Bertrand Toen, https://arxiv.org/pdf/math/0604504.pdf
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Examples:
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\({ \mathcal{M}_{g, n} }(X)\) the stack of stable maps to $X\in {\mathsf{sm}}\mathop{\mathrm{proj}}{\mathsf{Var}}_{/ {{\mathbf{C}}}} $
- Can take \({ \mathcal{M}_{g, n} }(X, \beta)\) for \(\beta \in H^2_{\mathrm{sing}}(X; {\mathbf{Z}})\) for those maps having \(\beta\) as a fundamental class, categorifies quantum cohomology (take \({\operatorname{HH}}\) or more generall \({\operatorname{THH}}\) to recover)
- \({\mathsf{Coh}}(X)\) the stack of coherent sheaves on a variety.
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\(K(G, n)\) Eilenberg-MacLane stacks
- Can generalize to sheaves of groups on a scheme
- Can define generalized cohomology \(H^n(X; A) = [X, K(A, n)] = \pi_0\mathop{\mathrm{Maps}}(F, K(A, n))\).
- \(S^1 = K({\mathbf{Z}}, 1)\) spheres
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Based loop stacks: for \(x: X\to Y\), given by \({\Omega}_x Y = X { {}^{ \scriptscriptstyle \underset{\scriptscriptstyle {Y} }{\times} ^{2} } }\), naturally forms a group stack.
- Defines homotopy sheaves by iterating \(\pi_n(Y; x) \coloneqq\Omega_x^{(n)}(Y)\)
- Stacks of abelian categories: for $B\in {}{k} \mathsf{Alg} $ (with conditions), the stack $k{\hbox{-}}{\mathsf{Ab}}\mathsf{Cat}{/ {B}} $ of \(k{\hbox{-}}\)linear abelian categories equivalent to ${}_{B}{\mathsf{Mod}} $.
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Betti stacks: for $X\in{\mathsf{sm}}\mathop{\mathrm{proj}}{\mathsf{Var}}_{/ {{\mathbf{C}}}} $, the constant stack \(X_B\) with values in \({\operatorname{Sing}}(X({\mathbf{C}}))\) (ssets of singular simplicies) and \(X_{\mathrm{dR}}\), by Riemann-Hilbert there is an equivalence of analytic stacks \(\mathop{\mathrm{Maps}}(X_B, {-})^{\mathrm{an}} { \, \xrightarrow{\sim}\, }\mathop{\mathrm{Maps}}(X_{\mathrm{dR}}, {-})^{\mathrm{an}}\).
- This is the start of nonabelian Hodge theory, the RHS is the “de Rham nonabelian cohomology” of \(X\) with coefficients in \(({-})\) (a “special” Artin stack). Admits Hodge and weight filtrations.
- \({ \mathsf{Vect} }_\bullet(X)\), the 1-stack of vector bundles over \(X\in {\mathsf{sm}}{\mathsf{Sch}}^{\mathrm{prop}}\)
- \(\operatorname{Pic}^0(X)\) the group stack of degree 0 line bundles
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\({ \mathcal{M}_{g, n} }(X)\) the stack of stable maps to $X\in {\mathsf{sm}}\mathop{\mathrm{proj}}{\mathsf{Var}}_{/ {{\mathbf{C}}}} $
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Tools in the theory:
- Intersection theory
- Six functor formalism
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\(\ell{\hbox{-}}\)adic cohomology
- Trace formulas
- Vanishing theorem
- Motivic cohomology
- Riemann-Roch
- Motivic integration
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Virtual fundamental classes (eg for Gromov-Witten theory)
- Can use to define Euler-characteristics, numerical invariants e.g. to count the number of (stable) sheaves with fixed data (similar to Casson invariants).
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Moduli problem: a functor \(F: \mathsf{C}\to {\mathsf{Set}}\) where \(F(X)\) should classify families of \(C\in \mathsf{C}\) parameterized by \(X\).
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Problem: elements in a set \(F(X)\) have a strict notion of equality, which is too strong. We usually want to classify up to isomorphism, equivalence, etc.
- Example weak equivalences: quasi-isomorphism of chain complexes, weak equivalence of spaces, categories up to equivalence.
- Problem: objects having nontrivial automorphisms makes the set of isomorphism classes ill-behaved.
- Solution: make target a groupoid instead of a set.
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Problem: elements in a set \(F(X)\) have a strict notion of equality, which is too strong. We usually want to classify up to isomorphism, equivalence, etc.
- Idea: stacks over a site \(\mathsf{C}\) are simplicial presheaves with descent.
- Artin stack: covered by a family of affine schemes \({\mathcal{U}}\rightrightarrows X\) where any \(Y\to X\) factors through \({\mathcal{U}}\) (the atlas). Equivalently, \({\mathcal{U}}\rightrightarrows X\) faithfully flat and locally finitely presented representable morphism.
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Application: geometric Langlands classicaly reads as a triangulated equivalence
\begin{align*}
{\mathsf{d}} {}_{{\mathcal{D}}}{\mathsf{Mod}} _{/ {{ \mathsf{Vect} }_n(X)}} \simeq{\mathsf{d}}_{\mathop{\mathrm{coh}}} \mathsf{Loc}^n_{/ {X}} = \mathop{\mathrm{Maps}}(X, \mathbf{B}\mkern-3mu \operatorname{GL} _n)
\end{align*}
- LHS: derived category of D-modules on the stack of degree \(n\) vector bundles over \(X\), RHS: stack of rank \(n\) flat vector bundles (local systems) over \(X\).
- Hot gossip as of 2005, c/o Lafforgue: the RHS needs to be the derived stack ${\mathbf{R}}\mathsf{Loc}^n_{/ {X}} $ to have a chance of being true.