--- date: 2023-02-15 14:37 aliases: ["Untitled"] --- Last modified: `=this.file.mday` --- - Tags: - #todo/untagged - Refs: - #todo/add-references - Links: - #todo/create-links --- # Symplectic Moment maps for symplectic manifolds: - $(M, \omega)$ symplectic, $R: G\to \Symp(X)$ a symplectic action by a Lie group. - Take $\beta: \lieg\dual \tensor\lieg\to \RR$ the natural pairing - Take a map $\rho: \lieg\to \Gamma(\T_X)$ where $\xi\in \lieg$ induces the vector field which is pointwise given by $$\rho(\xi)_p = \dd{}{t}\mid_{t=0} \exp(t\xi)\cdot p$$ for $p \in M$ where $\exp:\lieg\to G$ and $\cdot$ is the $G\dash$action $R$ on $M$. - Define $\mu: M\to \lieg\dual$ the almost unique map satisfying $d(x\mapsto \beta(\mu(x), \xi)) = \iota_{\rho(\xi)}\omega$. - I.e. $\iota_{\rho(\xi)}\omega$ is closed and exact and thus "integrable", $\mu$ is its antiderivative. Alternatively - $\xi \in \Gamma(\T_M)$ is a Hamiltonian vector field if $\iota_\xi \omega$ is exact. - $H\in C^\infty(M, \RR)$ is a Hamiltonian function for $\xi$ if $\iota_\xi \omega = dH$. - For $\psi: G\to \Aut(M)$ a smooth action, $\mu: M\to \lieg\dual$ satisfies: - The dual function $\mu^\xi: M\to \RR$ where $\mu^\xi(p) \da \inp{\mu(p)}{\xi}$ satisfies $d\mu^\xi = \iota_\xi \omega$ - $\mu$ is equivariant wrt the $G$ action and the coadjoint action on $\lieg\dual$: ![](attachments/2023-02-14-coadjoint.png) - Note: $\Ad: G\to \Aut(\lieg)$ is $g\mapsto \Ad_g$ which is $(d\Psi_g)_e$ for $\Psi_g(h) = ghg\inv$ and $\Ad^*: G\to \Aut(\lieg\dual)$ is the formal adjoint satisfying $\inp{\Ad^*_g a}{b} = \inp{a}{\Ad^*_{g\inv} b}$. - For $G$ a torus, $\Ad^*$ is trivial and this becomes $G\dash$invariance of $\mu$: $\mu \circ \psi_g = \mu$, and $\lieg\dual \cong \RR^{\dim G}$. # Symplectic Examples - Example - Take $(X,\omega) = (S^2, d\theta \wedge dh)$, let $S^1 \actson X$ by $z\dash$axis rotations, so for $s\in S^1$ take $(\theta, h) \mapsto (\theta + s, h)$. - The associated vector field is $\xi = \dd{}{\theta}$. - Solve $d\mu^\xi = \iota_\xi \omega \implies \iota_\xi(d\theta \wedge dh) = (\iota_\xi d\theta) \wedge dh - d\theta \wedge (\iota_\xi dh) = dh$. - Use $(\iota_\xi d\theta) \wedge dh - d\theta \wedge (\iota_\xi dh) =d\theta (\dd{}{\theta}) \wedge dh - d\theta \wedge dh(\dd{}\theta) = 1\wedge dh - d\theta\wedge 0$. - So $h$ is the height function and $\im \mu = [-1, 1]$. - ![](attachments/2023-02-14-sphere.png) - **Theorem (convexity)**: if $X$ is compact, connected, with a Hamiltonian torus action $\psi: T^d\actson X$ with moment map $\mu: X\to \liet\dual \cong \RR^d$, $\mu(X)$ is the convex hull of $\mu(\Fix(\psi))$ where $\Fix(\psi) \da \ts{p\in X\st \psi_g(p) = p\, \forall g\in G} = \Intersect_{g\in G}\Fix(\psi_g)$. Thus the image is a convex polytope called the **moment polytope**. - **Theorem (Delzant)**: The moment map induces a bijection between symplectic toric manifolds (up to $T\dash$equivariant symplectomorphisms) and Delzant polytopes (up to homothety, so $\SL_n(\ZZ)$). - A convex polytope $P\subseteq \RR^n$ with a fixed lattice $\Lambda$ is **Delzant** iff - Simplicity: Each vertex has valence $\dim P$ at each vertex $p_i$ - Rationality: All edges adjacent to $p$ are of the form $p_i+tv_i$ for some $v_i \in \Lambda$. - Smoothness: the $v_i$ can be choosen to form a $\ZZ\dash$basis of $\Lambda$. # AG Setup - Generalize: the image is lattice in $\ZZ^1\subseteq \RR^1$, and $S^2\cong \CP^1$. ![](attachments/2023-02-14-toricvars.png) - Write $\PP^2 = \ts{\abs{z_1}^2 + \abs{z_2}^2 + \abs{z_3}^2 = 1}/S^1 \subseteq \CC^3$, then $x+y+z=1 \implies z=1-x-y$ with $z\geq 0$ is a triangle. - **Definition**: a toric variety over $\mathbb{C}$ is a complex algebraic variety with an action of $\left(\mathbb{C}^{\times}\right)^n$ and a dense open subset isomorphic to $\left(\mathbb{C}^{\times}\right)^n$ carrying the regular action. That is, a toric variety is an algebraic torus orbit closure. - If $P$ is a polytope, points $u\in P \intersect M$ correspond to $\chi^u$ and assemble to a morphism $\phi: X_P \to \PP^{r-1}$. - Defines a moment map $\mu: X\to M_\RR$ defined by $$\mu(x) = {1\over \sum\abs{\chi^u(x)}} \sum_{u\in P\intersect M}\abs{\chi^u(x)}u$$ which induces a homeomorphism $X_{\geq } \cong P$ where $X_P$ retracts onto $X_{\geq}$. - **Theorem**: if $X_P$ is the toric variety associated to a polytope $P$, then $X_P\to X_P/(S^1)\cartpower{n}$ is the moment map to $P \subseteq (\liet^n)\dual \cong \RR^n$. - Constructing affine varieties from cones: - $\sigma \subset N_\RR$ a cone. - $\sigma\dual \subset M_\RR$ its dual cone - $S_\sigma$ the monoid it generates - $\CC[S_\sigma] = \ts{\sum_{m\in S_\sigma} c_m \chi^m \st c_m\in \CC, c_m=0 \, a.e.}$ its monoid algebra $\in \calg$ - $X_{\sigma} \da \spec \CC[S_\sigma]$ - From a polytope: ![](attachments/2023-02-14-image-closure.png) # Examples - Example: $e_2, 2e_1 -e_2$ goes to $e_1 + 2e_2, e_1$ with three generators $e_1, e_1+e_2, e_1 + 2e_2$ corresponding to $\CC[x, xy, xy^2] = \CC[x_1, x_2, x_3]/\gens{x_1x_3 - x_2^2}$, so the equation $xz=y$ in $\CC^3$. ![](attachments/2023-02-15-duals.png) ![](attachments/2023-02-15-plot.png) ![](attachments/2023-02-15-ex.png) # Properties Properties: - Complete if $\abs{\Sigma} = \Lambda_\RR$. - Always true in dimension 2, so 4-folds. - Polytope = convex hull of a finite number of points in $\Lambda_\RR$. - Polytopal if $\Sigma$ is spanned by the faces of a polytope. E.g. $\PP^2$. - $\Sigma$ smooth if every cone $\sigma_i = \Cone(S_i)$ is smooth, so $S_i$ is the $\ZZ\dash$basis of a sublattice of $N$. - **Theorems**: - $\Sigma$ complete $\iff X_\Sigma$ compact - $\Sigma$ smooth $\iff X_{\Sigma}$ smooth - $\Sigma$ polytopal $\iff X_\Sigma$ projective - **Fact**: if $S_i$ is the $\RR\dash$basis of a subspace of $N_\RR$, then $\sigma_i$ is **simplicial**. - Simplicial iff orbifold (at worst quotient singularities). - ![](attachments/2023-02-14simplicialnot.png) - **Fact**: every toric variety has a resolution of singularities by another toric variety. Blowups. - **Fact**: $K_{X} = -\sum D_{\rho_i}$. - Picard groups: ![](attachments/2023-02-14-m.png) - Generally $M\injects \ZZ^{\Sigma(1)} \surjects \Cl(X)$ so $\rho = \size \Sigma(1) - \dim_\ZZ M$. - **Lattices**: - $N \da \Hom_\Grp(\CC\units, (\CC\units)^n)$ is the cocharacter group, 1 parameter subgroups - $N$ forms a lattice: ![](attachments/2023-02-14-lattice1.png) - $N_\RR \da N\tensor_\ZZ \RR, N_\CC$, etc. - $M \da \Hom_\Grp((\CC\units)^n, \CC\units)$ is the group of characters: - ![](attachments/2023-02-14-characters1.png) - ![](attachments/2023-02-14-pairing.png) - Cones: ![](attachments/2023-02-14cone.png) - Orbit-Cone: ![](attachments/2023-02-14orbcone.png) # More AG Examples - Examples: - $\PP^1$: ![](attachments/2023-02-14p1.png) - $\CC^2$ and $\Bl_1 \CC^2$: ![](attachments/2023-02-14-bl1.png) - Hirzebruch surface: ![](attachments/2023-02-14hirz.png) - $\PP^2$: ![](attachments/2023-02-14-cones.png) - ![](attachments/2023-02-14coords.png) - Quadric cone ![](attachments/2023-02-14quadcone.png) - $\PP^1\times \PP^1$. # Constructions Polar duals: ![](attachments/2023-02-15-polar.png) ![](attachments/2023-02-15-reflex.png) ![](attachments/2023-02-15-polar-1.png) - Appearance in physics: gauged linear sigma models. - QFT: maps $\Sigma\to \CC^n$ with a compact $T^d$ action - "Ground states" (set potential energy to zero) identify with fibers of the moment map for $\CC_n$ mod $T^d$ (gauge equivalence) produces $X_P$ a toric variety. - Duality for $P, P^\circ$ gives rise to mirror symmetry. - $n=2$ and $X_P = \PP^1$ is the Hopf fibration ![](attachments/2023-02-15-scfts.png) Dwork![](attachments/2023-02-15-asdasdsa.png) # Ongoing - Almost toric fibrations: ![](attachments/2023-02-14-polytope.png) - How to give connect sum presentation for a complex surface?