Abstract
We introduce the notion of a pseudotoric structure on a symplectic manifold, generalizing the notion of a toric structure. We show that such a pseudotoric structure can exist on toric and nontoric symplectic manifolds. For the toric manifolds, it describes deformations of the standard toric Lagrangian fibrations; for the nontoric ones, it gives Lagrangian fibrations with singularities that are very close to the toric fibrations. We present examples of toric manifolds with different pseudotoric structures and prove that certain nontoric manifolds (smooth complex quadrics) have such structures. In the future, introducing this new structure can be useful for generalizing the geometric quantization and mirror symmetry methods that work well in the toric case to a broader class of Fano varieties.
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References
M. Audin, Torus Actions on Symplectic Manifolds (Progr. Math., Vol. 93), Birkhäuser, Basel (2004).
N. N. Nekhoroshev, Funct. Anal. Appl., 28, No. 2, 128–129 (1994).
S. A. Belyov and N. A. Tyurin, “On non toric fibrations on lagrangian tori of toric Fano varieties,” Preprint No. MPIM-2009-14, Max Planck Inst. Math., Bonn (2009).
T. Delzant, Bull. Soc. Math. France, 116, 315–339 (1988).
N. A. Tyurin, “Geometric quantization and algebraic Lagrangian geometry,” in: Surveys in Geometry and Number Theory: Reports on Contemporary Russian Mathematics (London Math. Soc. Lect. Note Ser., Vol. 338, N. Young, ed.), Cambridge Univ. Press, Cambridge (2007), pp. 279–318.
A. Ashtekar and T. Schilling, “Geometric formulation of quantum mechanics,” in: On Einstein's Path (A. Harvey, ed.), Springer, New York (1999), pp. 23–65.
N. A. Tyurin, Theor. Math. Phys., 158, 1–16 (2009).
D. Auroux, J. Gökova Geom. Topol., 1, 51–91 (2007).
S. Belyov, “Nonlinear quantum systems,” Master's thesis, Joint Inst. Nucl.Res., Dubna (2009).
S. Belyov, “Pseudotoric fibrations of nontoric del Pezzo surfaces,” (in preparation).
N. A. Tyurin, Theor. Math. Phys., 150, 278–287 (2007).
N. Hitchin, “Lectures on special Lagrangian submanifolds,” in: Winter School on Mirror Symmetry, Vector Bundles, and Lagrangian Submanifolds (AMS/IP Stud. Adv. Math., Vol. 23, C. Vafa and S.-T. Yau, eds.), Amer. Math. Soc., Providence, R. I. (2001), pp. 151–182.
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 162, No. 3, pp. 307–333, March, 2010.
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Tyurin, N.A. Pseudotoric Lagrangian fibrations of toric and nontoric fano varieties. Theor Math Phys 162, 255–275 (2010). https://doi.org/10.1007/s11232-010-0021-7
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DOI: https://doi.org/10.1007/s11232-010-0021-7