References

top
  1. [1] M. Andreatta & J. A. Wiśniewski, A note on nonvanishing and applications, Duke Math. J.72 (1993), 739–755. Zbl0853.14003MR1253623
  2. [2] E. Arbarello, M. Cornalba, P. A. Griffiths & J. Harris, Geometry of algebraic curves. Vol. I, Grund. Math. Wiss. 267, Springer, 1985. Zbl0559.14017MR770932
  3. [3] A. Beauville, Fano threefolds and K 3 surfaces, in The Fano Conference, Univ. Torino, Turin, 2004, 175–184. Zbl1096.14034MR2112574
  4. [4] O. Debarre, A. Iliev & L. Manivel, On the period map for prime Fano threefolds of degree ten, preprint arXiv:0812.3670, to appear in J. Algebraic Geom. Zbl1250.14029MR2846678
  5. [5] O. Debarre & C. Voisin, Hyper-Kähler fourfolds and Grassmann geometry, J. reine angew. Math. 649 (2010), 63–87. Zbl1217.14028MR2746467
  6. [6] R. Donagi & E. Markman, Spectral covers, algebraically completely integrable, Hamiltonian systems, and moduli of bundles, in Integrable systems and quantum groups (Montecatini Terme, 1993), Lecture Notes in Math. 1620, Springer, 1996, 1–119. Zbl0853.35100MR1397273
  7. [7] D. Eisenbud, S. Popescu & C. Walter, Enriques surfaces and other non-Pfaffian subcanonical subschemes of codimension 3, Comm. Algebra28 (2000), 5629–5653. Zbl0983.14018MR1808593
  8. [8] N. P. Gushelʼ, Fano varieties of genus 6 , Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), 1159–1174; English translation: Math. USSR Izv. 21 (1983), 445–459. Zbl0554.14014
  9. [9] A. Iliev & L. Manivel, Prime Fano threefolds and integrable systems, Math. Ann.339 (2007), 937–955. Zbl1136.14026MR2341908
  10. [10] A. Iliev & L. Manivel, Cubic hypersurfaces and integrable systems, Amer. J. Math.130 (2008), 1445–1475. Zbl1162.14032MR2464024
  11. [11] A. Kuznetsov & D. Markushevich, Symplectic structures on moduli spaces of sheaves via the Atiyah class, J. Geom. Phys.59 (2009), 843–860. Zbl1181.14049MR2536849
  12. [12] D. Logachev, Fano threefolds of genus 6 , preprint arXiv:math/0407147. Zbl1263.14040MR2989233
  13. [13] D. Markushevich, An integrable system of K 3 -Fano flags, Math. Ann.342 (2008), 145–156. Zbl1144.14039MR2415319
  14. [14] D. Markushevich, Integrable systems from intermediate Jacobians of fivefolds, preprint, 2009. Zbl1296.14032
  15. [15] S. Mukai, Curves, K 3 surfaces and Fano 3 -folds of genus ≤ 10 , in Algebraic geometry and commutative algebra, Vol. I, Kinokuniya, 1988, 357–377. Zbl0701.14044MR977768
  16. [16] S. Mukai, Moduli of vector bundles on K 3 surfaces and symplectic manifolds, Sūgaku Expositions1 (1988), 139–174. Zbl0685.14021MR922020
  17. [17] K. G. O’Grady, Involutions and linear systems on holomorphic symplectic manifolds, Geom. Funct. Anal.15 (2005), 1223–1274. Zbl1093.53081
  18. [18] K. G. O’Grady, Irreducible symplectic 4-folds and Eisenbud-Popescu-Walter sextics, Duke Math. J.134 (2006), 99–137. Zbl1105.14051
  19. [19] K. G. O’Grady, Dual double EPW-sextics and their periods, Pure Appl. Math. Q.4 (2008), 427–468. Zbl1152.14010MR2400882
  20. [20] K. G. O’Grady, Irreducible symplectic 4-folds numerically equivalent to ( K 3 ) [ 2 ] , Commun. Contemp. Math.10 (2008), 553–608. Zbl1216.14040MR2444848
  21. [21] E. Sernesi, Deformations of algebraic schemes, Grund. Math. Wiss. 334, Springer, 2006. Zbl1102.14001MR2247603

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.