Algebraic versus symplectic… toric manifolds

As the second and probably final (for the moment at least) comparative post between algebraic and symplectic geometry, I thought I’d compare their respective toric geometries and related combinatorics. ‘Symplectic’ as terminology was proposed by Weyl to reflect the similarities with complex geometry by replacing the Latin word ‘complex’ with its Greek equivalent. As this anecdote suggests, there is much shared between the two!

In the toric world, there are at least three ways to construct a toric manifold from some combinatorial data; for instance a polytope or a fan. One of them is by using lattice points (= sections for an algebraic geometer) to embed the relevant torus into some big projective space and take the closure. If you like, this is just applying Proj to the toric ample divisor corresponding to the polytope. Given a Delzant polytope \Delta with lattice points as vertices – notice that this can sometimes (always?) be achieved by suitable scaling of the symplectic form – one can perform this process to obtain a toric variety, which will be a smooth subvariety of some projective space by the Delzant condition. In particular, it inherits a symplectic form from the restriction of the Fubini-Study form as a complex submanifold of a projective space.

The torus action it also carries turns out to be Hamiltonian with respect to this symplectic structure as witnessed by its moment map, which has moment polytope… \Delta. That is, by Delzant’s theorem (which says that there’s a unique way to reconstruct a symplectic toric manifold from a polytope) the algebraic and symplectic constructions of a toric manifold from a lattice Delzant polytope agree.

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