Skip to main content
Top

2015 | OriginalPaper | Chapter

Reflexive Polytopes and Lattice-Polarized K3 Surfaces

Author : Ursula Whitcher

Published in: Calabi-Yau Varieties: Arithmetic, Geometry and Physics

Publisher: Springer New York

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

We review the standard formulation of mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, and compare this construction to a description of mirror symmetry for K3 surfaces which relies on a sublattice of the Picard lattice. We then show how to combine information about the Picard group of a toric ambient space with data about automorphisms of the toric variety to identify families of K3 surfaces with high Picard rank.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Literature
1.
go back to reference Batyrev, V.V.: Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties. J. Algebr. Geom. 3(3), 493–535 (1994)MathSciNetMATH Batyrev, V.V.: Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties. J. Algebr. Geom. 3(3), 493–535 (1994)MathSciNetMATH
2.
go back to reference Cox, D., Katz, S.: Mirror Symmetry and Algebraic Geometry. American Mathematical Society, Providence (1999)CrossRefMATH Cox, D., Katz, S.: Mirror Symmetry and Algebraic Geometry. American Mathematical Society, Providence (1999)CrossRefMATH
3.
go back to reference Cox, D.A., Little, J.B., Schenck, H.K.: Toric Varieties. Volume 124 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2011) Cox, D.A., Little, J.B., Schenck, H.K.: Toric Varieties. Volume 124 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2011)
4.
go back to reference Dolgachev, I.V.: Mirror symmetry for lattice polarized K3 surfaces. J. Math. Sci. 81(3), 2599–2630 (1996). Algebraic Geometry, 4 Dolgachev, I.V.: Mirror symmetry for lattice polarized K3 surfaces. J. Math. Sci. 81(3), 2599–2630 (1996). Algebraic Geometry, 4
6.
go back to reference Hosono, S., Lian, B.H., Oguiso, K., Yau, S.-T.: Autoequivalences of derived category of a K3 surface and monodromy transformations. J. Algebr. Geom. 13(3), 513–545 (2004)MathSciNetCrossRefMATH Hosono, S., Lian, B.H., Oguiso, K., Yau, S.-T.: Autoequivalences of derived category of a K3 surface and monodromy transformations. J. Algebr. Geom. 13(3), 513–545 (2004)MathSciNetCrossRefMATH
7.
go back to reference Karp, D., Lewis, J., Moore, D., Skjorshammer, D., Whitcher, U.: On a family of K3 surfaces with \(\mathcal{S}_{4}\) symmetry. In: Arithmetic and Geometry of K3 Surfaces and Calabi-Yau Threefolds. Springer, New York (2013) Karp, D., Lewis, J., Moore, D., Skjorshammer, D., Whitcher, U.: On a family of K3 surfaces with \(\mathcal{S}_{4}\) symmetry. In: Arithmetic and Geometry of K3 Surfaces and Calabi-Yau Threefolds. Springer, New York (2013)
9.
go back to reference Mavlyutov, A.: Semiample hypersurfaces in toric varieties. (2000). arXiv:math.AG/9812163 v2 Mavlyutov, A.: Semiample hypersurfaces in toric varieties. (2000). arXiv:math.AG/9812163 v2
11.
go back to reference Narumiya, N., Shiga, H.: The mirror map for a family of K3 surfaces induced from the simplest 3-dimensional reflexive polytope. In: Proceedings on Moonshine and Related Topics, 1999, Montréal. Volume 30 of CRM Proceedings Lecture Notes, pp. 139–161. American Mathematical Society, Providence (2001) Narumiya, N., Shiga, H.: The mirror map for a family of K3 surfaces induced from the simplest 3-dimensional reflexive polytope. In: Proceedings on Moonshine and Related Topics, 1999, Montréal. Volume 30 of CRM Proceedings Lecture Notes, pp. 139–161. American Mathematical Society, Providence (2001)
12.
go back to reference Nikulin, V.: Finite automorphism groups of Kähler K3 surfaces. Trans. Mosc. Math. Soc. (38), 71–135 (1980) Nikulin, V.: Finite automorphism groups of Kähler K3 surfaces. Trans. Mosc. Math. Soc. (38), 71–135 (1980)
13.
go back to reference Oguiso, K.: Picard numbers in a family of hyperkähler manifolds – a supplement to the article of R. Borcherds, L. Katzarkov, T. Pantev, N. I. Shepherd-Barron. arXiv.org:math/0011258 (2000) Oguiso, K.: Picard numbers in a family of hyperkähler manifolds – a supplement to the article of R. Borcherds, L. Katzarkov, T. Pantev, N. I. Shepherd-Barron. arXiv.org:math/0011258 (2000)
14.
go back to reference Peters, C., Stienstra, J.: A pencil of K3-surfaces related to Apéry’s recurrence for ζ(3) and Fermi surfaces for potential zero. In: Arithmetic of Complex Manifolds, 1988, Erlangen. Volume 1399 of Lecture Notes in Mathematics, pp. 110–127. Springer, Berlin (1989) Peters, C., Stienstra, J.: A pencil of K3-surfaces related to Apéry’s recurrence for ζ(3) and Fermi surfaces for potential zero. In: Arithmetic of Complex Manifolds, 1988, Erlangen. Volume 1399 of Lecture Notes in Mathematics, pp. 110–127. Springer, Berlin (1989)
15.
go back to reference Rohsiepe, F.: Lattice polarized toric K3 surfaces. (2004). arXiv:hep-th/0409290 v1 Rohsiepe, F.: Lattice polarized toric K3 surfaces. (2004). arXiv:hep-th/0409290 v1
17.
go back to reference Verrill, H.A.: Root lattices and pencils of varieties. J. Math. Kyoto Univ. 36(2), 423–446 (1996)MathSciNet Verrill, H.A.: Root lattices and pencils of varieties. J. Math. Kyoto Univ. 36(2), 423–446 (1996)MathSciNet
Metadata
Title
Reflexive Polytopes and Lattice-Polarized K3 Surfaces
Author
Ursula Whitcher
Copyright Year
2015
Publisher
Springer New York
DOI
https://doi.org/10.1007/978-1-4939-2830-9_3

Premium Partner