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• Tags:
  • #AG #MOC #higher-algebra/DAG #AG/moduli-spaces
• Refs:
  • http://www.ams.org/notices/200304/what-is.pdf
  • https://www.math.uni-bielefeld.de/~rehmann/ECM/cdrom/3ecm/pdfs/pant3/fantechi.pdf
  • Homotopy theory for stacks
  • Jarod Alpers: https://sites.math.washington.edu/~jarod/moduli.pdf
  • https://arxiv.org/pdf/1708.08124.pdf
  • https://people.math.harvard.edu/~gaitsgde/grad_2009/Sorger.pdf#page=11
  • Course on algebraic stacks: http://individual.utoronto.ca/groechenig/stacks.html and http://individual.utoronto.ca/groechenig/stacks.pdf #resources/full-courses
  • Seminar notes on stacks
  • Masterclass: deformation theory, algebraic stacks
• Links:
  • Classical ideas:
    • scheme
    • proper morphism
    • hypercovering
    • flat family
    • a stack is a category fibered in groupoids
    • Quot schemes
    • orbifold
    • foliated manifold
  • Types of moduli spaces:
    • fine moduli space
    • coarse moduli space
  • Types of moduli stacks:
    • algebraic space
    • Deligne-Mumford stack
    • Artin stack
    • gerbe
  • Specific stacks:
    • Unsorted/quotient stack
    • moduli stack of Higgs bundles
    • moduli space of curves
      • moduli stack of elliptic curves
    • moduli stack of abelian varieties
    • Hilbert scheme
    • representation stack
  • Common uses:
    • Gromov-Witten invariants
    • Lagrangian Floer cohomology
    • symplectic field theory
    • contact homology
    • Fukaya categories
    • string topology
  • stackification
  • How to realize a stack as a homotopy quotient?
  • level structure
  • Projective moduli space
  • Separated moduli problem
  • Flat family
  • Hilbert polynomial
  • Unsorted/topological stack
  • In derived geometry:
    • derived stack
    • stable infinity category
    • Tangent space to a functor
    • cotangent complex
    • loop stack
  • compactly supported cohomology for quotient stacks
  • moduli space
  • motivation for stacks
  • QCoh(B) is equivalent to Rep(G)
  • automorphisms necessitate stacks

stacks MOC

Idea: stacks are geometrically modeled on sites \(\mathsf{S}\), and e.g. \({\mathsf{Grpd}}\) is a stack modeled on \(\mathsf{S} = {\mathsf{Set}}\) with the discrete topology.

Write \(D\) for the dual numbers.

In moduli problems

Definitions

Informal definition:

In terms of a pseudofunctor:

• An algebraic space is a pair \((X, R)\) with \(R \subseteq X{ {}^{ \scriptscriptstyle\times^{2} } }\) an equivalence relation whose projections \(p_i: R\to X\) are etale morphisms. Idea: replace being locally isomorphic to affine space in the Zariski topology with the finer etale topology.

• A prestack is a functor \({\mathsf{Aff}}{\mathsf{Sch}}_{/k}^{\operatorname{op}}\to {\mathsf{hoType}}\)

  • Source: should interpret as the infinity category of derived rings over \(k\)…?
  • Target: the infinity category of spaces, i.e. homotopy types.

• The prestack of quasicoherent sheaves over ${\mathsf{Sch}}_{/ {S}} $ is a stack wrt the fpqc topology.

• A stack is a functor $M: \mathsf{C}\to {\mathsf{Sch}}_{/ {S}} $ that satisfies effective descent.

• A 1-stack of groupoids on \(\mathsf{C}\) is a category fibered in groupoids \({\mathcal{X}}\to \mathsf{C}\) satisfying certain descent conditions. This form a category ${\mathsf{St}}^1 \leq { \underset{\infty}{ \mathsf{Cat}} }{} _{/ { \mathsf{C} }} $, so morphisms are cones (of functors) over \(\mathsf{C}\). For morphisms, \(f\simeq g\) means there is a natural transformation from \(f\) to \(g\) commuting with the projections to \(\mathsf{C}\), so one can form a homotopy category f 1-stacks.

• A smooth proper stack is essentially a compact orbifold.

In terms of sheaves

See site.

Artin Stack

Algebraic Stacks

Geometric stacks

As in the case of the cotangent complex:

Sheaves on Stacks

Quotient stacks

Examples

• \({{\mathbf{B}}G}\): defined as \({\mathsf{G}{\hbox{-}}\mathsf{Torsors}} \leq G{\hbox{-}}{\mathsf{Set}}\) in terms of torsors. See BG, constructed as the quotient stack \([{\operatorname{pt}}/G]\).
• For any \(X\in G{\hbox{-}}{\mathsf{Set}}\), \({\mathbf{B}}GX = [X/G]\) whose objects are ${\mathsf{G}{\hbox{-}}\mathsf{Torsors}}{/ {X}} \leq G{\hbox{-}}{\mathsf{Set}}{/ {X}} $.
• \({\mathsf{Rep}}(G)\) can be interepreted as a category of sheaves on the stack \({{\mathbf{B}}G}\).

Quotient stacks and Picard stacks, in terms of torsors:

Exercises

• Show that \([{\mathbf{A}}^1_{/ {k}} /{\mathbf{Z}}]\) for \(\operatorname{ch}k = 0\) is an algebraic space that is not quasiseparated.

Results

• Working with a moduli stack instead of a moduli space allows pretending the space is smooth and admits a universal family.

• Neat trick from algebraic geometry: for a stack \(X/G\) where $X \in{\mathsf{Var}}_{/ {{\mathbf{C}}}} $ and \(G \in {\mathsf{Fin}}{\mathsf{Grp}}\), \begin{align*} H^*(X/G; {\mathbf{Q}}) \cong H^*(X; {\mathbf{Q}})^G \end{align*} where the RHS denotes the taking the \(G{\hbox{-}}\)invariants. Seems to only work over \({\mathbf{Q}}\). The quotient is scheme-theoretic. The actual definition involves equivariant cohomology.

• fpqc stack \(\implies\) fppf, etale stack