Abstract
We use local Hamiltonian torus actions to degenerate a symplectic manifold to a normal crossings symplectic variety in a smooth one-parameter family. This construction, motivated in part by the Gross–Siebert and B. Parker’s programs, contains a multifold version of the usual (two-fold) symplectic cut construction and in particular splits a symplectic manifold into several symplectic manifolds containing normal crossings symplectic divisors with shared irreducible components in one step.
Similar content being viewed by others
References
Abramovich, D., Chen, Q.: Stable logarithmic maps to Deligne–Faltings pairs II. Asian J. Math. 18(3), 465–488 (2014)
Audin, M.: Torus Actions on Symplectic Manifolds, 2nd edn. Progress in Mathematics, vol. 93. Birkhäuser Verlag, Basel (2004)
Canas da Silva, A.: Lectures on Symplectic Geometry. Lecture Notes in Mathematics, vol. 1764. Springer, Berlin (2001) (revised 2006)
Chen, Q.: Stable logarithmic maps to Deligne–Faltings pairs I. Ann. Math. 180(2), 455–521 (2014)
Farajzadeh Tehrani, M., McLean, M., Zinger, A.: The smoothability of normal crossings symplectic varieties (2014). arXiv:1410.2573v2
Farajzadeh Tehrani, M., McLean, M., Zinger, A.: Normal crossings singularities for symplectic topology. Adv. Math. 339, 672–748 (2018)
Farajzadeh Tehrani, M., Zinger, A.: On the multifold symplectic sum and cut constructions (in preparation)
Gompf, R.: A new construction of symplectic manifolds. Ann. Math. 142(3), 527–595 (1995)
Gross, M., Siebert, B.: Affine manifolds, log structures, and mirror symmetry. Turk. J. Math. 27(1), 33–60 (2003)
Gross, M., Siebert, B.: Logarithmic Gromov–Witten invariants. J. Am. Math. Soc. 26(2), 451–510 (2013)
Lerman, E.: Symplectic cuts. Math. Res. Lett. 2(3), 247–258 (1995)
Marsden, J., Weinstein, A.: Reduction of symplectic manifolds with symmetry. Rep. Math. Phys. 5(1), 121–130 (1974)
Meyer, K.: Symmetries and integrals in mechanics. In: Dynamical Systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), pp. 259–272. Academic Press, New York (1973)
Milnor, J.: Topology from the Differentiable Viewpoint. Princeton Landmarks in Mathematics. Princeton University Press, Princeton (1997)
Munkres, J.: Topology, 2nd edn. Pearson (2000)
Parker, B.: Exploded fibrations. In: Proceedings Gökova Geometry-Topology Conference 2006, pp. 52–90. Gökova Geometry/Topology Conference (GGT), Gökova (2007)
Parker, B.: Gromov–Witten invariants of exploded manifolds (2011). arXiv:1102.0158
Popa, A.: Two-point Gromov–Witten formulas for symplectic toric manifolds (2012). arXiv:1206.2703
Warner, F.: Foundations of Differentiable Manifolds and Lie Groups. Graduate Texts in Mathematics, vol. 94. Springer, New York (1983)
Author information
Authors and Affiliations
Corresponding author
Additional information
A. Zinger was partially supported by NSF Grant 1500875 and MPIM.
Rights and permissions
About this article
Cite this article
Farajzadeh Tehrani, M., Zinger, A. Normal Crossings Degenerations of Symplectic Manifolds. Peking Math J 2, 275–351 (2019). https://doi.org/10.1007/s42543-019-00017-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s42543-019-00017-y