Delzant's theorem

In differential geometry, a field of mathematics, Delzant's theorem provides a complete classification of symplectic toric manifolds in term of a special class of polytopes. It was proved by Thomas Delzant in 1988.

Preliminaries

A symplectic toric manifold is a compact connected symplectic manifold $(M^{2n},\omega )$ together with an effective Hamiltonian action of the $n$-dimensional torus $\mathbb {T} ^{n}=\mathbb {S} ^{1}\times \ldots \times \mathbb {S} ^{1}$ on $M$, with moment map denoted by $\mu :M\to \mathbb {R} ^{n}$.

The convexity theorem, proved independently in 1982 by Atiyah and by Guillemin and Sternberg using Morse theory, states that the image $\mu (M)\subset \mathbb {R} ^{n}$ is the convex hull of the images of the fixed points of the $\mathbb {T} ^{n}$-action. It is therefore a convex polytope, also called the moment polytope.

A Delzant polytope is a special kind of convex polytope in $\mathbb {R} ^{n}$, satisfying the following three properties

• it is a simple polytope, i.e. for each vertex $x$, exactly $n$ edges meet at $v$;
• it is a rational polytope, i.e. for each vertex $x$, the edges at $x$ are of the form $x+tu_{i}^{x}$, with $t\geq 0$ and $u_{i}^{x}\in \mathbb {Q}$;
• it is smooth polytope, i.e. for each vertex $x$, the corresponding vectors $u_{i}^{x}$ form a $\mathbb {Z}$-basis of $\mathbb {Z} ^{n}$.

Statement

Delzant's theorem states that there is a bijective correspondence between $2n$-dimensional symplectic toric manifolds (up to torus-equivariant symplectomorphism) and Delzant polytopes in $\mathbb {R} ^{n}$:$$\{{\text{Toric manifolds}}\}\leftrightarrow \{{\text{Delzant polytopes}}\},\quad (M^{2n},\omega ,\mathbb {T} ^{n},\mu )\mapsto \mu (M).$$More precisely, the moment polytope of every symplectic toric manifold is a Delzant polytope, every Delzant polytope is the moment polytope of such a manifold, and any two such manifolds with equivalent moment polytopes (up to translations and $\mathrm {GL} (n,\mathbb {Z} )$-transformations) admit a torus-equivariant symplectomorphism between them. The proof of this result is based on symplectic reduction.

Examples

Projective spaces

The complex projective space $\mathbb {CP} ^{n}$ is a $2n$-dimensional toric manifold together with the Fubini-Study symplectic form, the action by $\mathbb {T} ^{n}=\mathbb {S} ^{1}\times \ldots \times \mathbb {S} ^{1}$ given by$$(e^{i\theta _{1}},\ldots ,e^{i\theta _{n}})\cdot [z_{0}:z_{1}:\ldots :z_{n}]:=[z_{0}:e^{i\theta _{1}}z_{1}:\ldots :e^{i\theta _{n}}z_{n}],$$and the moment map$${\displaystyle \mu$$ :\mathbb {CP} ^{n}\to \mathbb {R} ^{n},\quad [z_{0}:z_{1}:\ldots :z_{n}]\mapsto {\frac {(|z_{1}|^{2},\ldots ,|z_{n}|^{2})}{|z_{0}|^{2}+|z_{1}|^{2}+\ldots +|z_{n}|^{2}}}.} Its associated Delzant polytope is the standard $n$-simplex in $\mathbb {R} ^{n}$.

In particular, the case $n=1$ can be identified with the $\mathbb {S} ^{1}$-action by rotations on $\mathbb {CP} ^{1}\cong \mathbb {S} ^{2}$, with moment map given by the height function; its associated Delzant polytope is a segment.

Products

The Delzant correspondence is compatible with products. For instance, the product $\mathbb {S} ^{2}\times \mathbb {S} ^{2}$ is a 4-dimensional toric manifold, being the product of two copies of the toric manifold $\mathbb {CP} ^{1}\cong \mathbb {S} ^{2}$ from above. Its associated Delzant polytope is the product of two segments, i.e. a rectangle in $\mathbb {R} ^{2}$.

Other examples

Any Hirzebruch surface is a 4-dimensional toric manifold; its associated Delzant polytope is a right-angled trapezium.