Exotic Deformations of Calabi-Yau Manifolds
Annales de l'Institut Fourier, Volume 63 (2013) no. 2, pp. 391-415.

We introduce Quantum Inner State manifolds (QIS manifolds) as (compact) 2n-dimensional symplectic manifolds (M,κ) endowed with a κ-tamed almost complex structure J and with a nowhere vanishing and normalized section ϵ of the bundle Λ J n,0 (M) satisfying the condition ¯ J ϵ=0.

We study the moduli space 𝔐 of QIS deformations of a given Calabi-Yau manifold, computing its tangent space and showing that 𝔐 is non obstructed. Finally, we present several examples of QIS manifolds.

On considère la classe des variétés QIS (Quantum Inner State variétés), à savoir la classe des variétés symplectiques, compactes et de dimension 2n, munies d’une structure presque complexe J modérée par k et d’une section ϵ du fibré Λ J n,0 (M), qui ne s’annule nulle part, normalisée et satisfaisant la condition ¯ J ϵ=0.

Le but du papier est d’étudier l’espace 𝔐 des modules des déformations QIS d’une variété de Calabi-Yau. À ce propos, on calcule l’espace tangent de 𝔐 et on montre que 𝔐 n’a pas d’obstructions. Plusieurs exemples de variétés QIS sont aussi exhibés.

DOI: 10.5802/aif.2764
Classification: 32G05, 53C15, 17B30
Keywords: tamed symplectic structure, Calabi-Yau manifold, quantum inner state structure, deformation, moduli space
Mot clés : variétés de Calabi-Yau
de Bartolomeis, Paolo 1; Tomassini, Adriano 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)
     author = {de Bartolomeis, Paolo and Tomassini, Adriano},
     title = {Exotic {Deformations} of {Calabi-Yau} {Manifolds}},
     journal = {Annales de l'Institut Fourier},
     pages = {391--415},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {63},
     number = {2},
     year = {2013},
     doi = {10.5802/aif.2764},
     zbl = {1293.32016},
     mrnumber = {3112516},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2764/}
AU  - de Bartolomeis, Paolo
AU  - Tomassini, Adriano
TI  - Exotic Deformations of Calabi-Yau Manifolds
JO  - Annales de l'Institut Fourier
PY  - 2013
SP  - 391
EP  - 415
VL  - 63
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2764/
DO  - 10.5802/aif.2764
LA  - en
ID  - AIF_2013__63_2_391_0
ER  - 
%0 Journal Article
%A de Bartolomeis, Paolo
%A Tomassini, Adriano
%T Exotic Deformations of Calabi-Yau Manifolds
%J Annales de l'Institut Fourier
%D 2013
%P 391-415
%V 63
%N 2
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2764/
%R 10.5802/aif.2764
%G en
%F AIF_2013__63_2_391_0
de Bartolomeis, Paolo; Tomassini, Adriano. Exotic Deformations of Calabi-Yau Manifolds. Annales de l'Institut Fourier, Volume 63 (2013) no. 2, pp. 391-415. doi : 10.5802/aif.2764. http://www.numdam.org/articles/10.5802/aif.2764/

[1] Auslander, L.; Green, L.; Hahn, F. Flows on homogeneous spaces, With the assistance of L. Markus and W. Massey, and an appendix by L. Greenberg. Annals of Mathematics Studies, No. 53, Princeton University Press, Princeton, N.J., 1963 | Zbl

[2] de Bartolomeis, Paolo Some constructions with Symplectic Manifolds and Lagrangian Submanifolds (preprint)

[3] de Bartolomeis, Paolo; Tomassini, Adriano On formality of some symplectic manifolds, Internat. Math. Res. Notices (2001) no. 24, pp. 1287-1314 | DOI | MR | Zbl

[4] de Bartolomeis, Paolo; Tomassini, Adriano On the Maslov index of Lagrangian submanifolds of generalized Calabi-Yau manifolds, Internat. J. Math., Volume 17 (2006) no. 8, pp. 921-947 | DOI | MR | Zbl

[5] Cheeger, Jeff; Gromoll, Detlef On the structure of complete manifolds of nonnegative curvature, Ann. of Math. (2), Volume 96 (1972), pp. 413-443 | DOI | MR | Zbl

[6] Deligne, Pierre; Griffiths, Phillip; Morgan, John; Sullivan, Dennis Real homotopy theory of Kähler manifolds, Invent. Math., Volume 29 (1975) no. 3, pp. 245-274 | DOI | MR | Zbl

[7] Fernández, Marisa; Gray, Alfred Compact symplectic solvmanifolds not admitting complex structures, Geom. Dedicata, Volume 34 (1990) no. 3, pp. 295-299 | DOI | MR | Zbl

[8] Fujiki, Akira; Schumacher, Georg The moduli space of Kähler structures on a real compact symplectic manifold, Publ. Res. Inst. Math. Sci., Volume 24 (1988) no. 1, pp. 141-168 | DOI | MR | Zbl

[9] Hasegawa, Keizo Complex and Kähler structures on compact solvmanifolds, J. Symplectic Geom., Volume 3 (2005) no. 4, pp. 749-767 http://projecteuclid.org/getRecord?id=euclid.jsg/1154467635 (Conference on Symplectic Topology) | DOI | MR | Zbl

[10] Hasegawa, Keizo A note on compact solvmanifolds with Kähler structures, Osaka J. Math., Volume 43 (2006) no. 1, pp. 131-135 http://projecteuclid.org/getRecord?id=euclid.ojm/1146242998 | MR | Zbl

[11] Hattori, Akio Spectral sequence in the de Rham cohomology of fibre bundles, J. Fac. Sci. Univ. Tokyo Sect. I, Volume 8 (1960), p. 289-331 (1960) | MR | Zbl

[12] Hitchin, Nigel Generalized Calabi-Yau manifolds, Q. J. Math., Volume 54 (2003) no. 3, pp. 281-308 | DOI | MR | Zbl

[13] Lafontaine, Jacques; Audin, Michèle Introduction: applications of pseudo-holomorphic curves to symplectic topology, Holomorphic curves in symplectic geometry (Progr. Math.), Volume 117, Birkhäuser, Basel, 1994, pp. 1-14 | MR

[14] Malcev, A. I. On a class of homogeneous spaces, Amer. Math. Soc. Translation, Volume 1951 (1951) no. 39, pp. 33 | MR

[15] Nakamura, Iku Complex parallelisable manifolds and their small deformations, J. Differential Geometry, Volume 10 (1975), pp. 85-112 | MR | Zbl

[16] Tian, Gang 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) (Adv. Ser. Math. Phys.), Volume 1, World Sci. Publishing, Singapore, 1987, pp. 629-646 | MR | Zbl

[17] Todorov, Andrey N. The Weil-Petersson geometry of the moduli space of SU (n3) (Calabi-Yau) manifolds. I, Comm. Math. Phys., Volume 126 (1989) no. 2, pp. 325-346 http://projecteuclid.org/getRecord?id=euclid.cmp/1104179854 | DOI | MR | Zbl

[18] Yan, Dong Hodge structure on symplectic manifolds, Adv. Math., Volume 120 (1996) no. 1, pp. 143-154 | DOI | MR | Zbl

[19] Yau, Shing Tung On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math., Volume 31 (1978) no. 3, pp. 339-411 | DOI | MR | Zbl

Cited by Sources: