Kyoto Journal of Mathematics

Brane involutions on irreducible holomorphic symplectic manifolds

Emilio Franco, Marcos Jardim, and Grégoire Menet

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In the context of irreducible holomorphic symplectic manifolds, we say that (anti)holomorphic (anti)symplectic involutions are brane involutions since their fixed point locus is a brane in the physicists’ language, that is, a submanifold which is either a complex or Lagrangian submanifold with respect to each of the three Kähler structures of the associated hyper-Kähler structure. Starting from a brane involution on a K3 or Abelian surface, one can construct a natural brane involution on its moduli space of sheaves. We study these natural involutions and their relation with the Fourier–Mukai transform. Later, we recall the lattice-theoretical approach to mirror symmetry. We provide two ways of obtaining a brane involution on the mirror, and we study the behavior of the brane involutions under both mirror transformations, giving examples in the case of a K3 surface and K3[2]-type manifolds.

Article information

Source
Kyoto J. Math., Volume 59, Number 1 (2019), 195-235.

Dates
Received: 5 July 2016
Accepted: 10 February 2017
First available in Project Euclid: 8 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1546916422

Digital Object Identifier
doi:10.1215/21562261-2018-0009

Mathematical Reviews number (MathSciNet)
MR3934628

Zentralblatt MATH identifier
07081627

Subjects
Primary: 14J28: $K3$ surfaces and Enriques surfaces
Secondary: 14J33: Mirror symmetry [See also 11G42, 53D37] 14J50: Automorphisms of surfaces and higher-dimensional varieties

Keywords
K3 surfaces involutions mirror symmetry

Citation

Franco, Emilio; Jardim, Marcos; Menet, Grégoire. Brane involutions on irreducible holomorphic symplectic manifolds. Kyoto J. Math. 59 (2019), no. 1, 195--235. doi:10.1215/21562261-2018-0009. https://projecteuclid.org/euclid.kjm/1546916422


Export citation

References

  • [1] P. S. Aspinwall and D. R. Morrison, “String theory on K3 surfaces” in Mirror Symmetry, II, AMS/IP Stud. Adv. Math. 1, Amer. Math. Soc., Providence, 1997, 703–716.
  • [2] M. F. Atiyah, K-theory and reality, Q. J. Math. 17 (1966), 367–386.
  • [3] D. Baraglia and L. P. Schaposnik, Higgs bundles and $(A,B,A)$-branes, Comm. Math. Phys. 331 (2014), no. 3, 1271–1300.
  • [4] C. Bartocci, U. Bruzzo, and D. Hernández Ruipérez, Fourier–Mukai and Nahm Transforms in Geometry and Mathematical Physics, Progr. Math. 276, Birkhaüser, Boston, 2009.
  • [5] A. Bayer, B. Hassett, and Y. Tschinkel, Mori cones of holomorphic symplectic varieties of K3 type, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), 941–950.
  • [6] A. Beauville, Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom. 18 (1983), no. 4, 755–782.
  • [7] A. Beauville, Antisymplectic involutions of holomorphic symplectic manifolds, J. Topol. 4 (2011), no. 2, 300–304.
  • [8] I. Biswas, Connections on principal bundles over Kähler manifolds with antiholomorphic involution, Forum Math. 17 (2005), no. 6, 871–884.
  • [9] I. Biswas, J. Huisman, and J. Hurtubise, The moduli space of stable vector bundles over a real algebraic curve, Math. Ann. 347 (2010), no. 1, 201–233.
  • [10] I. Biswas and G. Wilkin, Anti-holomorphic involutive isometry of hyper-Kähler manifolds and branes, J. Geom. Phys. 88 (2015), 52–55.
  • [11] S. Boissière, Automorphismes naturels de l’espace de Douady de points sur une surface, Canad. J. Math. 64 (2012), no. 1, 3–23.
  • [12] S. Boissière, C. Camere, and A. Sarti, Classification of automorphisms on a deformation family of hyper-Kähler four-folds by $p$-elementary lattices, Kyoto J. Math. 56 (2016), no. 3, 465–499.
  • [13] C. Camere, Lattice polarized irreducible holomorphic symplectic manifolds, Ann. Inst. Fourier (Grenoble) 66 (2016), no. 2, 687–709.
  • [14] I. V. Dolgachev, Mirror symmetry for lattice polarized $\mathrm{K3}$ surfaces, J. Math. Sci. 81 (1996), no. 3, 2599–2630.
  • [15] J. Fogarty, Algebraic families on an algebraic surface, Amer. J. Math. 90 (1968), 511–521.
  • [16] E. Franco, M. Jardim, and S. Marchesi, Branes in the moduli space of framed sheaves, Bull. Sci. Math. 141 (2017), no. 4, 353–383.
  • [17] B. Hassett and Y. Tschinkel, Hodge theory and Lagrangian planes on generalized Kummer fourfolds, Mosc. Math. J. 13 (2013), no. 1, 33–56.
  • [18] D. Huybrechts, Compact hyper-Kähler manifolds: Basic results, Invent. Math. 135 (1999), no. 1, 63–113.
  • [19] D. Huybrechts, “Moduli spaces of hyperkähler manifolds and mirror symmetry” in Intersection Theory and Moduli, ICTP Lect. Notes 19, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004, 185–247.
  • [20] D. Huybrechts, Lectures on K3 Surfaces, Cambridge Stud. Adv. Math. 158, Cambridge Univ. Press, Cambridge, 2016.
  • [21] D. Huybrechts and M. Lehn, The Geometry of Moduli Spaces of Sheaves, 2nd ed., Cambridge Math. Libr., Cambridge Univ. Press, Cambridge, 2010.
  • [22] A. Kapustin and E. Witten, Electric-magnetic duality and the geometric Langlands program, Commun. Number Theory Phys. 1 (2007), no. 1, 1–236.
  • [23] E. Looijenga and C. Peters, Torelli theorems for Kähler K3 surfaces, Compos. Math. 42 (1980/81), no. 2, 145–186.
  • [24] E. 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.
  • [25] E. Markman, “A survey of Torelli and monodromy results for holomorphic-symplectic varieties” in Complex and Differential Geometry, Springer Proc. Math. 8, Springer, Heidelberg, 2011, 257–322.
  • [26] G. Mongardi, Symplectic involutions on deformations of $K3^{[2]}$, Cent. Eur. J. Math. 10 (2012), no. 4, 1472–1485.
  • [27] G. Mongardi, On natural deformations of symplectic automorphisms of manifolds of $\mathrm{K3}^{[n]}$ type, C. R. Math. Acad. Sci. Paris 351 (2013), no. 13–14, 561–564.
  • [28] G. Mongardi, A note on the Kähler and Mori cones of hyperkähler manifolds, Asian J. Math. 19 (2015), no. 4, 583–591.
  • [29] G. Mongardi, K. Tari, and M. Wandel, Prime order automorphisms of generalised Kummer fourfolds, Manuscripta Math. 155 (2018), no. 3–4, 449–469.
  • [30] G. Mongardi and M. Wandel, Automorphisms of O’Grady’s manifolds acting trivially on cohomology, Algebr. Geom. 4 (2017), no. 1, 104–119.
  • [31] S. Mukai, Symplectic structure on the moduli space of sheaves on an abelian or K3 surface, Invent. Math. 77 (1984), no. 1, 101–116.
  • [32] S. Mukai, “On the moduli space of bundles on $\mathrm{K3}$ surfaces, I” in Vector Bundles on Algebraic Varieties (Bombay, 1984), Tata Inst. Fund. Res. Stud. Math. 11, Tata Inst. Fund. Res., Bombay, 1987, 341–413.
  • [33] V. V. Nikulin, Finite groups of automorphisms of Kählerian K3 surfaces (in Russian), Trudy Moskov. Mat. Obshch. 38 (1979), 75–137;
  • [34] V. V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 43, no. 1 (1979), 111–177, 238; English translation in Math. USSR-Izv. 14 (1980), no. 1, 103–167.
  • [35] V. V. Nikulin, Quotient-groups of groups of automorphisms of hyperbolic forms of subgroups generated by $2$-reflections (in Russian), Dokl. Akad. Nauk SSSR 248, no. 6 (1979), 1307–1309; English translation in Soviet Math. Dokl. 20 (1979), no. 5, 1156–1158.
  • [36] V. V. Nikulin, “Discrete reflection groups in Lobachevsky spaces and algebraic surfaces” in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, 1987, 654–671.
  • [37] F. Schaffhauser, Real points of coarse moduli schemes of vector bundles on a real algebraic curve, J. Symplectic Geom. 10 (2012), no. 4, 503–534.
  • [38] B. van Geemen and A. Sarti, Nikulin involutions on K3 surfaces, Math. Z. 255 (2007), no. 4, 731–753.