04.11.11 · algebraic-geometry / toric

Algebraic moment map and the polytope

shipped3 tiersLean: none

Anchor (Master): Fulton §4.2 + §5.2 (cohomology and the moment polytope); Cox-Little-Schenck §12.2 + §12.3; Audin *Topology of Torus Actions on Symplectic Manifolds* Ch. VII-VIII; Atiyah 1982 *Bull. London Math. Soc.* 14, 1-15 (convexity and commuting Hamiltonians); Guillemin-Sternberg 1982 *Invent. Math.* 67, 491-513 (convexity properties of the moment map); Guillemin-Sternberg 1984 *Geometric Quantization and Multiplicities of Group Representations* (Invent. Math. 67); Kempf-Ness 1979 *Lecture Notes in Math.* 732 (length functions in algebraic transformation groups, GIT-stability and the Kempf-Ness theorem); Kirwan 1984 *Cohomology of Quotients in Symplectic and Algebraic Geometry* (Princeton MN 31); Mumford-Fogarty-Kirwan 1994 *Geometric Invariant Theory* (Ergebnisse 34, 3rd ed.); Delzant 1988 *Bull. SMF* 116, 315-339 (symplectic-side classification); Brion 1987 *Bull. SMF* 115, 455-472 (image of the moment map for projective varieties)

Intuition [Beginner]

The algebraic moment map is a single picture that compresses the polytope-fan dictionary into a geometric drawing. Given a projective toric variety $X_{P}$ from a lattice polytope $P$ , the moment map is a continuous map $μ_{T} : X_{P} \to P$ that takes every point of the variety to a point of the polytope. The image of the entire variety is exactly the polytope, and the preimages of points inside the polytope are exactly the orbits of the compact torus on the variety. The polytope is therefore not just a combinatorial encoding of the variety — it is literally a shadow of the variety, painted in the character space.

Why bother with this picture? Because it joins two stories that develop separately. On the algebraic side, lattice polytopes encode projective toric varieties through the polytope-fan dictionary 04.11.10: the polytope determines a variety together with an ample line bundle, and the lattice points of the polytope are a basis for global sections of the line bundle. On the symplectic side, the moment map theorem of Atiyah and of Guillemin and Sternberg says that the image of any Hamiltonian torus action on a compact symplectic manifold is a convex polytope. The two stories meet in the projective toric case: the polytope from algebraic geometry and the polytope from symplectic geometry are the same polytope.

A concrete example shows the picture in action. The projective plane $P^{2}$ has the unit triangle as its polytope, and the compact torus $T_{R} = (S^{1})^{2}$ acts by rotating two of the three homogeneous coordinates. The moment map sends each point $[X_{0} : X_{1} : X_{2}]$ to the point $(∣ X_{1} ∣^{2}, ∣ X_{2} ∣^{2}) / (∣ X_{0} ∣^{2} + ∣ X_{1} ∣^{2} + ∣ X_{2} ∣^{2})$ in the triangle. The three vertices of the triangle are the images of the three fixed points. The three edges are the images of the three coordinate lines. The interior is the image of the open dense torus orbit. The polytope drawing of $P^{2}$ is the literal image of $P^{2}$ under the moment map.

Visual [Beginner]

A three-panel diagram of the algebraic moment map. Left panel: the projective plane $P^{2}$ drawn as an abstract toric variety with three fixed points labelled $p_{0}, p_{1}, p_{2}$ , three coordinate lines $ℓ_{01}, ℓ_{02}, ℓ_{12}$ , and the open torus orbit shaded. Middle: the moment-map arrow $μ_{T} : P^{2} \to Δ^{2}$ . Right: the unit triangle $Δ^{2} = conv ((0, 0), (1, 0), (0, 1))$ in the character space $M_{R} = R^{2}$ , with the three vertices labelled by the images of the fixed points, the three edges labelled by the images of the coordinate lines, and the interior labelled by the image of the open orbit.

The picture captures the moment map as a stratified drawing: the orbit-cone correspondence from 04.11.06 sits inside the moment-map picture as the rule that face $F$ of the polytope $\leftrightarrow$ orbit $O_{F}$ of the variety. The vertices of the polytope are the images of $T$ -fixed points; the open faces of the polytope are the images of the open torus orbits. Every projective toric variety carries its polytope picture in this way, and the picture is exact in the strong sense that the image is the polytope on the nose.

Worked example [Beginner]

Draw the moment map of the projective plane explicitly. Take $X = P^{2}$ with the polytope $P = conv ((0, 0), (1, 0), (0, 1))$ , the unit triangle. The compact torus $T_{R} = (S^{1})^{2}$ acts on $P^{2}$ by $$ (e^{i \theta_1}, e^{i \theta_2}) \cdot [X_0 : X_1 : X_2] = [X_0 : e^{i \theta_1} X_1 : e^{i \theta_2} X_2]. $$

Step 1. Identify the lattice-point basis. The three lattice points of $P$ are $(0, 0), (1, 0), (0, 1)$ , and the corresponding sections of $O_{P^{2}} (1)$ are $X_{0}, X_{1}, X_{2}$ respectively. The polytope $P$ sits inside the character space $M_{R} = R^{2}$ , with $(0, 0)$ corresponding to the constant character (associated with $X_{0}$ ), $(1, 0)$ corresponding to the character $(t_{1}, t_{2}) \mapsto t_{1}$ (associated with $X_{1}$ ), and $(0, 1)$ corresponding to $(t_{1}, t_{2}) \mapsto t_{2}$ (associated with $X_{2}$ ).

Step 2. Define the moment map. For each point $[X_{0} : X_{1} : X_{2}] \in P^{2}$ , set $$ \mu_T([X_0 : X_1 : X_2]) = \frac{|X_0|^2 \cdot (0, 0) + |X_1|^2 \cdot (1, 0) + |X_2|^2 \cdot (0, 1)}{|X_0|^2 + |X_1|^2 + |X_2|^2} = \frac{(|X_1|^2, |X_2|^2)}{|X_0|^2 + |X_1|^2 + |X_2|^2}. $$ This is a convex combination of the three lattice points $(0, 0), (1, 0), (0, 1)$ with non-negative weights adding to $1$ , hence lands in the convex hull $P$ of the lattice points.

Step 3. Image of the three fixed points. The $T$ -fixed points of $P^{2}$ are $p_{0} = [1 : 0 : 0], p_{1} = [0 : 1 : 0], p_{2} = [0 : 0 : 1]$ . Compute $μ_{T} (p_{0}) = ((0, 0)) = (0, 0)$ , $μ_{T} (p_{1}) = ((1, 0)) = (1, 0)$ , $μ_{T} (p_{2}) = ((0, 1)) = (0, 1)$ . The three vertices of the triangle are the images of the three fixed points.

Step 4. Image of the three coordinate lines. The coordinate line $ℓ_{01} = {[X_{0} : X_{1} : 0]}$ (the line where $X_{2} = 0$ ) maps under $μ_{T}$ to points of the form $((∣ X_{1} ∣^{2}, 0) / (∣ X_{0} ∣^{2} + ∣ X_{1} ∣^{2}))$ , which traces the segment from $(0, 0)$ to $(1, 0)$ as $∣ X_{1} ∣^{2} /∣ X_{0} ∣^{2}$ varies from $0$ to $\infty$ . Similarly $ℓ_{02}$ maps to the segment from $(0, 0)$ to $(0, 1)$ , and $ℓ_{12}$ maps to the segment from $(1, 0)$ to $(0, 1)$ . The three edges of the triangle are the images of the three coordinate lines.

Step 5. Image of the open torus orbit. The open torus orbit is the set of points where all three coordinates are nonzero. For each interior point $(a, b)$ with $a, b > 0$ and $a + b < 1$ , a unique compact-torus orbit in the open orbit maps to $(a, b)$ : choose a representative whose three squared moduli, divided by their total, equal $1 - a - b$ , $a$ , and $b$ , with the three phase factors then varying freely modulo the overall projective equivalence. The moment-map fibre over an interior point of $P$ is a single compact-torus orbit.

What this tells us. The moment map sends $P^{2}$ surjectively onto the unit triangle $P$ , with fibres exactly the compact-torus orbits on $P^{2}$ . Vertices of $P$ are fixed points of $T_{R}$ , edges of $P$ are images of one-dimensional orbits, and the interior of $P$ is the image of the open two-dimensional orbit. The polytope is the moment-map shadow of the toric variety, with the face structure of the polytope matching the orbit structure of the variety. This is the universal pattern of the algebraic moment map for projective toric varieties.

Check your understanding [Beginner]

Exercise (easy, multiple choice).

The algebraic moment map $μ_{T} : X_{P} \to P$ for a projective toric variety from a lattice polytope $P$ has image equal to:

A. A single point of $P$ . B. The lattice points $P \cap M$ . C. The boundary of the polytope (only). D. The entire polytope $P$ .

Hint

The image of the moment map of a Hamiltonian torus action on a compact symplectic manifold is a convex polytope (Atiyah 1982; Guillemin-Sternberg 1982). For the toric case, the polytope is exactly $P$ .

Answer

D. The image of the algebraic moment map $μ_{T} : X_{P} \to M_{R}$ is exactly the polytope $P$ . Vertices of $P$ are images of fixed points, faces are images of orbit closures, and the interior of $P$ is the image of the open torus orbit. Option A is wrong because the image has the full dimension of $P$ ; B is wrong because the image includes interior points, not just lattice points; C is wrong because the image is the entire convex polytope, not only its boundary.

Exercise (easy, true-false).

True or false: the preimage $μ_{T}^{- 1} (v)$ of a vertex $v$ of the polytope $P$ under the algebraic moment map is a single point of $X_{P}$ .

Hint

Vertices of the polytope correspond, by the orbit-cone correspondence, to fixed points of the compact-torus action. A fixed point is a single $T$ -orbit of dimension zero.

Answer

True. Each vertex $v$ of $P$ is the moment-map image of a unique $T$ -fixed point $p_{v} \in X_{P}$ . The orbit-cone correspondence from 04.11.06 identifies vertices of $P$ with the zero-dimensional orbits of the torus action — these are exactly the fixed points. The moment-map fibre over $v$ is the singleton ${p_{v}}$ , equivalently a zero-dimensional compact-torus orbit. For higher-dimensional faces $F \subseteq P$ , the fibres of $μ_{T}$ over interior points of $F$ are compact tori of dimension $dim F$ .

Formal definition [Intermediate+]

Let $N$ be a lattice of rank $n$ with dual lattice $M = Hom_{Z} (N, Z)$ , as in 04.11.01. Let $T = N \otimes_{Z} C^{*} ≅ (C^{*})^{n}$ be the complex algebraic torus, with compact subgroup $T_{R} = N \otimes_{Z} S^{1} ≅ (S^{1})^{n} \subset T$ . Let $P \subseteq M_{R}$ be a full-dimensional lattice polytope and let $(X_{P}, L_{P})$ be the polarised projective toric variety from 04.11.10.

Definition (Demazure projective embedding). By the Demazure character formula 04.11.10, the global sections $H^{0} (X_{P}, L_{P}) = ⨁_{m \in P \cap M} C \cdot s_{m}$ form a $T$ -representation with $∣ P \cap M ∣$ weight-one summands indexed by lattice points of $P$ . The Demazure projective embedding is the closed immersion $$ \iota_P : X_P \hookrightarrow \mathbb{P}\big(H^0(X_P, L_P)^\vee\big) = \mathbb{P}^{|P \cap M| - 1}, $$ sending $x \in X_{P}$ to the hyperplane ${s \in H^{0} (X_{P}, L_{P}) : s (x) = 0}$ in the dual space. In coordinates indexed by lattice points $m \in P \cap M$ , the embedding reads $ι_{P} (x) = [s_{m} (x)]_{m \in P \cap M}$ .

Definition (algebraic moment map). The algebraic moment map of the projective toric variety $X_{P}$ is the map $$ \mu_T : X_P(\mathbb{C}) \to M_\mathbb{R}, \qquad \mu_T(x) := \sum_{m \in P \cap M} \frac{|s_m(x)|^2}{\sum_{m' \in P \cap M} |s_{m'}(x)|^2} \cdot m. $$ The map is continuous, $T_{R}$ -invariant (each $∣ s_{m} (x) ∣^{2}$ is $T_{R}$ -invariant since $T_{R}$ acts on $s_{m}$ by a unit-modulus character), and lands in the convex hull of ${m : m \in P \cap M} = P$ .

Definition (symplectic moment map; equivalent characterisation). Equip $P^{∣ P \cap M ∣ - 1}$ with the Fubini-Study symplectic form $ω_{FS}$ , and pull back via $ι_{P}$ to obtain a symplectic form $ω_{P} = ι_{P}^{*} ω_{FS}$ on the smooth locus of $X_{P}$ . The compact torus $T_{R}$ acts on $(X_{P}, ω_{P})$ by Hamiltonian symplectomorphisms (the action lifts to the polarisation, with Hamiltonian generators arising from the lifted infinitesimal action). The symplectic moment map of this Hamiltonian action equals the algebraic moment map $μ_{T}$ defined above, up to an additive translation; both define the same map modulo a possible shift by a fixed element of $M_{R}$ .

Definition (face stratification). For each face $F \subseteq P$ , write $O_{F}$ for the corresponding $T$ -orbit in $X_{P}$ under the orbit-cone correspondence of 04.11.06 (face $F$ of $P$ $\leftrightarrow$ cone $σ_{F}^{\lor}$ in the normal fan $Σ_{P}$ $\leftrightarrow$ orbit $O_{F}$ of codimension $dim F$ ). The face stratification of $X_{P}$ partitions the variety as $X_{P} = ⨆_{F} O_{F}$ , where the disjoint union runs over all faces of $P$ .

Counterexamples to common slips

"The moment map is defined only when $X_{P}$ is smooth." It is not. The algebraic moment map via the lattice-point formula above makes sense on every projective toric variety $X_{P}$ , including singular ones (e.g., weighted projective spaces, conifolds, toric varieties from non-Delzant polytopes). The symplectic moment map via the Fubini-Study form needs smoothness for $ω_{P}$ to be a genuine symplectic form, but the algebraic formula extends through singular points by continuity.
"The moment-map fibre over an interior point of $P$ is a single point." It is not. The fibre over an interior point of $P$ is an entire $T_{R}$ -orbit, equivalently a compact torus $(S^{1})^{n}$ . Only over vertices of $P$ does the fibre degenerate to a single point (a $T$ -fixed point). The general rule, matching the orbit-cone correspondence: fibres over interior points of a $k$ -dimensional face $F \subseteq P$ are compact tori of dimension $k$ .
"The moment map depends only on the variety $X_{P}$ , not on the polarisation $L_{P}$ ." It depends on both. Different ample polarisations of the same $X_{P}$ give different polytopes $P, P^{'}$ (with the same normal fan), and the moment maps land in different polytopes. The moment map is canonically attached to the polarised projective toric variety $(X_{P}, L_{P})$ , not to $X_{P}$ alone.
"The image of the moment map is the lattice points of $P$ ." It is the entire continuous polytope $P$ , not just $P \cap M$ . The lattice points $P \cap M$ are the moment-map images of certain special points (corresponding to weight vectors of the $T$ -action), but the full image fills out the convex polytope.

Key theorem with proof [Intermediate+]

The signature theorem of this unit identifies the image of the algebraic moment map with the polytope $P$ , with full stratification of fibres by faces. This is the projective-toric incarnation of the Atiyah-Guillemin-Sternberg convexity theorem 05.04.03, specialised to the case where the symplectic manifold is a projective toric variety with Fubini-Study form from the Demazure embedding.

Theorem (image of the algebraic moment map; Atiyah 1982 + Guillemin-Sternberg 1982 + Fulton 1993 §4.2). Let $X_{P}$ be a projective toric variety from a full-dimensional lattice polytope $P \subseteq M_{R}$ , and let $μ_{T} : X_{P} (C) \to M_{R}$ be the algebraic moment map of the compact-torus action.

(a) Image equals polytope. $μ_{T} (X_{P} (C)) = P$ as subsets of $M_{R}$ .

(b) Vertices equal fixed-point images. $μ_{T} (X_{P}^{T_{R}}) = Vert (P)$ , i.e., the moment-map image of the $T_{R}$ -fixed-point set equals the set of vertices of $P$ . The correspondence is a bijection: each vertex $v \in Vert (P)$ has a unique fixed-point preimage $p_{v} \in X_{P}$ .

(c) Fibre = compact-torus orbit. For every face $F \subseteq P$ and every point $\overset{m}{ˉ} \in relint (F)$ , the preimage $μ_{T}^{- 1} (\overset{m}{ˉ})$ is a single compact-torus orbit $T_{R} \cdot x$ inside the algebraic orbit $O_{F}$ of 04.11.06, with $dim_{R} (μ_{T}^{- 1} (\overset{m}{ˉ})) = dim F$ . In particular, fibres over interior points of $P$ are top-dimensional compact tori $(S^{1})^{n}$ , and fibres over vertices are points.

(d) Convexity (Atiyah-Guillemin-Sternberg). The image $μ_{T} (X_{P} (C))$ is convex: it is the convex hull of the moment-map images of the $T_{R}$ -fixed points, equivalently $conv (Vert (P)) = P$ .

Proof.

The argument has four steps. First, identify the moment-map formula with the explicit lattice-point formula via the Demazure embedding. Second, compute the image of the fixed points. Third, identify fibres with compact-torus orbits inside algebraic orbits. Fourth, invoke convexity to close.

Step 1: lattice-point formula on $X_{P}$ . By the Demazure character formula 04.11.10, $H^{0} (X_{P}, L_{P}) = ⨁_{m \in P \cap M} C \cdot s_{m}$ with each $s_{m}$ a $T$ -semi-invariant of weight $m$ . The Demazure projective embedding $ι_{P} : X_{P} ↪ P^{∣ P \cap M ∣ - 1}$ sends $x$ to $[s_{m} (x)]_{m \in P \cap M}$ . The standard moment map of the compact torus $T_{R}$ acting on $P^{N - 1}$ (with $N = ∣ P \cap M ∣$ , $T_{R}$ acting on each coordinate by the character with weight $m$ ) is the map $$ \mu_{\mathrm{FS}}([z_m]{m}) = \sum{m} \frac{|z_m|^2}{\sum_{m'} |z_{m'}|^2} \cdot m, $$ the barycentric coordinates on the lattice-point set ${m \in P \cap M}$ in $M_{R}$ . Pulling back along $ι_{P}$ gives $$ \mu_T(x) = \mu_{\mathrm{FS}}(\iota_P(x)) = \sum_{m} \frac{|s_m(x)|^2}{\sum_{m'} |s_{m'}(x)|^2} \cdot m, $$ which is the algebraic moment map of the definition above. This identifies the algebraic moment map with the Fubini-Study moment map composed with the Demazure embedding.

Step 2: image of the fixed points. A $T_{R}$ -fixed point of $X_{P}$ is a point $x$ such that $T_{R} \cdot x = {x}$ . Under the Demazure embedding, $ι_{P} (x) = [z_{m}]_{m}$ is a fixed point of the $T_{R}$ -action on $P^{N - 1}$ , equivalent to $[z_{m}]_{m}$ being a coordinate point of $P^{N - 1}$ — that is, all $z_{m} = 0$ except for a single index $m = v$ . By the orbit-cone correspondence of 04.11.06, $T$ -fixed points of $X_{P}$ are in bijection with maximal cones $σ_{v}$ of $Σ_{P}$ , equivalently with vertices $v \in Vert (P)$ . For the fixed point $p_{v}$ corresponding to vertex $v$ : $ι_{P} (p_{v}) = [0 : \dots : 1 : \dots : 0]$ (a $1$ in position $v$ , zeros elsewhere), hence $μ_{T} (p_{v}) = (1 \cdot v + 0 \cdot rest) /1 = v$ .

So $μ_{T} (p_{v}) = v$ for each vertex $v \in Vert (P)$ , and the image of the fixed-point set $X_{P}^{T_{R}}$ is exactly $Vert (P)$ . The correspondence $v \leftrightarrow p_{v}$ is the bijection of (b).

Step 3: fibres of the moment map. Pick a face $F \subseteq P$ and an interior point $\overset{m}{ˉ} \in relint (F)$ . The fibre $μ_{T}^{- 1} (\overset{m}{ˉ})$ is the set of $x \in X_{P}$ with $μ_{T} (x) = \overset{m}{ˉ}$ . Since $\overset{m}{ˉ}$ is in the relative interior of $F$ , the barycentric coordinates expressing $\overset{m}{ˉ}$ as a convex combination of $P \cap M$ are supported on $F \cap M$ — that is, only lattice points $m \in F$ contribute. Equivalently, $s_{m} (x) = 0$ for $m \in / F$ , and the support condition cuts out the algebraic orbit $O_{F}$ corresponding to $F$ under the orbit-cone correspondence: $O_{F} = {x \in X_{P} : s_{m} (x) \neq = 0 ⟺ m \in F}$ (this is the standard description of toric orbits in terms of which lattice-point sections vanish).

Within $O_{F}$ , the moment-map fibre $μ_{T}^{- 1} (\overset{m}{ˉ}) \cap O_{F}$ is the set of points with prescribed magnitudes $∣ s_{m} (x) ∣^{2} / (\sum_{m^{'} \in F} ∣ s_{m^{'}} (x) ∣^{2})$ matching the barycentric coordinates of $\overset{m}{ˉ}$ in $F$ . Magnitudes prescribed leaves the phases of the $s_{m} (x)$ free, modulo the overall projective equivalence — equivalently, the fibre over $\overset{m}{ˉ} \in relint (F)$ is a single $T_{R}$ -orbit of dimension $dim F$ inside $O_{F}$ . The algebraic orbit $O_{F}$ has complex dimension $dim F$ , and its compact-torus orbit is a real torus of the same real dimension $dim F$ . In particular, for $F$ a vertex ( $dim F = 0$ ) the fibre is a point; for $F$ the full polytope ( $dim F = n$ ) the fibre is the top-dimensional compact torus $(S^{1})^{n}$ .

Step 4: convexity and image identification. The image $μ_{T} (X_{P})$ contains all vertices of $P$ (Step 2) and contains all interior points (Step 3 with $F = P$ produces a fibre over every $\overset{m}{ˉ} \in relint (P)$ inside the open torus orbit). By the Atiyah-Guillemin-Sternberg convexity theorem 05.04.03 applied to the compact symplectic manifold $X_{P}$ (in the smooth case; the general case follows by continuity at singular toric strata), the image $μ_{T} (X_{P})$ is convex.

Since the image contains the vertices and is convex, it contains the convex hull of the vertices, which is exactly $P$ . Conversely, the image is contained in $P$ by the barycentric-coordinate definition: each $μ_{T} (x)$ is a convex combination of points of $P \cap M \subseteq P$ , hence lies in $P$ . Therefore $μ_{T} (X_{P}) = P$ . This proves (a) and (d). $□$

Bridge. The image-equals-polytope theorem builds toward 05.04.05 (Duistermaat-Heckman, which exactly computes the pushforward measure $(μ_{T})_{*} ω_{P}^{n} / n!$ on $P$ as the Lebesgue measure on $P$ in the toric case, with a normalisation constant equal to the volume of the compact torus) and 04.11.12 (the cohomology ring of $X_{P}$ as the Stanley-Reisner ring of $P$ , where the moment-map fibration is the geometric input to the toric Stanley-Reisner presentation), and the central insight is that the polytope $P$ is the complete shadow of $X_{P}$ under the compact-torus action — every algebraic-geometric invariant of $X_{P}$ has a polytope-side computation through the moment map. This bridge appears again in 04.11.13 (toric intersection theory and mixed volumes, where intersection numbers compute as mixed volumes of polytopes via the moment-map identification), in 04.10.06 (Kempf-Ness theorem on the algebraic-geometric side, where the GIT quotient is identified with the symplectic quotient $μ_{T}^{- 1} (0) / T_{R}$ ), and in 04.11.16 (reflexive polytopes and Batyrev mirror symmetry, where the moment-map polytope is the combinatorial input to the Calabi-Yau hypersurface construction).

Putting these together, the algebraic moment map promotes the polytope-fan dictionary from a static combinatorial encoding to a dynamic geometric drawing: the polytope $P$ is not just a label of the polarised projective toric variety $(X_{P}, L_{P})$ , but is literally drawn inside $M_{R}$ as the image of $X_{P}$ under a continuous, surjective, $T_{R}$ -invariant map whose fibres recover the orbit-cone correspondence. The bridge is the Atiyah-Guillemin-Sternberg convexity theorem, which guarantees that the image is the convex polytope $P$ on the nose, and the Kempf-Ness theorem, which identifies the algebraic GIT description of $X_{P}$ with the symplectic-reduction description in terms of moment-map level sets.

Exercises [Intermediate+]

Exercise 3 (medium, symbolic).

For the Hirzebruch surface $F_{1} = P (O_{P^{1}} \oplus O_{P^{1}} (1))$ with the trapezoid polytope $P = conv ((0, 0), (1, 0), (1, 1), (0, 2))$ from 04.11.10 Exercise 5, identify the four $T$ -fixed points and their moment-map images.

Hint

The four vertices of the trapezoid are the moment-map images of the four $T$ -fixed points. The fixed points correspond to the four maximal cones of the normal fan, equivalently to the four vertices of $P$ .

Answer

The trapezoid has four vertices: $v_{1} = (0, 0), v_{2} = (1, 0), v_{3} = (1, 1), v_{4} = (0, 2)$ . These correspond to the four $T$ -fixed points $p_{1}, p_{2}, p_{3}, p_{4}$ of $F_{1}$ via the orbit-cone correspondence: $v_{i} \leftrightarrow$ maximal cone $σ_{i}$ of $Σ_{P} \leftrightarrow$ zero-dimensional orbit $O_{σ_{i}} = {p_{i}}$ of $X_{P} = F_{1}$ .

Geometrically, the four fixed points are the intersections of the two coordinate sections (the negative-curve section $S$ with $S^{2} = - 1$ and the positive section $S + F$ with $(S + F)^{2} = 1$ ) with the two fibre lines (over the two fixed points of $P^{1}$ ). The moment-map image of each $p_{i}$ is the corresponding vertex $v_{i}$ of $P$ .

The compact torus $T_{R} = (S^{1})^{2}$ acts on $F_{1}$ via its embedding into $P^{4}$ by the line bundle $L_{P} = F + 2 σ$ (which has $5$ global sections corresponding to the $5$ lattice points of $P$ ). The Demazure embedding $F_{1} ↪ P^{4}$ realises the moment-map formula in coordinates indexed by the $5$ lattice points of $P$ , with image equal to the trapezoid.

Rubric: full credit for the four vertices identified as moment-map images of the four fixed points, with the orbit-cone correspondence explicit.

Exercise 4 (medium, symbolic).

For $X_{P}$ a smooth projective toric variety, prove that the algebraic moment map sends the open torus orbit $T \subset X_{P}$ surjectively onto the interior $int (P)$ of the polytope, with fibres exactly the compact-torus orbits $T_{R} \subset T$ .

Hint

A point of the open orbit has all lattice-point sections nonzero, so its moment-map image is an interior convex combination of $P \cap M$ — that is, an interior point of $P$ . Conversely, every interior point of $P$ is achievable by choosing appropriate magnitudes for the sections.

Answer

The open torus orbit $T \subset X_{P}$ is the locus where all lattice-point sections $s_{m}$ for $m \in P \cap M$ are nonzero (the open subset of $X_{P}$ avoiding all toric divisors). For $x \in T$ , every $s_{m} (x) \neq = 0$ , so the barycentric coordinates $λ_{m} (x) := ∣ s_{m} (x) ∣^{2} / (\sum_{m^{'}} ∣ s_{m^{'}} (x) ∣^{2})$ are all strictly positive, and $\sum_{m} λ_{m} = 1$ . The moment-map image $μ_{T} (x) = \sum_{m} λ_{m} (x) \cdot m$ is therefore a convex combination of the lattice points $P \cap M$ with all weights strictly positive. By Carathéodory's theorem (every interior point of the convex hull of a finite set has a strict-positive-weights representation when the set spans), the image $μ_{T} (T)$ is exactly the interior $int (P)$ .

Surjectivity onto $int (P)$ . Given any $\overset{m}{ˉ} \in int (P)$ , write $\overset{m}{ˉ} = \sum_{m} λ_{m} \cdot m$ with $λ_{m} > 0$ and $\sum_{m} λ_{m} = 1$ (possible because the lattice points of $P$ — in particular the vertices — span $P$ as a convex polytope, and any interior point is in the relative interior of any full-dimensional facet decomposition). Take $x \in T$ with $∣ s_{m} (x) ∣^{2} = λ_{m}$ (rescaling overall to fix the projective ambiguity); then $μ_{T} (x) = \overset{m}{ˉ}$ .

Fibres are compact-torus orbits. Given $\overset{m}{ˉ} \in int (P)$ , the fibre $μ_{T}^{- 1} (\overset{m}{ˉ}) \cap T$ consists of points $x \in T$ with prescribed magnitudes $∣ s_{m} (x) ∣^{2} / \sum_{m^{'}} ∣ s_{m^{'}} (x) ∣^{2} = λ_{m}$ for each $m \in P \cap M$ . The magnitudes are determined; the phases of $s_{m} (x) \in C^{*}$ remain free modulo the overall projective equivalence. The phase freedom is exactly the $T_{R}$ -action: changing the phase of each $s_{m} (x)$ by a unit-modulus complex number $e^{i θ_{m}}$ corresponds to acting on $x$ by the compact-torus element $(e^{i θ_{m}})$ (in the dual coordinates). The fibre $μ_{T}^{- 1} (\overset{m}{ˉ}) \cap T$ is therefore a single compact-torus orbit, a real torus of dimension $n = dim T_{R}$ . $□$

Rubric: full credit for the surjectivity argument via Carathéodory, the fibre identification, and the dimension count.

Exercise 5 (medium, short-answer).

Explain why the algebraic moment map of a projective toric variety $X_{P}$ has image equal to the polytope $P$ as a special case of the Atiyah-Guillemin-Sternberg convexity theorem 05.04.03. Identify the symplectic manifold, the Hamiltonian action, and the moment map in the general framework.

Hint

The compact manifold is $X_{P}$ with the pulled-back Fubini-Study form $ω_{P} = ι_{P}^{*} ω_{FS}$ . The Hamiltonian action is the compact-torus action $T_{R} ↷ X_{P}$ . The moment map is the algebraic moment map $μ_{T}$ .

Answer

Symplectic manifold. $X_{P}$ , equipped with the symplectic form $ω_{P}$ obtained by pulling back the Fubini-Study symplectic form $ω_{FS}$ on $P^{∣ P \cap M ∣ - 1}$ via the Demazure embedding $ι_{P}$ . The pull-back $ω_{P} = ι_{P}^{*} ω_{FS}$ is a Kähler form on the smooth locus of $X_{P}$ , with the toric singularities (if any) along the boundary toric strata.

Hamiltonian action. The compact torus $T_{R} = (S^{1})^{n}$ acts on $X_{P}$ by Hamiltonian symplectomorphisms: the action on $P^{∣ P \cap M ∣ - 1}$ via the Demazure-embedding weight decomposition restricts to a Hamiltonian action with respect to $ω_{FS}$ , and pulls back to a Hamiltonian action on $(X_{P}, ω_{P})$ .

Moment map. The symplectic moment map $μ : X_{P} \to t^{*} = M_{R}$ of the Hamiltonian $T_{R}$ -action equals the algebraic moment map $μ_{T}$ defined by the lattice-point formula, up to an additive constant fixed by the convention on the splitting $H^{0} (X_{P}, L_{P}) = ⨁_{m} C \cdot s_{m}$ .

Atiyah-Guillemin-Sternberg theorem 05.04.03. For any Hamiltonian action of a compact connected torus on a compact connected symplectic manifold, the image of the moment map is a convex polytope, equal to the convex hull of the moment-map images of the fixed-point set. Applied here: the fixed points of $T_{R}$ on $X_{P}$ correspond to vertices of $P$ via the orbit-cone correspondence, with $μ_{T} (p_{v}) = v$ by Step 2 of the main theorem. The convex hull of the vertices is exactly $P$ , so $μ_{T} (X_{P}) = P$ .

Connection with the toric structure. The AGS theorem gives convexity of the image in general; the toric structure of $X_{P}$ adds the stronger statement that the image is the specific polytope $P$ from which $X_{P}$ was constructed. The polytope-fan dictionary 04.11.10 guarantees that the AGS polytope of the Hamiltonian $T_{R}$ -action on $(X_{P}, ω_{P})$ coincides with the combinatorial polytope $P$ .

Rubric: full credit for identifying the symplectic manifold, the Hamiltonian action, the moment map, and the invocation of AGS to conclude the convex-image equals the convex hull of fixed-point images.

Exercise 6 (medium, numeric).

The barycentre of the polytope $P$ is the moment-map image of a particular point of $X_{P}$ — the "balanced point" where all lattice-point sections have equal magnitude. For $P$ the unit triangle in $R^{2}$ , compute the barycentre and the corresponding point of $P^{2}$ .

Hint

The barycentre of the convex hull of three points is the arithmetic mean of the three points. The corresponding point of $P^{2}$ has all three coordinate moduli equal.

Answer

Barycentre. The unit triangle $P = conv ((0, 0), (1, 0), (0, 1))$ has barycentre $$ \bar P = \frac{(0, 0) + (1, 0) + (0, 1)}{3} = (1/3, 1/3). $$

Corresponding point. The "balanced point" $x_{*} \in P^{2}$ where all three coordinate moduli are equal has $∣ X_{0} ∣ = ∣ X_{1} ∣ = ∣ X_{2} ∣$ , with the phases free modulo overall projective equivalence. The canonical representative is $x_{*} = [1 : 1 : 1]$ . Compute $μ_{T} (x_{*}) = (1 \cdot (0, 0) + 1 \cdot (1, 0) + 1 \cdot (0, 1)) /3 = (1/3, 1/3)$ , matching the barycentre of $P$ .

The barycentre-balanced-point correspondence holds in general: for any projective toric variety $X_{P}$ , the moment-map image of the "all-sections-equal" point is the barycentre of $P$ in the metric induced by the lattice. Rubric: full credit for the barycentre $(1/3, 1/3)$ and the point $[1 : 1 : 1]$ with the moment-map verification.

Exercise 7 (hard, short-answer).

State the Kempf-Ness theorem identifying the GIT quotient of $X_{P}$ with a symplectic quotient via the moment map. Explain how the polytope $P$ enters both sides.

Hint

Kempf-Ness 1979 identifies the GIT quotient $X / / G_{C}$ of a projective variety with linearised reductive group action with the symplectic quotient $μ^{- 1} (0) / G_{R}$ of the compact form. For toric varieties from the Cox construction, $X_{P} = C^{∣ P \cap M ∣} / /_{L} (C^{*})^{k}$ for an appropriate $k$ and character $L$ .

Answer

Kempf-Ness theorem (Kempf-Ness 1979 Lecture Notes in Math. 732; Mumford-Fogarty-Kirwan 1994 §8.2). Let $W$ be a finite-dimensional Hermitian complex vector space with a linear action of a complex reductive group $G_{C} = K^{C}$ (the complexification of a compact group $K$ ), and let $L = C_{χ}$ be a $G_{C}$ -linearisation by a character $χ$ . The GIT quotient $W / /_{χ} G_{C}$ — defined as the proj of the $χ$ -semi-invariant ring — is homeomorphic to the symplectic quotient $μ_{K}^{- 1} (0) / K$ , where $\mu_K : W \to \mathfrak{k}^ $i s t h e m o m e n t ma p o f t h e$ K $- a c t i o n w i t h r es p ec tt o t h es t an d a r d H er mi t ian K \overset{a}{¨} h l er f or m o n$ W $, s hi f t e d b y t h ec ha r a c t er$ \chi$.*

Toric case (Cox 1995; CLS §14.4). For a projective toric variety from the polytope $P$ , the Cox construction realises $X_{P}$ as a GIT quotient: $$ X_P = (\mathbb{C}^{|\Sigma_P(1)|} \setminus Z(\Sigma_P)) /!/\chi G, $$ where $G = Hom (Pic (X_{P}), C^{*}) \subset (C^{*})^{∣ Σ_{P} (1) ∣}$ acts on the coordinate ring $\mathbb{C}[x\rho : \rho \in \Sigma_P(1)] $b y P i c a r d - g r a d in g c ha r a c t er s,$ Z(\Sigma_P) $i s t h e i r r e l e v an tl oc u se n co d e d b y t h e f an, an d$ \chi $i s t h ec ha r a c t er o f$ G $cor r es p o n d in g t o t h e am pl e - l in e - b u n d l ec l a ss$ [L_P] \in \mathrm{Pic}(X_P)$.

Kempf-Ness applied. The compact subgroup $K = G \cap (S^{1})^{∣ Σ_{P} (1) ∣}$ acts on $C^{∣ Σ_{P} (1) ∣}$ , and the Kempf-Ness theorem identifies $$ X_P = \mathbb{C}^{|\Sigma_P(1)|} /!/\chi G\mathbb{C} \cong \mu_K^{-1}(c)/K, $$ where $μ_{K}$ is the moment map of the $K$ -action and $c$ is the centre-of-mass parameter encoding the character $χ$ , equivalently the ample-line-bundle polarisation. The level set $μ_{K}^{- 1} (c) \subset C^{∣ Σ_{P} (1) ∣}$ is a real manifold (a real submanifold of the affine space), and the symplectic quotient is the smooth real-quotient by $K$ .

Polytope on both sides. The polytope $P$ enters the GIT side as the lattice-point support of the ample line bundle $L_{P}$ (via the Demazure character formula $H^{0} (X_{P}, L_{P}) = ⨁_{m \in P \cap M} C \cdot χ^{m}$ ). It enters the symplectic side as the image of the residual compact-torus action on $X_{P} = μ_{K}^{- 1} (c) / K$ : the original compact-torus action on $C^{∣ Σ_{P} (1) ∣}$ descends to a residual $T_{R} = (S^{1})^{n}$ action on the symplectic quotient (the residual torus being the quotient of $(S^{1})^{∣ Σ_{P} (1) ∣}$ by $K$ ), and the moment map of this residual action is exactly the algebraic moment map $μ_{T} : X_{P} \to P$ studied in this unit.

The polytope is therefore the bridging object in both directions: on the GIT side, $P$ is the polytope of the ample line bundle $L_{P}$ in the Picard-graded coordinate ring; on the symplectic side, $P$ is the image of the moment map of the residual compact-torus action. The Kempf-Ness theorem identifies these two roles of $P$ as the two faces of the same geometric object. $□$

Rubric: full credit for the Kempf-Ness theorem statement, the Cox-construction toric GIT presentation, the residual-torus identification, and the polytope role on both sides.

Exercise 8 (hard, short-answer).

For a smooth projective toric variety $X_{P}$ , the Duistermaat-Heckman theorem 05.04.05 computes the pushforward measure $(μ_{T})_{*} ω_{P}^{n} / n!$ on $P$ as the Lebesgue measure on $P$ (up to a normalising constant). State the Duistermaat-Heckman formula in the toric case and explain what each side measures.

Hint

The pushforward measure of the symplectic volume form $ω_{P}^{n} / n!$ under the moment map $μ_{T}$ is a measure on $P \subset M_{R}$ . The Duistermaat-Heckman theorem says this measure has a piecewise-polynomial density; in the toric case the density is constant.

Answer

Duistermaat-Heckman in the toric case (Guillemin-Sternberg 1984 §32; Fulton §4.4). Let $X_{P}$ be a smooth projective toric variety with polytope $P$ and Fubini-Study symplectic form $ω_{P}$ . The pushforward of the Liouville volume form under the moment map is $$ (\mu_T)_* \big(\omega_P^n / n!\big) = (2\pi)^n \cdot \mathbf{1}_P \cdot dm, $$ where $1_{P}$ is the indicator function of $P$ and $d m$ is the Lebesgue measure on $M_{R}$ normalised by the lattice volume (so that the fundamental parallelepiped of $M$ has Lebesgue volume $1$ ).

Interpretation of each side.

Left side. The Liouville volume form $ω_{P}^{n} / n!$ on $X_{P}$ is the symplectic volume form, with total integral the symplectic volume of $X_{P}$ — algebraically equal to the degree of $X_{P}$ under the Demazure embedding by $L_{P}$ , equivalently to $vol (P) \cdot n!$ by the toric Bernstein-Kushnirenko theorem 04.11.14. Pushing forward to $P$ via $μ_{T}$ produces a measure on $P$ supported on the image of $μ_{T}$ .

Right side. The Lebesgue measure $1_{P} \cdot d m$ on $M_{R}$ restricted to $P$ has total integral $vol (P)$ in the lattice-normalised Lebesgue measure. Multiplying by $(2 π)^{n}$ gives total integral $(2 π)^{n} \cdot vol (P)$ , matching the symplectic volume of $X_{P}$ up to the $(2 π)^{n}$ factor coming from the compact-torus orbit volume (each fibre over an interior point of $P$ is a compact torus $(S^{1})^{n}$ of volume $(2 π)^{n}$ ).

Polynomial density and toric simplification. The Duistermaat-Heckman theorem in general states that the pushforward measure has piecewise-polynomial density on the moment-map image. In the toric case, the density is the constant $(2 π)^{n}$ , because every compact-torus fibre has the same volume $(2 π)^{n}$ (every interior fibre is the full compact torus $(S^{1})^{n}$ ). The Lebesgue-density form is the simplest case of the Duistermaat-Heckman formula and is the toric specialisation.

Connections. The Duistermaat-Heckman formula on toric varieties is the symplectic-side incarnation of the Ehrhart-polynomial calculation $dim H^{0} (X_{P}, L_{P}^{\otimes k}) = ∣ k P \cap M ∣$ from 04.11.10. Both formulas express algebraic-geometric invariants of $X_{P}$ as lattice-point or Lebesgue-volume invariants of $P$ . The Brion-Vergne lattice-point formula generalises both, expressing equivariant Euler characteristics of toric line bundles in terms of fixed-point residue formulas. $□$

Rubric: full credit for the Duistermaat-Heckman formula in the toric case, the interpretation of each side as symplectic volume vs. lattice-Lebesgue measure, and the connection to the Ehrhart polynomial of $P$ .

Advanced results [Master]

The algebraic moment map as a Kähler-stratified picture

Theorem (refined moment-map stratification; Fulton §4.2; Audin Ch. VII §3). Let $X_{P}$ be a smooth projective toric variety with polytope $P$ . The algebraic moment map $μ_{T} : X_{P} \to P$ restricts to a locally-product topological fibration over the relative interior of each face of $P$ , with fibre the compact torus $T_{R}^{F} = (S^{1})^{d i m F}$ for the face $F$ . The stratification refines the orbit-cone correspondence to a stratified-fibration picture, with:

(i) Vertices: $μ_{T}^{- 1} (v) = {p_{v}}$ , a single $T$ -fixed point.

(ii) Edges: $μ_{T}^{- 1} (relint (e)) = O_{e} ≅ T_{R}^{1} \times (interval)$ , a circle bundle over the interval $relint (e)$ .

(iii) Higher faces: $μ_{T}^{- 1} (relint (F)) = O_{F}$ topologically a $T_{R}^{F}$ -bundle over $relint (F)$ .

(iv) Open face: $μ_{T}^{- 1} (int (P)) = T \subset X_{P}$ topologically $T_{R}^{n} \times int (P)$ .

The stratification turns the polytope $P$ into a stratification base, with strata indexed by faces of $P$ and total spaces $O_{F}$ in $X_{P}$ stacked over the strata. The moment-map fibration over each stratum is a principal $T_{R}^{F}$ -bundle; the bundle is in fact a product over each open stratum because the relative interior of a face is contractible.

A direct consequence: the algebraic moment map $μ_{T}$ is a complete moment map in the sense of Lerman 1995, meaning the topology of $X_{P}$ is recoverable from the topology of $P$ together with the face-stratification fibration data. Specifically, $X_{P}$ is the iterated mapping torus of the moment-map fibration over each face, with the gluing data encoded by the face inclusions of $P$ .

The Kempf-Ness theorem and the GIT-symplectic correspondence

Theorem (Kempf-Ness for toric varieties; Kempf-Ness 1979 LNM 732; Cox 1995 J. Algebraic Geom. 4). Let $X_{P}$ be a projective toric variety from the polytope $P$ , presented via the Cox construction $X_{P} = (C^{∣ Σ_{P} (1) ∣} ∖ Z (Σ_{P})) / /_{χ} G$ . Then $X_{P}$ is homeomorphic to the symplectic quotient $$ X_P \cong \mu_G^{-1}(c)/G_{\mathbb{R}}, $$ where $\mu_G : \mathbb{C}^{|\Sigma_P(1)|} \to \mathrm{Lie}(G_\mathbb{R})^ $i s t h e m o m e n t ma p o f t h eco m p a c t s u b g r o u p$ G_\mathbb{R} \subset G $a c t in g l in e a r l y o n$ \mathbb{C}^{|\Sigma_P(1)|} $, an d$ c \in \mathrm{Lie}(G_\mathbb{R})^ $i s t h ece n t r a l e l e m e n t cor r es p o n d in g t o t h e am pl e - l in e - b u n d l e l in e a r i s a t i o n$ L_P$.

The Kempf-Ness theorem is the algebraic-geometric / symplectic-geometric bridge for the toric setup. The GIT side ( $X_{P}$ as a quotient of an affine space by a reductive group) connects to the polytope-fan dictionary by the Cox construction. The symplectic side ( $X_{P}$ as a symplectic-reduction quotient) connects to the moment-map calculus by the standard symplectic-reduction recipe. The two presentations are homeomorphic via the Kempf-Ness theorem, and the residual compact-torus action on the symplectic quotient is exactly the action whose moment map is $μ_{T} : X_{P} \to P$ .

The Kempf-Ness theorem extends to non-toric reductive group actions, where the GIT quotient and the symplectic quotient agree as topological spaces but may differ as schemes (the GIT quotient is naturally a scheme; the symplectic quotient is naturally a topological space, with a smooth structure when the action is free). For toric varieties, both quotients are naturally schemes and the identification is at the level of complex-analytic spaces.

The image-of-the-moment-map theorem for general projective varieties (Brion 1987)

Theorem (image of the moment map for $T$ -projective varieties; Brion 1987 Bull. SMF 115). Let $X \subset P^{N}$ be a projective variety carrying a $T = (\mathbb{C}^)^n $- a c t i o n l i f t e d t o$ \mathcal{O}{\mathbb{P}^N}(1) $, an d l e t$ \mu_T : X \to M\mathbb{R} $b e t h e a l g e b r ai c m o m e n t ma p o f t h eco m p a c t - t or u s a c t i o n . T h e n t h e ima g e$ \mu_T(X) $i s a co n v e x r a t i o na l p o l y t o p e, c a l l e d t h e * * m o m e n tp o l y t o p e * * o f$ X$.*

This generalises the toric case from "image is the polytope of the toric variety" to "image is some rational polytope of the projective variety", with the moment polytope an invariant of the polarised variety. For toric $X$ the moment polytope is the polytope $P$ of the polytope-fan dictionary; for non-toric $X$ the moment polytope is generally a proper sub-polytope of the convex hull of $T$ -weights of $H^{0} (X, O (1))$ .

The Brion image theorem extends to spherical varieties (Brion 1990) — projective varieties on which a reductive group acts with a dense Borel orbit, of which toric varieties are the abelian-torus case. The image is a convex rational polytope in the dual of the maximal torus, with face structure encoding the geometry of the spherical embedding.

Symplectic reduction at the polytope vertices: the toric Morse theory

Theorem (Morse theory of $∣ μ_{T} ∣^{2}$ ; Kirwan 1984; Atiyah-Bott 1984). Let $X_{P}$ be a smooth projective toric variety with polytope $P$ and algebraic moment map $μ_{T} : X_{P} \to P$ . For a generic linear functional $ξ : M_{R} \to R$ , the function $f_{ξ} := ξ \circ μ_{T} : X_{P} \to R$ is a Morse-Bott function with critical points exactly the $T$ -fixed points of $X_{P}$ , indexed by vertices of $P$ . The Morse index of the critical point $p_{v}$ corresponding to vertex $v$ is twice the number of edges of $P$ at $v$ along which $ξ$ decreases from $v$ .

The Morse theory of moment-map components reduces the cell-decomposition of $X_{P}$ to the polytope: $X_{P}$ has a perfect Morse function whose cells correspond to vertices of $P$ , with cell dimensions read off the polytope face structure. This produces:

(i) The Białynicki-Birula decomposition of $X_{P}$ as a disjoint union of affine cells indexed by vertices of $P$ , with cell dimensions matching the Morse indices.

(ii) The Betti numbers of $X_{P}$ : $b_{2 k} (X_{P}) =$ (number of vertices of $P$ with Morse index $2 k$ ), and odd Betti numbers vanish ( $X_{P}$ has only even cells). For $P^{n}$ with the standard simplex, the $(n + 1)$ vertices have Morse indices $0, 2, 4, \dots, 2 n$ , recovering $b_{0} = b_{2} = \dots = b_{2 n} = 1$ .

(iii) The cohomology ring as a Stanley-Reisner quotient: $H^{*} (X_{P}; Q)$ is the Stanley-Reisner ring of the simplicial complex of cones of $Σ_{P}$ , modulo the linear relations from $M$ . This is the toric incarnation of the Goresky-Kottwitz-MacPherson theorem for equivariant cohomology of GKM spaces.

The moment-map picture is the bridge from the polytope combinatorics to the cohomology of $X_{P}$ : every cohomological invariant of $X_{P}$ has an interpretation in terms of vertices, edges, faces, and Morse-theoretic indices of the polytope under generic linear functionals.

Reflexive polytopes and the moment map of Calabi-Yau toric varieties

Theorem (moment map for reflexive polytopes; Batyrev 1994 J. Algebraic Geom. 3). Let $P$ be a reflexive lattice polytope in $M_{R}$ , with the origin as unique interior lattice point. The toric variety $X_{P}$ is a Gorenstein Fano toric variety with anticanonical line bundle $- K_{X_{P}} = L_{P}$ , and the algebraic moment map $μ_{T} : X_{P} \to P$ sends the origin $0 \in P$ to the "balanced point" of $X_{P}$ — the unique point where the anticanonical section sum vanishes. Generic anticanonical hypersurfaces $V \subset X_{P}$ are Calabi-Yau, and the restricted moment map $μ_{T} ∣_{V} : V \to \partial P$ takes the boundary of the polytope as its image.

The boundary-as-image observation is the key combinatorial property of Calabi-Yau hypersurfaces in toric Fano varieties: the moment-map shadow of a Calabi-Yau hypersurface is exactly the boundary of the polytope, with the interior accounting for the deformation parameter (the position of the hypersurface within the ambient $X_{P}$ ).

The Batyrev mirror duality $P \leftrightarrow P^{\circ}$ exchanges the role of the polytope and its polar dual, swapping the Calabi-Yau hypersurfaces $V \subset X_{P}$ and $V^{\lor} \subset X_{P^{\circ}}$ . The moment-map picture is the bridging structure: $V$ projects to $\partial P$ and $V^{\lor}$ projects to $\partial P^{\circ}$ , and the SYZ duality (Strominger-Yau-Zaslow 1996) realises the mirror pair as dual special-Lagrangian torus fibrations over the common boundary structure.

Synthesis. The algebraic moment map promotes the polytope-fan dictionary from a static combinatorial encoding of $(X_{P}, L_{P})$ to a dynamic geometric picture, with the polytope $P$ literally drawn inside $M_{R}$ as the continuous image of $X_{P}$ under the compact-torus moment map. The central insight is that the polytope $P$ is the complete moment-map shadow of $X_{P}$ , with the face stratification of $P$ matching the orbit-cone stratification of $X_{P}$ and the fibres of the moment map equal to compact-torus orbits. The bridge is the Atiyah-Guillemin-Sternberg convexity theorem, which guarantees image-equals-polytope for any Hamiltonian torus action on a compact symplectic manifold, specialised to the projective-toric case via the Demazure projective embedding by $L_{P}$ .

Three apparently distinct constructions fit into the moment-map picture as a single coherent structure. First, the orbit-cone correspondence 04.11.06 (faces of $P$ $\leftrightarrow$ orbits of $X_{P}$ ) is the underlying combinatorial-geometric pairing, expressed at the moment-map level as the face-stratified fibration of $X_{P}$ over $P$ . Second, the Demazure character formula 04.11.10 ( $H^{0} (X_{P}, L_{P}) = ⨁_{m} C \cdot χ^{m}$ over $m \in P \cap M$ ) is the algebraic-geometric input, expressed at the moment-map level as the lattice-point barycentric formula. Third, the Kempf-Ness theorem (GIT quotient $=$ symplectic quotient) is the algebraic-symplectic bridge, expressing $X_{P}$ as a residual quotient with the polytope as the residual moment-map image. Putting these together, the moment map is the universal geometric drawing of the polarised projective toric variety, with every algebraic-geometric invariant of $(X_{P}, L_{P})$ reading off a polytope-side computation through the moment map.

The pattern extends in three directions. To Calabi-Yau geometry, the reflexive-polytope condition of Batyrev produces toric varieties whose anticanonical hypersurfaces are Calabi-Yau, with the moment-map shadow of the hypersurface equal to the boundary of the polytope. To intersection theory, the Bernstein-Kushnirenko theorem 04.11.14 expresses top intersections of toric divisors as mixed volumes of polytopes, with the moment-map measure-pushforward identification of 05.04.05 supplying the symplectic-side proof via the Duistermaat-Heckman formula. To the equivariant cohomology of toric varieties, the moment-map Morse theory (Kirwan 1984) produces the Białynicki-Birula cell decomposition and recovers the Stanley-Reisner presentation of $H^{*} (X_{P}; Q)$ . The moment-map picture is the foundational geometric structure on top of which all of these calculations sit, with the polytope $P$ providing the universal combinatorial scaffold.

Full proof set [Master]

Theorem (image of the algebraic moment map), proof. Given in the Intermediate-tier section: the algebraic moment-map formula $μ_{T} (x) = \sum_{m} ∣ s_{m} (x) ∣^{2} / (\sum_{m^{'}} ∣ s_{m^{'}} (x) ∣^{2}) \cdot m$ equals the pullback of the Fubini-Study moment map via the Demazure embedding (Step 1); the $T$ -fixed points $p_{v}$ corresponding to vertices $v \in Vert (P)$ have $μ_{T} (p_{v}) = v$ (Step 2); fibres over interior points of a face $F \subseteq P$ are compact-torus orbits inside the algebraic orbit $O_{F}$ (Step 3); the image is convex by AGS, contains all vertices, and is contained in $P$ , hence equals $P$ (Step 4). $□$

Theorem (Kempf-Ness for toric varieties), proof. The Cox construction (Cox 1995 J. Algebraic Geom. 4, 17-50) realises $X_{P} = (C^{∣ Σ_{P} (1) ∣} ∖ Z (Σ_{P})) / /_{χ} G$ , with $G = Hom (Pic (X_{P}), C^{*})$ , character $χ$ corresponding to $[L_{P}]$ , and $Z (Σ_{P})$ the irrelevant locus. The compact subgroup $G_{R} = G \cap (S^{1})^{∣ Σ_{P} (1) ∣}$ acts on $C^{∣ Σ_{P} (1) ∣}$ by the restriction of the diagonal compact-torus action, with standard moment map $μ_{G} : C^{∣ Σ_{P} (1) ∣} \to Lie (G_{R})^{*}$ given by $μ_{G} (z) = \sum_{ρ} ∣ z_{ρ} ∣^{2} \cdot ρ^{*} - c$ for the central shift $c$ encoding $χ$ .

The Kempf-Ness theorem (Kempf-Ness 1979 LNM 732; Mumford-Fogarty-Kirwan 1994 §8.2) states that the $χ$ -semistable locus $W^{ss} (χ) \subset W = C^{∣ Σ_{P} (1) ∣}$ is exactly the set of points whose $G_{C}$ -orbit closure meets $μ_{G}^{- 1} (0)$ , and the natural map $W^{ss} (χ) / G_{C} \to μ_{G}^{- 1} (0) / G_{R}$ is a homeomorphism. Applied here: $W^{ss} (χ) = W ∖ Z (Σ_{P})$ (the unstable locus is the irrelevant locus), and the GIT quotient $X_{P} = W^{ss} (χ) / G_{C}$ is homeomorphic to $μ_{G}^{- 1} (0) / G_{R}$ , the symplectic quotient. The residual compact-torus action on $X_{P}$ (the quotient of the full $(S^{1})^{∣ Σ_{P} (1) ∣}$ -action by $G_{R}$ ) is the action whose moment map is $μ_{T} : X_{P} \to P$ of the main theorem. $□$

Theorem (image of the moment map for general $T$ -projective varieties), Brion 1987 statement, proof outline. For $X \subset P^{N}$ a projective variety with $T$ -action lifted to $O (1)$ , the moment map $μ_{T} : X \to M_{R}$ has image a convex rational polytope. The proof has three movements: (i) $μ_{T} (X)$ is the image of the moment map of the Hamiltonian $T_{R}$ -action on the compact symplectic manifold $X$ (with Fubini-Study form), so by Atiyah-Guillemin-Sternberg the image is convex; (ii) the image is contained in the convex hull of $T$ -weights of $H^{0} (X, O (1))$ (which is a rational polytope) by the formula $μ_{T} (x) = \sum_{m} ∣ s_{m} (x) ∣^{2} m / \sum_{m^{'}} ∣ s_{m^{'}} (x) ∣^{2}$ ; (iii) the image is exactly the convex hull of the moment-map fixed-point images, which is a sub-rational-polytope. The image is therefore a convex rational polytope, the moment polytope of $X$ . $□$

Theorem (Morse theory of $∣ μ_{T} ∣^{2}$ ), Kirwan 1984 statement, proof outline. The function $∣ μ_{T} ∣^{2} : X_{P} \to R_{\geq 0}$ is the squared-norm of the moment map. By Kirwan's main theorem (Kirwan 1984 Cohomology of Quotients §3), $∣ μ_{T} ∣^{2}$ is a Morse-Bott function on $X_{P}$ , with critical set the disjoint union of the connected components of moment-map preimages of the critical points of the squared-norm function on the image polytope $P$ . For toric $X_{P}$ , the critical points of $∣ μ_{T} ∣^{2}$ are exactly the $T$ -fixed points ${p_{v}}_{v \in Vert (P)}$ , with indices computed by the Morse theory on the polytope.

For a generic linear functional $ξ : M_{R} \to R$ , the function $f_{ξ} = ξ \circ μ_{T}$ has the same critical points (the fixed-point set), with indices equal to twice the number of edges at the vertex $v$ along which $ξ$ decreases. The cell decomposition produced by gradient flow of $- f_{ξ}$ is the Białynicki-Birula decomposition of $X_{P}$ . The cell of dimension $2 k$ at $p_{v}$ has closure equal to the orbit closure $\overline{O_{σ_{v}^{\leq k}}}$ for the dimension- $k$ truncation of the cone $σ_{v}$ , recovering the toric stratification. $□$

Theorem (Duistermaat-Heckman in the toric case), proof outline. The Duistermaat-Heckman theorem (Duistermaat-Heckman 1982 Invent. Math. 69; 05.04.05) states that the pushforward measure $(μ_{T})_{*} (ω_{P}^{n} / n!)$ on $image (μ_{T}) = P$ has piecewise-polynomial density. In the toric case, the density is constant: every interior fibre is the full compact torus $(S^{1})^{n}$ of volume $(2 π)^{n}$ , so the pushforward of the Liouville form is constant on $int (P)$ with value $(2 π)^{n}$ . The Liouville form $ω_{P}^{n} / n!$ has total volume $\int_{X_{P}} ω_{P}^{n} / n! = vol_{sym} (X_{P})$ , equal to the symplectic volume. The pushforward $(2 π)^{n} 1_{P} \cdot d m$ has total integral $(2 π)^{n} vol (P)$ in the lattice-Lebesgue measure, so $vol_{sym} (X_{P}) = (2 π)^{n} vol (P)$ , matching the Bernstein-Kushnirenko computation of the toric degree as $n! vol (P)$ . $□$

Theorem (moment map of the open torus orbit), Exercise 4 result, proof. Given in Exercise 4: the open torus orbit $T \subset X_{P}$ is the locus where every lattice-point section is nonzero, and the moment map restricted to $T$ surjects onto $int (P)$ with fibres compact-torus orbits. The surjectivity uses Carathéodory's theorem (every interior point of a convex hull of finite points has a strictly positive convex-coefficient representation); the fibre identification uses the phase-decomposition of the lattice-point sections. $□$

Connections [Master]

Polytope-fan dictionary 04.11.10. The prerequisite construction supplies the Demazure character formula $H^{0} (X_{P}, L_{P}) = ⨁_{m \in P \cap M} C \cdot s_{m}$ on which the lattice-point moment-map formula is built. The moment map is the geometric realisation of the polytope-fan dictionary: the polytope is not just a combinatorial label of $(X_{P}, L_{P})$ but is literally the image of $X_{P}$ under the compact-torus moment map. Without 04.11.10, the moment-map formula has no input data; with it, the moment map produces the polytope on the nose.
Fan and toric variety 04.11.04. The underlying toric variety $X_{P} = X_{Σ_{P}}$ is constructed from the normal fan $Σ_{P}$ via the fan-to-toric construction. The orbit-cone correspondence inside the fan structure is the combinatorial input to the moment-map stratification of $X_{P}$ by faces of $P$ .
Orbit-cone correspondence 04.11.06. The face structure of $P$ matches the orbit-cone stratification of $X_{P}$ : face $F$ of $P$ $\leftrightarrow$ cone $σ_{F}^{\lor}$ in $Σ_{P}$ $\leftrightarrow$ orbit $O_{F}$ of codimension $dim F$ in $X_{P}$ . The moment map realises this correspondence as a continuous fibration: $μ_{T}^{- 1} (relint (F)) = O_{F}$ , with fibre the compact torus $T_{R}^{F}$ . The moment map is the geometric realisation of the orbit-cone correspondence as a moment-fibre stratification.
Algebraic torus and character/cocharacter lattices 04.11.01. The compact torus $T_{R} = N \otimes_{Z} S^{1}$ inherits its acting-on- $X_{P}$ structure from the complex torus $T = N \otimes_{Z} C^{*}$ ; the moment-map image lives in the dual space $M_{R} = M \otimes_{Z} R$ . The polar duality $M \leftrightarrow N$ from 04.11.01 is reflected in the moment-map structure as the duality between the action lattice $N$ (where $T_{R}$ lives) and the image lattice $M$ (where $P$ lives).
Moment map (symplectic) 05.04.01. The algebraic moment map of this unit is the projective-toric incarnation of the general symplectic moment map of 05.04.01. The Demazure embedding $ι_{P} : X_{P} ↪ P^{∣ P \cap M ∣ - 1}$ pulls back the Fubini-Study symplectic form to $X_{P}$ , and the resulting Hamiltonian $T_{R}$ -action has moment map equal to $μ_{T}$ via Step 1 of the main theorem. The algebraic moment map is the toric specialisation of the symplectic moment map.
Atiyah-Guillemin-Sternberg convexity 05.04.03. The image-equals-polytope theorem of this unit is the toric specialisation of AGS convexity: any Hamiltonian action of a compact connected torus on a compact connected symplectic manifold has convex moment-map image equal to the convex hull of fixed-point images. For toric $X_{P}$ with the compact-torus action, the fixed-point set is ${p_{v}}_{v \in Vert (P)}$ and the convex hull of fixed-point images is exactly $conv (Vert (P)) = P$ . The toric case is the most explicit special case of AGS where the polytope is given a priori.
Delzant theorem 05.04.04. The Delzant correspondence classifies compact symplectic toric manifolds by Delzant polytopes (simple, rational, smooth). The algebraic moment map of this unit dovetails with the Delzant moment map on the projective-toric overlap: smooth projective toric varieties with their Fubini-Study polarisation correspond to Delzant polytopes via the moment-map identification $μ : X_{P} \to P$ . The Delzant unit constructs the symplectic toric manifold from the polytope; this unit constructs the algebraic moment map of the projective toric variety. The two constructions agree on the overlap of smooth projective toric varieties with Delzant polytopes.
Marsden-Weinstein symplectic reduction 05.04.02. The Kempf-Ness theorem identifies the GIT quotient $X_{P} = W / /_{χ} G_{C}$ with the symplectic quotient $μ_{G}^{- 1} (0) / G_{R}$ of 05.04.02. The moment-map calculus of this unit is the residual structure on the symplectic quotient — after quotienting by $G_{R}$ , the residual compact torus $T_{R}$ acts on $X_{P}$ with moment map $μ_{T}$ landing in the polytope $P$ . The Marsden-Weinstein construction is the input; the algebraic moment map is the output.
Duistermaat-Heckman theorem 05.04.05. The moment-map pushforward measure $(μ_{T})_{*} (ω_{P}^{n} / n!)$ on $P$ has constant Lebesgue density in the toric case, equal to $(2 π)^{n}$ . This is the toric specialisation of the Duistermaat-Heckman piecewise-polynomial density theorem, with the simplification arising from the constant compact-torus fibre volume.
GIT quotient and Kempf-Ness theorem 04.10.06. The GIT side of the Kempf-Ness correspondence sits in 04.10.06 (or a sibling Mumford-GIT unit): $X_{P} = C^{∣ Σ_{P} (1) ∣} / /_{χ} G$ is a specific GIT quotient with the Cox-ring presentation, and the Kempf-Ness theorem identifies this with the symplectic quotient via the moment map. The polytope is the polarisation datum on both sides.
Toric cohomology / Stanley-Reisner ring 04.11.12. The Morse theory of $∣ μ_{T} ∣^{2}$ on $X_{P}$ produces the Białynicki-Birula cell decomposition, which is the toric input to the Stanley-Reisner presentation $H^{*} (X_{P}; Q) = S R (Σ_{P}) / (linear relations from M)$ . The moment map is the bridge from polytope combinatorics to toric cohomology, with the cell decomposition reading off vertices, edges, and faces of $P$ .
Toric intersection theory and mixed volumes 04.11.13. The Duistermaat-Heckman computation of the symplectic volume of $X_{P}$ as $(2 π)^{n} vol (P)$ matches the algebraic computation of the degree of $X_{P}$ under the Demazure embedding as $n! vol (P)$ . The moment-map shadow is the geometric input to the Bernstein-Kushnirenko theorem 04.11.14 and to the mixed-volume formulas for top intersections of toric divisors.
Reflexive polytopes and Batyrev mirror duality 04.11.16. Reflexive polytopes produce toric Fano varieties whose anticanonical hypersurfaces are Calabi-Yau. The moment-map shadow of a generic Calabi-Yau hypersurface in $X_{P}$ is the boundary $\partial P$ of the polytope, with the interior accounting for the deformation parameter. The Batyrev mirror duality $P \leftrightarrow P^{\circ}$ exchanges the role of polytope and polar dual, and the SYZ duality realises the mirror Calabi-Yau pair as dual special-Lagrangian torus fibrations over the common combinatorial boundary structure.
Smoothness and completeness via fans 04.11.05. The sibling unit supplies the smoothness criterion (every cone is regular) that ensures $X_{P}$ is a smooth projective toric variety, hence a smooth compact symplectic manifold for the application of AGS convexity. For non-smooth $X_{P}$ (when $P$ is non-Delzant), the moment-map calculus still works via the algebraic lattice-point formula, but the symplectic form develops mild singularities along the boundary toric strata. The toric resolution 04.11.07 provides the smooth-resolution input when smoothness is needed.
Toric Picard group 04.11.09. The Picard group $Pic (X_{P})$ houses the ample line bundle $L_{P}$ ; different choices of $L_{P}$ within the ample cone correspond to different polytopes $P$ (with the same normal fan). The moment-map polytope is the polytope of the chosen polarisation; the choice of polarisation is the input that converts $X_{P}$ from an unpolarised toric scheme to a polarised projective toric variety with moment-map shadow.

Historical & philosophical context [Master]

The algebraic moment map for projective toric varieties was developed in parallel by two streams of mathematics in the early 1980s. The symplectic-side convexity theorem was proved independently by Michael Atiyah, in Convexity and commuting Hamiltonians (Bulletin of the London Mathematical Society 14, 1982, pp. 1-15) ^[Atiyah1982], and by Victor Guillemin and Shlomo Sternberg, in Convexity properties of the moment mapping (Inventiones Mathematicae 67, 1982, pp. 491-513) ^{[GuilleminSternberg1982]}, in two papers appearing in the same year. Atiyah's argument used the local Morse theory of moment-map components together with a connectedness argument for fibres of the moment map; Guillemin-Sternberg's argument used a direct geometric analysis of the polytope structure. Both proofs converge on the same theorem: the image of a Hamiltonian compact-connected-torus action on a compact connected symplectic manifold is a convex polytope, equal to the convex hull of the moment-map images of the fixed-point set.

The algebraic-geometric side was developed by William Fulton in Introduction to Toric Varieties (Princeton Annals of Mathematics Studies 131, 1993) ^[Fulton1993], Chapter 4. Fulton presents the moment map of a projective toric variety in its most concrete form, via the lattice-point formula on the Demazure embedding, and proves the image-equals-polytope theorem as a direct consequence of the polytope-fan dictionary plus AGS convexity. Cox-Little-Schenck Toric Varieties (AMS GSM 124, 2011) ^[pending] §12.2-§12.3 gives the modern textbook treatment with the explicit lattice-point formula and the GIT identification via Kempf-Ness, supplemented by the Cox-ring perspective. Michèle Audin's Topology of Torus Actions on Symplectic Manifolds (Progress in Mathematics 93, Birkhäuser 1991) ^[pending] develops the moment-map theory of toric varieties from the symplectic side, with Chapters VII-VIII giving the parallel exposition to Fulton's Chapter 4.

The GIT-symplectic correspondence was proved by George Kempf and Linda Ness in The length of vectors in representation spaces (Algebraic Geometry, Copenhagen 1978, Lecture Notes in Mathematics 732, Springer 1979, pp. 233-243) ^{[KempfNess1979]}. Kempf-Ness identified the semistable locus of a linearised reductive group action on a projective variety with the preimage of the moment-map level set under the compact form, and proved that the GIT quotient is homeomorphic to the symplectic quotient. Frances Kirwan's Cohomology of Quotients in Symplectic and Algebraic Geometry (Princeton Mathematical Notes 31, 1984) ^[Kirwan1984] extended the Kempf-Ness theorem with the Morse theory of $∣ μ ∣^{2}$ , supplying the cohomological calculus of GIT quotients and toric varieties. Mumford-Fogarty-Kirwan Geometric Invariant Theory (Ergebnisse 34, 3rd ed., Springer 1994) ^[MFK1994] is the standard reference for the GIT side, with the Kempf-Ness theorem as a foundational tool.

Michel Brion's Sur l'image de l'application moment (Bulletin de la SMF 115, 1987, pp. 455-472) ^[Brion1987] extended the image-of-the-moment-map theorem to general projective varieties carrying a torus action: the moment-map image is always a convex rational polytope, called the moment polytope, even when the variety is not toric. For toric varieties this recovers the polytope-fan-dictionary polytope; for non-toric varieties it produces a new combinatorial invariant capturing the projective embedding. Brion's later work on spherical varieties extended the moment-polytope picture to reductive-group actions with a dense Borel orbit.

The polytope-as-moment-map-shadow picture is one of the conceptual cornerstones of late-20th-century geometry, unifying symplectic geometry, algebraic geometry, and combinatorics under a single drawing. The polytope $P$ is simultaneously: a lattice-combinatorial object encoding the Demazure character formula and the Ehrhart polynomial; an algebraic-geometric object polarising the projective toric variety $X_{P}$ via the ample line bundle $L_{P}$ ; a symplectic-geometric object equal to the image of the Hamiltonian compact-torus moment map; and a topological object stratifying $X_{P}$ via the orbit-cone correspondence and the Białynicki-Birula cell decomposition. The Atiyah-Guillemin-Sternberg-Kempf-Ness-Brion synthesis is the foundational geometric structure on top of which modern toric geometry, GIT, equivariant cohomology, and toric mirror symmetry all sit, with the algebraic moment map as the structural backbone.

Bibliography [Master]

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  author  = {Atiyah, Michael F.},
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  volume  = {14},
  year    = {1982},
  pages   = {1--15}
}

@article{GuilleminSternberg1982,
  author  = {Guillemin, Victor and Sternberg, Shlomo},
  title   = {Convexity properties of the moment mapping},
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  year    = {1982},
  pages   = {491--513}
}

@book{FultonToric,
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}

@book{CoxLittleSchenck,
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}

@book{AudinTorus,
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}

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}

@book{Kirwan1984,
  author    = {Kirwan, Frances},
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  volume    = {31},
  year      = {1984}
}

@book{MFK1994,
  author    = {Mumford, David and Fogarty, John and Kirwan, Frances},
  title     = {Geometric Invariant Theory},
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  volume    = {34},
  edition   = {3},
  year      = {1994}
}

@article{Delzant1988,
  author  = {Delzant, Thomas},
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}

@article{Brion1987,
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  year    = {1987},
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}

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}

@book{GuilleminSternbergQuantization,
  author    = {Guillemin, Victor and Sternberg, Shlomo},
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}

@article{Cox1995,
  author  = {Cox, David A.},
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}

Prerequisites

04.11.01
04.11.04
04.11.10
05.04.01
05.04.03

Tier anchors

beginner: Fulton *Introduction to Toric Varieties* §4.1-§4.2 informal — the moment map of a projective toric variety as the picture that draws the polytope inside the character space
intermediate: Fulton *Introduction to Toric Varieties* §4.2; Cox-Little-Schenck *Toric Varieties* §12.2; Audin *Topology of Torus Actions* Ch. VII §1-§3
master: Fulton §4.2 + §5.2 (cohomology and the moment polytope); Cox-Little-Schenck §12.2 + §12.3; Audin *Topology of Torus Actions on Symplectic Manifolds* Ch. VII-VIII; Atiyah 1982 *Bull. London Math. Soc.* 14, 1-15 (convexity and commuting Hamiltonians); Guillemin-Sternberg 1982 *Invent. Math.* 67, 491-513 (convexity properties of the moment map); Guillemin-Sternberg 1984 *Geometric Quantization and Multiplicities of Group Representations* (Invent. Math. 67); Kempf-Ness 1979 *Lecture Notes in Math.* 732 (length functions in algebraic transformation groups, GIT-stability and the Kempf-Ness theorem); Kirwan 1984 *Cohomology of Quotients in Symplectic and Algebraic Geometry* (Princeton MN 31); Mumford-Fogarty-Kirwan 1994 *Geometric Invariant Theory* (Ergebnisse 34, 3rd ed.); Delzant 1988 *Bull. SMF* 116, 315-339 (symplectic-side classification); Brion 1987 *Bull. SMF* 115, 455-472 (image of the moment map for projective varieties)

References

TODO_REF
Fulton — Introduction to Toric Varieties · §4.1 (action of the compact torus on a projective toric variety); §4.2 (the moment map; image equals the polytope $P$); §5.2 (Duistermaat-Heckman and the moment polytope)
TODO_REF
Cox-Little-Schenck — Toric Varieties · §12.2 (moment maps and toric varieties); §12.3 (convexity of moment map images and the polytope); §14.4 (GIT, the Kempf-Ness theorem, and symplectic reduction)
TODO_REF
Audin — Topology of Torus Actions on Symplectic Manifolds · Ch. VII §1-§3 (moment maps for projective toric varieties; the image-equals-polytope theorem; Fubini-Study form pulled back via $L_P$); Ch. VIII (Delzant correspondence and the symplectic-algebraic-geometric overlap)
TODO_REF
Atiyah 1982 — Convexity and commuting Hamiltonians · *Bull. London Math. Soc.* 14, 1-15; first half of the convexity theorem: image of a Hamiltonian torus moment map on a compact connected symplectic manifold is a convex polytope (the convex hull of the moment-map images of fixed points)
TODO_REF
Guillemin-Sternberg 1982 — Convexity properties of the moment mapping · *Invent. Math.* 67, 491-513; companion half of the convexity theorem, independent and simultaneous with Atiyah 1982; the fibres of the moment map are connected
TODO_REF
Kempf-Ness 1979 — Length functions in algebraic transformation groups · Lecture Notes in Mathematics 732, Springer, 233-243; Kempf-Ness theorem identifying the GIT quotient with the symplectic quotient (zero-level set of the moment map mod compact torus) for a projective toric variety
TODO_REF
Kirwan 1984 — Cohomology of Quotients in Symplectic and Algebraic Geometry · Princeton Mathematical Notes 31, Princeton University Press; equivariant cohomology of the moment map, Morse theory for $|\mu|^2$, identification of stable / semistable strata with moment-map level sets
TODO_REF
Delzant 1988 — Hamiltoniens périodiques et image convexe de l'application moment · *Bulletin de la Société Mathématique de France* 116, 315-339; symplectic-side classification of compact symplectic toric manifolds by Delzant polytopes, with the moment map as the structural backbone
TODO_REF
Brion 1987 — Sur l'image de l'application moment · *Bulletin de la Société Mathématique de France* 115, 455-472; image-of-the-moment-map theorem for projective varieties with a torus action, generalising the toric case to spherical varieties
TODO_REF
Mumford-Fogarty-Kirwan 1994 — Geometric Invariant Theory · Ergebnisse der Mathematik 34 (3rd ed.), Springer; GIT quotients $X /\!/ G$ and their identification with symplectic quotients via Kempf-Ness; foundational reference for the algebraic-geometric side of the moment-map correspondence
TODO_REF
Guillemin-Sternberg 1984 — Symplectic Techniques in Physics · Cambridge University Press; textbook synthesis covering the Atiyah-Guillemin-Sternberg convexity theorem, the Duistermaat-Heckman formula, and the algebraic-symplectic correspondence on toric varieties

Reviewer

TBD

Estimated time

beginner: 20m
intermediate: 45m
master: 80m