Toric structures on bundles of projective spaces

DOI10.4310/JSG.2014.V12.N4.A3zbMATH Open1319.53088arXiv1202.3422OpenAlexW2122161761MaRDI QIDQ2347212

Andrew Fanoe

Publication date: 27 May 2015

Published in: The Journal of Symplectic Geometry (Search for Journal in Brave)

Abstract: Recently, extending work by Karshon, Kessler and Pinsonnault, Borisov and McDuff showed that a given symplectic manifold $(M,omega)$ has a finite number of distinct toric structures. Moreover, McDuff also showed a product of two projective spaces $$ with any given symplectic form has a unique toric structure provided that $r,sgeq2$. In contrast, the product $$ can be given infinitely many distinct toric structures, though only a finite number of these are compatible with each given symplectic form $omega$. In this paper we extend these results by considering the possible toric structures on a toric symplectic manifold $(M,omega)$ with $dim{H}^2(M)=2$. In particular, all such manifolds are $$ bundles over $$ for some $r,s$. We show that there is a unique toric structure if $r<s$, and also that if $r,sgeq2$ then $M$ has at most finitely many distinct toric structures that are compatible with any symplectic structure on $M$. Thus, in this case the finiteness result does not depend on fixing the symplectic structure. We will also give other examples where $(M,omega)$ has a unique toric structure, such as the case where $(M,omega)$ is monotone.

Full work available at URL: https://arxiv.org/abs/1202.3422

zbMATH Keywords

toric varietiespolytopessymplectic manifolds

Mathematics Subject Classification ID

Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) (52B20) Toric varieties, Newton polyhedra, Okounkov bodies (14M25) Symplectic manifolds (general theory) (53D05)

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