Exotic Deformations of Calabi-Yau Manifolds

Paolo de Bartolomeis[1]; Adriano Tomassini[2]

  • [1] Università di Firenze Dipartimento di Matematica e Informatica “Ulisse Dini” Viale Morgagni 67/a 50134 Firenze (Italy)
  • [2] Università di Parma Dipartimento di Matematica e Informatica Parco Area delle Scienze 53/A 43124 Parma (Italy)

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 2, page 391-415
  • ISSN: 0373-0956

Abstract

top
We introduce Quantum Inner State manifolds (QIS manifolds) as (compact) -dimensional symplectic manifolds endowed with a -tamed almost complex structure and with a nowhere vanishing and normalized section of the bundle satisfying the condition .We study the moduli space of QIS deformations of a given Calabi-Yau manifold, computing its tangent space and showing that is non obstructed. Finally, we present several examples of QIS manifolds.

How to cite

top

de Bartolomeis, Paolo, and Tomassini, Adriano. "Exotic Deformations of Calabi-Yau Manifolds." Annales de l’institut Fourier 63.2 (2013): 391-415. <http://eudml.org/doc/275436>.

@article{deBartolomeis2013,
abstract = {We introduce Quantum Inner State manifolds (QIS manifolds) as (compact) $2n$-dimensional symplectic manifolds $(M,\kappa )$ endowed with a $\kappa $-tamed almost complex structure $J$ and with a nowhere vanishing and normalized section $\epsilon $ of the bundle $\Lambda ^\{n,0\}_J(M)$ satisfying the condition $\overline\{\partial \}_J\epsilon =0$.We study the moduli space $\mathfrak\{M\}$ of QIS deformations of a given Calabi-Yau manifold, computing its tangent space and showing that $\mathfrak\{M\}$ is non obstructed. Finally, we present several examples of QIS manifolds.},
affiliation = {Università di Firenze Dipartimento di Matematica e Informatica “Ulisse Dini” Viale Morgagni 67/a 50134 Firenze (Italy); Università di Parma Dipartimento di Matematica e Informatica Parco Area delle Scienze 53/A 43124 Parma (Italy)},
author = {de Bartolomeis, Paolo, Tomassini, Adriano},
journal = {Annales de l’institut Fourier},
keywords = {tamed symplectic structure; Calabi-Yau manifold; quantum inner state structure; deformation; moduli space; Calabi Yau manifold},
language = {eng},
number = {2},
pages = {391-415},
publisher = {Association des Annales de l’institut Fourier},
title = {Exotic Deformations of Calabi-Yau Manifolds},
url = {http://eudml.org/doc/275436},
volume = {63},
year = {2013},
}

TY - JOUR
AU - de Bartolomeis, Paolo
AU - Tomassini, Adriano
TI - Exotic Deformations of Calabi-Yau Manifolds
JO - Annales de l’institut Fourier
PY - 2013
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 2
SP - 391
EP - 415
AB - We introduce Quantum Inner State manifolds (QIS manifolds) as (compact) $2n$-dimensional symplectic manifolds $(M,\kappa )$ endowed with a $\kappa $-tamed almost complex structure $J$ and with a nowhere vanishing and normalized section $\epsilon $ of the bundle $\Lambda ^{n,0}_J(M)$ satisfying the condition $\overline{\partial }_J\epsilon =0$.We study the moduli space $\mathfrak{M}$ of QIS deformations of a given Calabi-Yau manifold, computing its tangent space and showing that $\mathfrak{M}$ is non obstructed. Finally, we present several examples of QIS manifolds.
LA - eng
KW - tamed symplectic structure; Calabi-Yau manifold; quantum inner state structure; deformation; moduli space; Calabi Yau manifold
UR - http://eudml.org/doc/275436
ER -

References

top
  1. L. Auslander, L. Green, F. Hahn, Flows on homogeneous spaces, (1963), Princeton University Press, Princeton, N.J. Zbl0106.36802
  2. Paolo de Bartolomeis, Some constructions with Symplectic Manifolds and Lagrangian Submanifolds Zbl1220.53064
  3. Paolo de Bartolomeis, Adriano Tomassini, On formality of some symplectic manifolds, Internat. Math. Res. Notices (2001), 1287-1314 Zbl1004.53068MR1866746
  4. Paolo de Bartolomeis, Adriano Tomassini, On the Maslov index of Lagrangian submanifolds of generalized Calabi-Yau manifolds, Internat. J. Math. 17 (2006), 921-947 Zbl1115.53053MR2261641
  5. Jeff Cheeger, Detlef Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. (2) 96 (1972), 413-443 Zbl0246.53049MR309010
  6. Pierre Deligne, Phillip Griffiths, John Morgan, Dennis Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), 245-274 Zbl0312.55011MR382702
  7. Marisa Fernández, Alfred Gray, Compact symplectic solvmanifolds not admitting complex structures, Geom. Dedicata 34 (1990), 295-299 Zbl0703.53030MR1066580
  8. Akira Fujiki, Georg Schumacher, The moduli space of Kähler structures on a real compact symplectic manifold, Publ. Res. Inst. Math. Sci. 24 (1988), 141-168 Zbl0655.32020MR944870
  9. Keizo Hasegawa, Complex and Kähler structures on compact solvmanifolds, J. Symplectic Geom. 3 (2005), 749-767 Zbl1120.53043MR2235860
  10. Keizo Hasegawa, A note on compact solvmanifolds with Kähler structures, Osaka J. Math. 43 (2006), 131-135 Zbl1105.32017MR2222405
  11. Akio Hattori, Spectral sequence in the de Rham cohomology of fibre bundles, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1960), 289-331 (1960) Zbl0099.18003MR124918
  12. Nigel Hitchin, Generalized Calabi-Yau manifolds, Q. J. Math. 54 (2003), 281-308 Zbl1076.32019MR2013140
  13. Jacques Lafontaine, Michèle Audin, Introduction: applications of pseudo-holomorphic curves to symplectic topology, Holomorphic curves in symplectic geometry 117 (1994), 1-14, Birkhäuser, Basel MR1274924
  14. A. I. Malcev, On a class of homogeneous spaces, Amer. Math. Soc. Translation 1951 (1951) MR39734
  15. Iku Nakamura, Complex parallelisable manifolds and their small deformations, J. Differential Geometry 10 (1975), 85-112 Zbl0297.32019MR393580
  16. Gang Tian, Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric, Mathematical aspects of string theory (San Diego, Calif., 1986) 1 (1987), 629-646, World Sci. Publishing, Singapore Zbl0696.53040MR915841
  17. Andrey N. Todorov, The Weil-Petersson geometry of the moduli space of (Calabi-Yau) manifolds. I, Comm. Math. Phys. 126 (1989), 325-346 Zbl0688.53030MR1027500
  18. Dong Yan, Hodge structure on symplectic manifolds, Adv. Math. 120 (1996), 143-154 Zbl0872.58002MR1392276
  19. Shing Tung Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math. 31 (1978), 339-411 Zbl0369.53059MR480350

NotesEmbed ?

top

You must be logged in to post comments.