Orbifold cohomology and quantum cohomology

of orbifolds

ICTP, Trieste, September 2002 lectures by Dan Abramovich

Part 1: Moduli of twisted stable maps

• with Angelo Vistoli

• other construction by W. Chen and Y. Ruan

Work over ℂ recall that nice moduli problems are typically Deligne-Mumford stacks admitting projective coarse moduli spaces.

Classic Examples:

• \overline{M}_{g,n}; \overline{M}_{g,n}(X, β)

Notation: \overline{M}_{g,n}(X, β) =: K_{g,n}(X, β)

• BG = {pt}/G = moduli of principal homogeneous G-spaces

• the quotient stack [V/G] Stacks have been introduced as “moduli objects” for families of schemes, sheaves etc.

People have now come to accept them as basic algebraic geometry objects.

Is K_{g,n}(X , β) likely to be interesting for a DM stack X? examples:

1. X = \overline{M}_{γ,ν} ⇝

K_{g,n}(X) = moduli of fibered surfaces: families of stable curves over stable curves

2. X = BG ⇝ moduli of principal G bundles on stable curves recall: Definition. A stable n-pointed map

(C, … x_{i} …, f : C → X)

of genus g and image class β is a morphism from a prestable n-pointed curve (C, x_{1}, …, x_{n}) to a variety X, such that

• the group Aut_{X}(f : C → X, x_{i}) of automorphisms of f fixing x_{i} is finite

• Now let X be a Deligne–Mumford stack. Preliminary Definition. A stable n-pointed map

(C, x_{i}, f : C → X)

is a morphism from a prestable n-pointed curve (C, x_{i}) to the stack X such that

• Aut_{X} (f : C → X) is finite

•

PROBLEM: NOT COMPLETE!! e.g. consider degeneration of a smooth curve of genus 2,

and map to BG, with G = (ℤ/r)^{4}. smooth fiber C_{η} has connected principal G bundles, one for each isomorphism

H_{1}(C_{η}, ℤ)/r →^{∼} G

C_{s} has H_{1}(C_{η}, ℤ) = ℤ^{3}, so no connected G bundle!!

Claim: this problem is resolved if we allow the degenerate curve to be a stack. Consider the example above:

\xymatrix{P_{η} \ar@{^{(}->}[r] \ar[d] & ? \ar[d] \\ C_{η} \ar@{^{(}->}[r] \ar[d] & C \ar[d] \\ \{η\} \ar@{^{(}->}[r] & S}

If we try to extend directly (e.g. normalization of C in P_{η}) we typically get ramification index r over C_{s}.

(i) after base change of order r may assume P_{η} extends over C_{sm}:

P_{η} ⊂ P_{sm}
↓  ↓
C_{η} ⊂ C_{sm} ⊂ C

(Purity of Branch Locus)

(ii) at node p we have in local analytic coordinates C_{p} : xy = t^{r} We can pick a “uniformization” V_{p} → C_{p} V_{p} : {ξη = t} where u = ξ^{r}, v = η^{r} V_{p} nonsingular. Purity ⟹ the bundle extends:

P_{p} → V_{p}

There is an action of μ_{r} on V_{p} which lifts to the bundle P_{p}. (iii) V_{p} → C_{p} with μ_{r} action is a chart for an orbifold structure C → C, namely a DM stack C with moduli space C moreover we have a morphism C → BG Twisted curves (n = 0) Definition. A twisted (pre-stable) curve is a Deligne–Mumford stack C with coarse moduli space C such that

1. C is a pre-stable curve,

2. the map C → C to the coarse moduli curve is an isomorphism away from singularities of C, and

3. C has at most nodes as singularities Local description in a family

\begin{aligned}& C : & xy = f(t)^{r} \\ C = [V/μ_{r}], & V : & ξη = f(t), & x = ξ^{r}, y = η^{r} \end{aligned}

Action of μ_{r} : (ξ, η) ↦ ( ζ_{r} · ξ , ζ_{r}^{a} · η) with gcd(a, r) = 1.

Terminology: r = 1: untwisted; r > 1: twisted. a ≡ −1 mod r: balanced; otherwise unbalanced. Note: if f(t) ≠ 0 then balanced. n > 0: Twisted curves at a marking One way to arrive at marked curves is by normalizing a nodal curves and “separating” a node in two markings.

Local description via parameters

\begin{aligned}& C : & x \\ C = [V/μ_{r}], & V : & ξ, & x = ξ^{r} \end{aligned}

Action of μ_{r} : ξ ↦ ζ_{r} · ξ

Global description: a marking is a closed substack which is a gerbe banded by μ_{r}. NOT NECESSARILY A SECTION Definition Let X be a D-M stack, X → X projective coarse moduli scheme. An n-pointed twisted stable map f : C → X to X of genus g, class β, is a diagram

\begin{matrix}C & →^{f} & X \\ ↓ & & ↓ \\ C & →^{\overline{f}} & X \end{matrix}

where

(i) C twisted curve with coarse moduli space C (ii) C → X representable

(no unnecessary branchings) (iii) C → X stable n-pointed map of genus g and class β

(⇔ automorphism group is finite)

Theorem: the moduli of n-pointed twisted stable maps of genus g and class β to X is a proper D-M stack K_{g,n}(X , β) admitting a projective coarse moduli scheme K_{g,n}(X , β). In fact, there is a commutative diagram of finite maps (not cartesian)

K_{g,n}(X , β) → K_{g,n}(X , β)
↓  ↓
K_{g,n}(X, β) → K_{g,n}(X, β)

Issues coming in proof:

(i) existence of bottom line in diagram (ii) valuative criterion for properness like for BG (iii) annoying technicalities Problem: a stack is a category, but this is apriori a 2-category. Claim: the 2-category of twisted curves is equivalent to the associated category.

Lemma Let F: X → Y be a morphism of Deligne–Mumford stacks over a scheme S. Assume that there exists a dense open representable substack (i.e. an algebraic space) U ⊆ X and an open representable substack V ⊆ Y such that F maps U into V. Then any automorphism of F is trivial.

proof aside

in fact we have: Proposition: the category of twisted curves is an algebraic stack.

Purity Lemma S a nonsingular surface, U ⊂ S, U = S {pt}. Given a diagram

[figure: X; U; f_{U}; S; ∃? f; \overline{f}; X]

Then the f_{U} extends uniquely to f : S → X (up to unique isomorphism...)

proof aside

Valuative criterion for properness (one case): Suppose S = SpecR, the spectrum of a discrete valuation ring; Generic point η = SpecK, special point s. Suppose C_{η}/K is an untwisted curve, C_{η} → X a (twisted) stable map. Then, after a base change S′ → S, there is an extension

\xymatrix{C_{η′} \ar@{^{(}->}[r] \ar[d] & C \ar[r] \ar[d] & X \\ \{η\} \ar@{^{(}->}[r] & S}

Steps:

(i) Kontsevich: extend the map to the coarse moduli space X. (ii) Properness of X: extend the map C_{η} → X over an open set containing the generic points of C_{s} (iii) Purity Lemma: the map extends over the smooth locus of C. (iv) local argument as for BG: lift the map to [V_{p}/μ_{r}] → X

(v) descent: replace [V_{p}/μ_{r}] by something representable.

Global description of a twisted marking Recall that a marking Σ^{C} ⊂ C is a gerbe. In fact, there is a nice canonical way to recover Σ^{C} ⊂ C from Σ^{C} ⊂ C, away from nodes. The point is that the pullback of Σ^{C} is r · Σ^{C}.

Definition. Let V be a scheme, L an invertible sheaf, s a global section of L. Define a stack \sqrt[r]{(L, s)} whose objects over a scheme T are

(f, M, t, ϕ),

where

• f : T → V is a morphism

• M is an invertible sheaf on V

• t is a section of M

• ϕ : M^{⊗r} →^{∼} f^{∗}L is an isomorphism, such that

• ϕ(t^{⊗r}) = s,

and arrows are fiber diagrams. Proposition Near Σ^{C}, the twisted curve C is isomorphic to \sqrt[r]{(L, s)} where L is O_{C}(Σ^{C}) and s is the defining section.