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.
Citation:
N. A. Tyurin, “Pseudotoric Lagrangian fibrations of toric and nontoric Fano varieties”, TMF, 162:3 (2010), 307–333; Theoret. and Math. Phys., 162:3 (2010), 255–275
This publication is cited in the following 10 articles:
Chan K., Leung N.C., Li Ch., “Pseudotoric Structures and Special Lagrangian Torus Fibrations on Certain Flag Varieties”, J. Geom. Phys., 146 (2019), UNSP 103489
N. A. Tyurin, “Pseudotoric structures: Lagrangian submanifolds and Lagrangian fibrations”, Russian Math. Surveys, 72:3 (2017), 513–546
N. A. Tyurin, “Pseudotoric structures on a hyperplane section of a toric manifold”, Theoret. and Math. Phys., 182:2 (2015), 159–172
N. A. Tyurin, “Pseudotoric Structures and Lagrangian Spheres in the Flag Variety F3”, Math. Notes, 96:3 (2014), 458–461
S. A. Belyov, N. A. Tyurin, “Pseudotoric structures on toric symplectic manifolds”, Theoret. and Math. Phys., 175:2 (2013), 571–579
Tyurin N.A., “Exotic Chekanov Tori in Toric Symplectic Varieties”, Xxth International Conference on Integrable Systems and Quantum Symmetries (Isqs-20), Journal of Physics Conference Series, 411, eds. Burdik C., Navratil O., Posta S., IOP Publishing Ltd, 2013, 012028
S. A. Belev, N. A. Tyurin, “Lifts of Lagrangian Tori”, Math. Notes, 91:5 (2012), 735–737
N. A. Tyurin, “Nonstandard Lagrangian tori and pseudotoric structures”, Theoret. and Math. Phys., 171:2 (2012), 700–703
N. A. Tyurin, “Chekanov tori and pseudotoric structures”, Russian Math. Surveys, 66:1 (2011), 181–182
N. A. Tyurin, “Special Lagrangian fibrations on the flag variety F3”, Theoret. and Math. Phys., 167:2 (2011), 567–576