--- date: 2022-01-15 21:49 modification date: Saturday 12th February 2022 01:03:28 title: orbifold aliases: [orbifolds] --- Last modified date: <%+ tp.file.last_modified_date() %> --- - Tags: - #higher-algebra/stacks #todo/too-long - Refs: - - Links: - [isotropy group](isotropy%20group) - [global homotopy theory](global%20homotopy%20theory) - [stacks MOC](Unsorted/stacks%20MOC.md) - [orbifold fundamental group](orbifold%20fundamental%20group.md) - [crepant resolution](crepant%20resolution) - [deformation](Unsorted/deformation.md) - [Thom form](Thom%20form.md) - [orbifold fundamental group](orbifold%20fundamental%20group) - [Chern-Ruan theory](Chern-Ruan%20theory.md) - [Unsorted/string topology](Unsorted/string%20topology.md) - [Bredon cohomology](Bredon%20cohomology.md) - [equivariant K theory](equivariant%20K%20theory) - [Twisted K theory](Unsorted/Twisted%20K%20theory.md) - [quantum cohomology](quantum%20cohomology.md) - [elliptic complex](Unsorted/elliptic%20complex.md) - [Gromov-Witten invariants](Gromov-Witten%20invariants) - [quantum cohomology](Unsorted/quantum%20cohomology.md) - [inertia orbifold](Unsorted/string%20topology.md) --- # orbifold # Definitions ![](attachments/Pasted%20image%2020220806140019.png) ![](attachments/Pasted%20image%2020220806140030.png) ![](attachments/Pasted%20image%2020220806140043.png) # Misc ![](attachments/Pasted%20image%2020220503233129.png) Idea: "differentiable" [Deligne-Mumford stack](Deligne-Mumford%20stack). Behave like smooth objects in some sense, even when the realization space is singular. Locally modeled as the quotient of a smooth manifold by a finite group. Special case: quotient spaces $M/G$ for $G$ a compact [Lie group](Unsorted/Lie%20group.md) acting smoothly on a smooth manifold. In this case $\ccxH(M/G) \cong \ccxH_G(M)$, where the first is [orbifold cohomology](orbifold%20cohomology) and the second is $G\dash$[equivariant homology](Unsorted/equivariant%20cohomology.md). Every classical $n$-orbifold $\mathcal{X}$ is diffeomorphic to a quotient orbifold for a smooth, effective, and "almost free" $O(n)$-action on a smooth manifold $M$. There is a notion of an *orbibundle* over an orbifold, and $\T M$ is an example. The sections correspond to $\Orth_n\dash$equivariant sections of the tangent bundle to the frame orbibundle $\Frame(M) \to M$. As a category: ![](attachments/Pasted%20image%2020220213021314.png) ![](attachments/Pasted%20image%2020220213021240.png) Pullbacks: ![](attachments/Pasted%20image%2020220213021450.png) # Classical concepts - Riemannian metrics: sections of $\Sym^2(\T X)$ - Almost-complex structures: bundle morphisms $J \in \Endo(\T X)$ with $J^2 = -\id$, so $J = \id^{1\over 4}$. - The de Rham complex: forms are $\drcomplex_X = \globsec{ \Extcomplex \T\dual X}$, exterior derivative is defined the same way. ![](attachments/Pasted%20image%2020220212220206.png) - Symplectic structures: $\omega \in \Omega^2_X$ - Dolbeaut cohomology: require all defining data to be holomorphic. - The canonical: $K_X \da \Extpower^{\dim M}\T\dual X\slice \CC$, whose fibers are of the form $\CC/G_x$ where $G_x$ acts by the determinant. Not necessarily a line bundle unless $G_x \in \SL_{\dim M}(\CC)$ for all $x$. - Calabi-Yau: $K_X$ is trivial. # Definitions - An $n$-dimensional complex orbifold $X$ is **Gorenstein** if all the local groups $G_{x}$ are subgroups of $S L_{n}(\mathbb{C})$. - Let $\mathcal{G}$ be a Lie groupoid. For a point $x \in G_{0}$, the set of all arrows from $x$ to itself is a Lie group, denoted by $G_{x}$ and called the **isotropy or local group** at $x$. - The set $t s^{-1}(x)$ of targets of arrows out of $x$ is called the **orbit** of $x$. - The **orbit space** $|\mathcal{G}|$ of $\mathcal{G}$ is the quotient space of $G_{0}$ under the equivalence relation $x \sim y$ if and only if $x$ and $y$ are in the same orbit. - Conversely, we call $\mathcal{G}$ a **groupoid presentation** of $|\mathcal{G}|$. - Definitions: for $\mathcal{G}$ a Lie groupoid, - $\mathcal{G}$ is **proper** if $(s, t): G_{1} \rightarrow G_{0} \times G_{0}$ is a proper map. Note that in a proper Lie groupoid $\mathcal{G}$, every isotropy group is compact. - $\mathcal{G}$ is a **foliation groupoid** if each [isotropy group](isotropy%20group) $G_{x}$ is discrete. - $\mathcal{G}$ is [etale](etale.md) if $s$ and $t$ are local diffeomorphisms. If $\mathcal{G}$ is an étale groupoid, we define its **dimension** $\operatorname{dim} \mathcal{G}=\operatorname{dim} G_{1}=\operatorname{dim} G_{0}$. - Regard a Lie group $G$ as a groupoid $\mcg$ having a single object, then $\mcg$ is a proper étale groupoid if and only if $G$ is finite. We call such groupoids **point orbifolds**, and denote them by $\pt^{G}$. - Two Lie groupoids $\mathcal{G}$ and $\mathcal{G}^{\prime}$ are **Morita equivalent** if there exists a span: third groupoid $\mathcal{H}$ and two equivalences $$ \mathcal{G} \swarrow \mathcal{H} \searrow \mathcal{G}^{\prime} . $$ - An **orbifold groupoid** is a proper étale Lie groupoid. - An orbifold groupoid $\mathcal{G}$ is **effective** if for every $x \in G_{0}$ there exists an open neighborhood $U_{x}$ of $x$ in $G_{0}$ such that the associated homomorphism $G_{x} \rightarrow \operatorname{Diff}\left(U_{x}\right)$ is injective. - An **orbifold groupoid** sometimes refers to a proper foliation Lie groupoids. Up to "Morita equivalence" this is equivalent. - The **inertia groupoid**: ![](attachments/Pasted%20image%2020220212224153.png) In terms of the [action groupoid](action%20groupoid): ![](attachments/Pasted%20image%2020220213021822.png) # Examples ![](attachments/Pasted%20image%2020220806140117.png) ## The Kummer surface ![](attachments/Pasted%20image%2020220212021959.png) ![](attachments/Pasted%20image%2020220212022025.png) ![](attachments/Pasted%20image%2020220213020123.png) ## The mirror quintic See [mirror quintic](mirror%20quintic.md) ![](attachments/Pasted%20image%2020220212022056.png) ## Weighted projective spaces See [weighted projective space](weighted%20projective%20space) ![](attachments/Pasted%20image%2020220212022201.png) ![](attachments/Pasted%20image%2020220212190058.png) ## Moduli of elliptic curves See [moduli stack of elliptic curves](Unsorted/moduli%20stack%20of%20elliptic%20curves.md): ![](attachments/Pasted%20image%2020220212022339.png) [complete intersections](complete%20intersections) of [toric](Unsorted/toric.md). Arithmetic orbifolds: ![](attachments/Pasted%20image%2020220212022901.png) ![](attachments/Pasted%20image%2020220212022911.png) # Notes - Note that every étale groupoid is a foliation groupoid. - A Lie groupoid is a foliation groupoid if and only if it is Morita equivalent to an etale groupoid. - If $\mathcal{G}$ is a Lie groupoid, then for any $x \in G_{0}$ the isotropy group $G_{x}$ is a Lie group. - If $\mathcal{G}$ is proper, then every isotropy group is a compact Lie group. - In particular, if $\mathcal{G}$ is a proper foliation groupoid, then all of its isotropy groups are finite. - Given an orbifold $\mathcal{X}$, with underlying space $X$, its structure is completely described by the Morita equivalence class of an associated effective orbifold groupoid $\mathcal{G}$ such that $|\mathcal{G}| \cong X .$ ![](attachments/Pasted%20image%2020220212184651.png) - Defining homotopy invariants: take the [classifying space](Unsorted/classifying%20space.md): ![](attachments/Pasted%20image%2020220212185614.png) ![](attachments/Pasted%20image%2020220212185835.png) ![](attachments/Pasted%20image%2020220212185846.png) ![](attachments/Pasted%20image%2020220212185744.png) - $K_{X}$ is an orbibundle with fibers of the form $\mathbb{C} / G_{x}$, where $G_{x}$ acts through the determinant. - The Gorenstein condition is necessary for a [crepant resolution](crepant%20resolution) to exist - Satisfied automatically by Calabi-Yau orbifolds. - Quotients by subgroups of [SL2](Unsorted/SL2.md): ![](attachments/Pasted%20image%2020220212213149.png) If $X$ is a Calabi-Yau orbifold and $(Y, f)$ is a [crepant resolution](crepant%20resolution) of $X$, then $Y$ has a family of [Ricci-flat](Ricci-flat.md) [Kahler metrics](Kahler%20metrics) which make it into a [Calabi-Yau manifold](Unsorted/Calabi-Yau%20manifold.md). In the particular case where $X$ is the quotient $\mathbb{T}^{4} /(\mathbb{Z} / 2 \mathbb{Z})$, then the Kummer construction gives rise to a crepant resolution that happens to be the [K3 surface](K3%20surface.md). ## Orbifold pi_1 Covers: ![](attachments/Pasted%20image%2020220212234616.png) ![](attachments/Pasted%20image%2020220212234805.png) ![](attachments/Pasted%20image%2020220212234541.png) One recovers the theory of [Hurwitz covers](Hurwitz%20covers) as the theory of representable orbifold morphisms from an orbifold Riemann surface to $\pt^{S_n}$. Any effective orbifold can be expressed as the quotient of a smooth manifold by an almost free action of a compact Lie group. Therefore, we can use methods from equivariant topology to study the K-theory of effective orbifolds ## Orbifold K_0 ![](attachments/Pasted%20image%2020220213024028.png) ![](attachments/Pasted%20image%2020220213024110.png) ![](attachments/Pasted%20image%2020220213024412.png) ![](attachments/Pasted%20image%2020220213024427.png) ![](attachments/Pasted%20image%2020220213024457.png) ## Orbifold Euler characteristic ![](attachments/Pasted%20image%2020220213024204.png) Setting $R\left(G_{\sigma}\right)$ to be the complex [representation ring](representation%20ring.md) of the stabilizer of $\sigma$ in $M$, ![](attachments/Pasted%20image%2020220213024314.png) # Bundles ![](attachments/Pasted%20image%2020220806140234.png) ![](attachments/Pasted%20image%2020220806140303.png)