toric geometry

In mathematics and theoretical physics, toric geometry is a set of methods in algebraic geometry in which complex manifolds are visualized as fiber bundles with multi-dimensional tori as fibers. For example, the complex projective plane CP² may be represented by three complex coordinates satisfying

|z_1|^2+|z_2|^2+|z_3|^2 = 1

where the sum has been chosen to account for the real rescaling part of the projective map, and the coordinates must be moreover identified by the following

U(1)

action:

(z_1,z_2,z_3)\approx e^{i\phi} (z_1,z_2,z_3)

The approach of toric geometry is to write

(x,y,z) = (|z_1|^2,|z_2|^2,|z_3|^2)

The coordinates

x,y,z

are non-negative, and they parameterize a triangle because

x+y+z=1\quad

i.e.

\quad z=1-x-y

The triangle is the toric base of the complex projective plane. The generic fiber is a two-torus parameterized by the phases of

z_1,z_2

; the phase of

z_3

can be chosen real and positive by the

U(1)

symmetry. However, the two-torus degenerates into three different circles on the boundary of the triangle i.e. at

x=0

y=0

z=0

because the phase of

z_1,z_2,z_3

becomes inconsequential, respectively. The precise orientation of the circles within the torus is usually depicted by the slope of the line intervals (the sides of the triangle, in this case). Many more complicated complex manifolds, for example del Pezzo surfaces, admit a toric description.

Abstract formulation

The origins of toric geometry were in particular compactification questions; but it was soon formulated as the geometry theory of algebraic varieties V defined by monomial sets of equations. The geometric equivalent to that is to have an action on V of an algebraic torus, with an open orbit. This is the theory of toric varieties or torus embeddings. Computationally they can be treated by means of the semigroup defined by the exponents in the monomials, making them particularly tractable. A toroidal embedding is a variety that is locally isomorphic to a toric variety. Here, locally is in the sense of differential geometry, not with respect to the Zariski topology. Toric geometry can also be used in relation with invariant theory (particularly geometric invariant theory), roughly in the way maximal torus theory is applied to Lie groups, but relating to moduli spaces rather than representation theory.

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