Minimal models of compact symplectic semitoric manifolds

JOURNAL OF GEOMETRY AND PHYSICS

卷:125

Article

Kane, D. M.¹  Palmer, J.²  Pelayo, A.¹ 

[1] Univ Calif San Diego, Dept Math, 9500 Gilman Dr 0112, La Jolla, CA 92093 USA

[2] Rutgers State Univ, Dept Math, Hill Ctr Busch Campus,110 Frelinghuysen Rd, Piscataway, NJ 08854 USA

关键词: Symplectic geometry;    Integrable systems;    SL2(Z);    Symplectic toric manifolds;    Fans;    Matrix calculus;   

DOI  :  10.1016/j.geomphys.2017.12.005

来源: Elsevier

【 摘 要 】

A symplectic semitoric manifold is a symplectic 4-manifold endowed with a Hamiltonian (S-1 x R)-action satisfying certain conditions. The goal of this paper is to construct a new symplectic invariant of symplectic semitoric manifolds, the helix, and give applications. The helix is a symplectic analogue of the fan of a nonsingular complete toric variety in algebraic geometry, that takes into account the effects of the monodromy near focus focus singularities. We give two applications of the helix: first, we use it to give a classification of the minimal models of symplectic semitoric manifolds, where minimal is in the sense of not admitting any blowdowns. The second application is an extension to the compact case of a well known result of V (u) over tilde Ngoc about the constraints posed on a symplectic semitoric manifold by the existence of focus-focus singularities. The helix permits to translate a symplectic geometric problem into an algebraic problem, and the paper describes a method to solve this type of algebraic problem. (C) 2018 Elsevier B.V. All rights reserved.

【 授权许可】

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