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• Tags:
  • #higher-algebra/stacks #todo/too-long
• Refs:
  • https://www.math.colostate.edu/~renzo/teaching/Orbifolds/Ruan.pdf
• Links:
  • isotropy group
  • global homotopy theory
  • stacks MOC
  • orbifold fundamental group
  • crepant resolution
  • deformation
  • Thom form
  • orbifold fundamental group
  • Chern-Ruan theory
  • Unsorted/string topology
  • Bredon cohomology
  • equivariant K theory
  • Twisted K theory
  • quantum cohomology
  • elliptic complex
  • Gromov-Witten invariants
  • quantum cohomology
  • inertia orbifold

orbifold

Definitions

Misc

Idea: “differentiable” Deligne-Mumford stack. 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 acting smoothly on a smooth manifold. In this case \({ { {H}^{\scriptscriptstyle \bullet}} }(M/G) \cong { { {H}^{\scriptscriptstyle \bullet}} }_G(M)\), where the first is orbifold cohomology and the second is \(G{\hbox{-}}\)equivariant homology.

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 \({\mathbf{T}}M\) is an example. The sections correspond to \({\operatorname{O}}_n{\hbox{-}}\)equivariant sections of the tangent bundle to the frame orbibundle \(\mathop{\mathrm{Frame}}(M) \to M\).

As a category: Pullbacks:

Classical concepts

• Riemannian metrics: sections of \(\operatorname{Sym}^2({\mathbf{T}}X)\)
• Almost-complex structures: bundle morphisms \(J \in { \operatorname{End} }({\mathbf{T}}X)\) with \(J^2 = -\operatorname{id}\), so \(J = \operatorname{id}^{1\over 4}\).
• The de Rham complex: forms are \({ { {\Omega}^{\scriptscriptstyle \bullet}} }_X = {{\Gamma}\qty{ { { { {\bigwedge}^{\scriptscriptstyle \bullet}} }^{\scriptscriptstyle \bullet}} {\mathbf{T}} {}^{ \vee }X} }\), exterior derivative is defined the same way.
• Symplectic structures: \(\omega \in \Omega^2_X\)
• Dolbeaut cohomology: require all defining data to be holomorphic.
• The canonical: $K_X \coloneqq\bigwedge\nolimits^{\dim M}{\mathbf{T}} {}^{ \vee }X_{/ {{\mathbf{C}}}} $, whose fibers are of the form \({\mathbf{C}}/G_x\) where \(G_x\) acts by the determinant. Not necessarily a line bundle unless \(G_x \in {\operatorname{SL}}_{\dim M}({\mathbf{C}})\) 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 \(G_{x}\) is discrete.
  • \(\mathcal{G}\) is etale 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 \({\mathcal{G}}\) having a single object, then \({\mathcal{G}}\) is a proper étale groupoid if and only if \(G\) is finite. We call such groupoids point orbifolds, and denote them by \({\operatorname{pt}}^{G}\).
• Two Lie groupoids \(\mathcal{G}\) and \(\mathcal{G}^{\prime}\) are Morita equivalent if there exists a span: third groupoid \(\mathcal{H}\) and two equivalences \begin{align*} \mathcal{G} \swarrow \mathcal{H} \searrow \mathcal{G}^{\prime} . \end{align*}
• 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: In terms of the action groupoid:

Examples

The Kummer surface

The mirror quintic

See mirror quintic

Weighted projective spaces

See weighted projective space

Moduli of elliptic curves

See moduli stack of elliptic curves:

complete intersections of toric.

Arithmetic orbifolds:

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 \(|\mathcal{G}| \cong X .\)
• Defining homotopy invariants: take the classifying space:
• \(K_{X}\) 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 to exist
  • Satisfied automatically by Calabi-Yau orbifolds.
• Quotients by subgroups of SL2: If \(X\) is a Calabi-Yau orbifold and \((Y, f)\) is a crepant resolution of \(X\), then \(Y\) has a family of Ricci-flat Kahler metrics which make it into a Calabi-Yau manifold. 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.

Orbifold pi_1

Covers:

One recovers the theory of Hurwitz covers as the theory of representable orbifold morphisms from an orbifold Riemann surface to \({\operatorname{pt}}^{S_n}\).

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

Orbifold Euler characteristic

Setting \(R\left(G_{\sigma}\right)\) to be the complex representation ring of the stabilizer of \(\sigma\) in \(M\),

Bundles