Geodesic
Classifying Toric and Semitoric Fans by Lifting Equations from $\mathrm{SL}_2({\mathbb Z})$
Symmetry, integrability and geometry: methods and applications, Tome 14 (2018) Cet article a éte moissonné depuis la source Math-Net.Ru

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We present an algebraic method to study four-dimensional toric varieties by lifting matrix equations from the special linear group $\mathrm{SL}_2(\mathbb{Z})$ to its preimage in the universal cover of $\mathrm{SL}_2(\mathbb{R})$. With this method we recover the classification of two-dimensional toric fans, and obtain a description of their semitoric analogue. As an application to symplectic geometry of Hamiltonian systems, we give a concise proof of the connectivity of the moduli space of toric integrable systems in dimension four, recovering a known result, and extend it to the case of semitoric integrable systems with a fixed number of focus-focus points and which are in the same twisting index class. In particular, we show that any semitoric system with precisely one focus-focus singular point can be continuously deformed into a system in the same isomorphism class as the Jaynes–Cummings model from optics.
Keywords: symplectic geometry; integrable system; semitoric integrable systems; toric integrable systems; focus-focus singularities; $\mathrm{SL}_2(\mathbb{Z})$. 
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     author = {Daniel M. Kane and Joseph Palmer and \'Alvaro Pelayo},
     title = {Classifying {Toric} and {Semitoric} {Fans} by {Lifting} {Equations} from $\mathrm{SL}_2({\mathbb Z})$},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2018},
     volume = {14},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a15/}
}
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Daniel M. Kane; Joseph Palmer; Álvaro Pelayo. Classifying Toric and Semitoric Fans by Lifting Equations from $\mathrm{SL}_2({\mathbb Z})$. Symmetry, integrability and geometry: methods and applications, Tome 14 (2018). http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a15/

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