Orbifold cohomology and quantum cohomology

of orbifolds

ICTP, Trieste, September 2002 lectures by Dan Abramovich

Part 2:

Algebraic Orbifold Quantum Products

• Work over ℂ

• This work with Tom Graber and Angelo Vistoli • Quantum products based on W. Chen + Y. Ruan

• global quotient case by Fantechi + Göttche.

Suppose that X is an algebraic stack, with moduli space X. Let C be a twisted curve Contraction Lemma: Given (possibly unstable)

\begin{aligned}C & → & X \\ ↓ & & ↓ \\ C & → & X \end{aligned}

Then if C → X can be stabilized, so does C → X.

Applications:

(a) functoriality in X → Y

(b) forgetful maps for untwisted markings and universal curve.

proof aside

Recall: Gromov-Witten theory Let X be a smooth projective variety. Consider the correspondence

\xymatrix{\overline{M}_{g,n+1}(X, β) \ar[r]^{e_{n+1}} \ar[d]_{e_{1}×···×e_{n}} & X \\ X^{n}}

Define

H^{∗}(X)^{n} ⟶ H^{∗}(X)
γ_{1} × ⋯ × γ_{n} ↦ ⟨γ_{1}, …, γ_{n}, ∗⟩_{g,β} ,

where ⟨γ_{1}, …, γ_{n}, ∗⟩_{g,β} = (e_{n+1})_{∗} (e_{1,... ,n}^{∗}(γ_{1} × ⋯ × γ_{n}) ∩ [\overline{M}_{g,n+1}(X, β)]^{v}) In genus zero these GW invariants satisfy WDVV relations, giving associativity.

“Small” quantum product is defined by

γ_{1} ∗ γ_{2} = ∑_{β∈H_{2}(X)} ⟨γ_{1}, γ_{2}, ∗⟩_{0,β} q^{β}.

(grading of q^{β} is 2c_{1}(X) · β) Associativity (assuming nothing odd):

∑_{β_{1}+β_{2}=β} ⟨ ⟨γ_{1}, γ_{2}, ∗⟩_{0,β_{1}}, γ_{3}, ∗ ⟩_{0,β_{2}} =

∑_{β_{1}+β_{2}=β} ⟨ ⟨γ_{1}, γ_{3}, ∗⟩_{0,β_{1}}, γ_{2}, ∗ ⟩_{0,β_{2}} .

A key step in proving associativity is a morphism from

\overline{M}_{0,3}(X, β_{1}) ×_{X} \overline{M}_{0,3}(X, β_{2})

to a “boundary divisor” in

\overline{M}_{0,4}(X, β_{1} + β_{2})

M_{2} → X

↙ ↘

M_{1} → X X
↙ ↘
X X

\xymatrix{M_{1} ×_{X} M_{2} \ar@{^{(}->}[r] & D_{(12|34)}(X) \ar[d] \ar[r] & \overline{M}_{0,4}(X, β) \ar[d] \\  & D_{(12|34)} \ar@{^{(}->}[r] & _{\overline{M}_{0,4}}}

In order to generalize this picture, we will need

1. an analogue of H^{∗}(X) or A^{∗}(X) for a smooth Deligne–Mumford stack,

2. an analogue of the evaluation maps e_{i} and e_{n+1},

3. an analogue of the virtual fundamental class, and

4. an analogue of the fiber product description of the boundary divisors.

We’ll start with the last Interpretation of boundary: suppose C =

C_{1} ∪^{p} C_{2}

Then Hom(C, X) =

Hom(C_{1}, X) ×_{Hom({p},X)} Hom(C_{2}, X)

Now say C is a nodal twisted curve.

C = C_{1} ∪^{℘} C_{2},

where

℘ ≃ B(μ_{r})

Hom(C, X) =

Hom(C_{1}, X) ×_{Hom(℘,X)} Hom(C_{2}, X)

So evaluation lands in

Hom ( B(μ_{r}) , X )

(and not just in X)

Inertia stack Here

X_{1} = ⋃_{r} HomRep(Bμ_{r}, X)

= \{(x, H, χ) | \begin{matrix}x ∈ Ob(X), \\ H ⊂ Aut x, \\ χ : H ˜→μ_{r} \end{matrix} \}

= \{(x, g) | \begin{matrix}x ∈ Ob(X), \\ g ∈ Aut x \end{matrix} \}
= I(X)

The points of the coarse moduli space I(X) are

( [x] , (g) )

where [x] ∈ X is the isom. class of an object x and

(g) is the conjugacy class of an element g ∈ Aut(x)

Let X be a smooth Deligne–Mumford stack of dimension d with projective coarse moduli space X

WE WORK WITH

H_{orb}^{∗}( X ) = H^{∗}( X_{1} )

Age Let ( [x] , (g) ) ∈ I(X) Can write locally X = [V / Γ], where Γ = Aut(x), and V smooth. Diagonalize the action of g on the tangent T_{x}(V), with eigenvalues

e^{(2π i)·r_{j}}, j = 1, …, d

with 0 ≤ r_{j} < 1. Definition: the age

a(x, g) = ∑_{i=1}^{d} r_{j}.

Claim: a(x, g) is constant on every connected component Z ⊂ I(X) (notation a(Z))

Claim: a(x, g) + a(x, g^{−1}) = dim X − dim Z

GRADING

Definition: The degree of an element α ∈ H^{i}(Z) in H_{orb}^{∗}(X) is defined to be

i + 2a(Z)

Note: dim H_{orb}^{i}(X) = dim H_{orb}^{d−i}(X).

Evaluation Unless g = 0, n = 3, β = 0, the markings of K_{g,n}(X, ·) are nontrivial grbes. Define \overline{M}_{g,n}(X, ·) = twisted stable maps with sections

= fibered product of universal gerbes over K_{g,n}(X, ·)

Have evaluation maps

\overline{M}_{g,n}(X, ·) ⟶^{e_{i}} X_{1}

recall that in

C = C_{1} ∪^{℘} C_{2},

we can naturally identify B(μ_{r}) ≃ ℘ ⊂ C_{2}, via the action of μ_{r} on the normal bundle to the marking in C_{2}. but then the action on the C_{1} side is opposite!

(ξ, η) ↦ ( ζ_{r} · ξ , ζ_{r}^{−1} · η)

Define involution

ι : X_{1} → X_{1}

via

g ↦ g^{−1}

Define:

ě_{i} = ι ∘ e_{i}

Deformations / Obstructions: general case (i) Twisted curves are unobstructed:

Ext^{2}(Ω_{C}^{1}, O_{C}) = 0

(ii) Obstructions of f : C → X lie in H^{1}( C , f^{∗}T_{X} )

(iii) Relative infinitesimal deformations are H^{0}( C , f^{∗}T_{X} )

Behrend - Fantechi, Kresch ⟹ virtual fundamental class [K_{g,n}(X, ·)]^{v} Because of sections need to use [\overline{M}_{g,n}(X, ·)]^{w} = r_{1} ⋯ r_{n} [\overline{M}_{g,n}(X, ·)]^{v} Special case: every deformation is unobstructed.

C ⟶^{f} X ; π ↓ ; \overline{M}

then [\overline{M}]^{v} = c_{top}(E) = c_{top} ( ℝ^{1}π_{∗} f^{∗}T_{X} )

If, moreover, C = [D/H] with f̃ : D → X, then

E = ( ℝ^{1}π̃_{∗} f̃^{∗}T_{X} )^{H}

3-point orbifold G–W invariants

\overline{M}_{0,3}(X , β) →^{ě_{3}} X_{1}
e_{1} × e_{2} ↓
X_{1} × X_{1}

for γ_{i} ∈ H_{orb}^{∗}(X) Define ⟨γ_{1}, γ_{2}, ∗⟩_{β} =

(ě_{3})_{∗} ( e_{1}^{∗}(γ_{1})e_{2}^{∗}(γ_{2}) ∩ [\overline{M}_{0,3}(X, β)]^{w} )

“Small” quantum product is again defined by

γ_{1} ∗ γ_{2} = ∑_{β∈H_{2}(X)} ⟨γ_{1}, γ_{2}, ∗⟩_{0,β} q^{β}.

Theorem: this is a graded skew-commutative associative ring.

grading of q^{β} is 2(c_{1}(X) · β)

Will only prove when every deformation is unobstructed, and curves smooth freely Associativity (assuming nothing odd):

∑_{β_{1}+β_{2}=β} ⟨ ⟨γ_{1}, γ_{2}, ∗⟩_{0,β_{1}}, γ_{3}, ∗ ⟩_{0,β_{2}} =

∑_{β_{1}+β_{2}=β} ⟨ ⟨γ_{1}, γ_{3}, ∗⟩_{0,β_{1}}, γ_{2}, ∗ ⟩_{0,β_{2}} .

M_{2} → X_{1}

↙ ↘

M_{1} → X_{1} X_{1}

↙ ↘ X_{1} X_{1} The product diagram

M_{1} ×_{X_{1}} M_{2} → M_{2}
↓  ↓
M_{1} → X_{1}

can be expanded:

M_{1} ×_{X_{1}} M_{2} → M_{2} → X_{1}
↓  ↓  ↓
M_{1} × M_{2} → X_{1} × M_{2} → X_{1} × X_{1}
↓  ↓
M_{1} → X_{1}

It follows that the coefficient in (γ_{1} ∗ γ_{2}) ∗ γ_{3} is

r_{×}^{2} e_{4 ∗}^{×} ( ( ∏_{i=1}^{3} e_{i}^{× ∗}(γ_{i})) · ℭ)

where ℭ = c_{top}( p_{1}^{∗}E_{1} ⊕ p_{2}^{∗}E_{2} ) c_{top}( N_{δ}/N_{p_{1}×p_{2}} )

Here E_{i} = obstruction bundle of f_{i} : C_{i} → X, N_{δ}/N_{p_{1}×p_{2}} = excess normal bundle

Can we make this “symmetric”?

The divisor diagram

M_{1} ×_{X_{1}} M_{2}

↓

M_{0,4}(X) ← D ×_{M_{0,4}} M_{0,4}(X) ⊃ D(X)
↓  ↓
M_{0,4} ⊃  D

So enough to show

ℭ = gl^{∗}c_{top}(E)

E = obstruction bundle of f_{0,4} : C_{0,4} → X.

We consider the normalization sequence 0 → f^{∗}T X → ν_{∗}f′^{∗}T X → (f^{∗}T X)|_{Σ} → 0.

Pushing forward to Y gives 0 → π_{∗}f^{∗}T X → π′_{∗}f′^{∗}T X → (π_{∗}(f^{∗}T X)|_{Σ})

→ gl^{∗}E → p_{1}^{∗}E_{1} ⊕ p_{2}^{∗}E_{2} → 0. Let Q be the bottom kernel. Need: Q = excess bundle.

Have on top 0 → T Y → p_{1}^{∗}T M_{1} ⊕ p_{2}^{∗}T M_{2}

→ π_{∗}((f^{∗}T X)|_{Σ}) → Q → 0

So we got 0 → N_{p_{1}×p_{2}} → π_{∗}((f^{∗}T X)|_{Σ}) → Q → 0

So the following suffices: Lemma.

π_{∗} ((f^{∗}T X)|_{Σ}) ≃ e_{×}^{∗} T X_{1}