Genus Zero Gromov-Witten Invariants.

(Lectures at the University of Nice in the Fall, 2000)

5. The Unreasonable Effectiveness of Localization.

The equivariant cohomology of a complex manifold X with a holomorphic action of the complex torus T = (C^{∗})^{n} is a graded ring H_{T}^{∗}(X, Q) together with a graded homomorphism from the cohomology ring of the classifying space:

π^{∗} : Q[t_{1}, ..., t_{n}] = H^{∗}(BT, Q) → H_{T}^{∗}(X, Q)

Two extreme examples are of particular interest:

• If the T -action is trivial, then H_{T}^{∗}(X, Q) = H^{∗}(X, Q)[t_{1}, ..., t_{n}] with the natural structure of a free module over Q[t_{1}, ..., t_{n}].

• If the T -action is free, then H_{T}^{∗}(X, Q) = H^{∗}(X/T, Q) and the maximal ideal m = ⟨t_{i}⟩ is the kernel of π^{∗}.

Equivariant cohomology shares many of the functorial properties of ordinary cohomology. For example, if

f : X → Y

is a T -equivariant holomorphic map, then:

• the (graded) pullback f^{∗} is always defined, and

• the pushforward f_{∗} is defined for proper maps f.

There are equivariant Chern classes

c_{d}^{T} (E) ∈ H_{T}^{2d}(X, Q)

for any holomorphic vector bundle E on X together with a lifting of the T action that is linear on the fibers of E. The tangent bundle T X has a canonical lifting, hence canonical equivariant Chern classes.

The connected components of the locus of fixed points are denoted:

i : F ↪ X

They are embedded (holomorphic) submanifolds and i is T -equivariant for the trivial T -action on F. The normal bundle N_{F/X} has a canonical lifting of the (trivial) T action on F, and thus has canonical equivariant Chern classes. In particular, the equivariant Euler class of F is the Chern class of degree equal to the codimension of F, and is denoted:

ϵ_{T} (F) ∈ H^{∗}(F, Q)[t_{1}, ..., t_{n}]

The constant (in t) term of ϵ_{T} (F) is the ordinary Euler class, which may well be zero. But the following result may be regarded as the most important feature of equivariant cohomology:

Theorem: ϵ_{T} (F) is always invertible in H_{T}^{∗}(F, Q)(t_{1}, ..., t_{n}).

Given a class c ∈ H_{T}^{∗}(X, Q), we define the residue of c on F to be:

\frac{i^{∗}c}{ϵ_{T} (F)} ∈ H_{T}^{∗}(F, Q)(t_{1}, ..., t_{n})

and an important corollary of the theorem is the following:

Functoriality of Residues: If f : X → Y is a T -equivariant proper map, fix a component j : G ↪ Y of the locus of fixed points of Y. Then for each equivariant cohomology class c ∈ H_{T}^{∗}(X, Q),

∑_{F ⊂f^{−1}(G)} f|_{F}_{∗} (\frac{i^{∗}c}{ϵ_{T} (F)}) = \frac{j^{∗}f_{∗}c}{ϵ_{T} (G)} ∈ H_{T}^{∗}(G, Q)(t_{1}, ..., t_{n})

i.e. the pushforward of the residues is the residue of the pushforward

Basic Observation: There are canonical embeddings:

i : \overline{M}_{0,n}(X, β) ↪ \overline{G}_{0,n}(X, β) := \overline{M}_{0,0}(X × (P^{1})^{n}, (β, 1^{n}))

that are T -equivariant for the trivial action on \overline{M}_{0,n}(X, β) and the standard action on (P^{1})^{n} with:

ϵ_{T} (\overline{M}_{0,n}(X, β)) = ∏_{i=1}^{n} t_{i}(t_{i} − ψ_{i})

The embedding is described set-theoretically as follows. Given a stable map f : C → X with marked points p_{1}, ..., p_{n} ∈ C, consider the curve C′ with no marked points but n additional smooth components C_{1}, ..., C_{n} meeting C at p_{1}, ..., p_{n}. There is a well-defined class of stable maps f′ : C′ → X × (P^{1})^{n} such that f′|_{C} = (f, 0, ..., 0) and f′|_{C_{i}} = (f(p_{i}), 0, ..., ϕ_{i}, ..., 0) for isomorphisms ϕ_{i} : C_{i} → P^{1}.

Thus we have a diagram:

\overline{G}_{0,n}(X, β)
↑
\overline{M}_{0,n}(X, β) →^{ev} X^{n}

and my philosophy is the following. If we could fill in with:

\overline{G}_{0,n}(X, β) →^{f} Y
i ↑  j ↑
\overline{M}_{0,n}(X, β) →^{ev} X^{n}

so that f is T -equivariant and X^{n} is fixed in Y, then we could apply the functoriality of residues to the class 1 to conclude that:

J_{β}(t_{1}, ..., t_{n}) + ∑_{F} ev_{∗} (\frac{1}{ϵ_{T} (F)}) = \frac{j^{∗}f_{∗}1}{ϵ_{T} (X^{n})}

summed over the other fixed loci F in f^{−1}(X^{n}).

In some cases this is indeed “unreasonably effective”. For instance:

Projective Space: If n = 1 and X = P^{m}, then we fill in with:

\overline{G}_{0,1}(P^{m}, d) →^{f} P^{(m+1)(d+1)−1}
i ↑  j ↑
\overline{M}_{0,1}(P^{m}, d) →^{ev}  P^{m}

so that f is birational and f^{−1}(P^{m}) = \overline{M}_{0,1}(P^{m}, d). The projective space P^{(m+1)(d+1)−1} is the space of m + 1 homogeneous polynomials of degree d in the variables x, y on P^{1}. A general element of \overline{G}_{0,1}(P^{m}, d) can be interpreted as a morphism ϕ : P^{1} → P^{m} of degree d. Since such a morphism is described by m + 1 homogeneous polynomials of degree d with no common factors, this gives a well-defined birational map:

\overline{G}_{0,1}(P^{m}, d) − − > P^{(m+1)(d+1)−1}

which extends to the map of the diagram.

The formula then follows from a computation:

ϵ_{T} (P^{m}) = ∏_{k=1}^{d}(H + kt)^{n+1}

The space \overline{G}_{0,1}(X, d) in general was studied by gauge theorists well before algebraic geometers got a hold of it. It can be regarded as Gromov’s “bubble-tree” compactification of the space Map(X, d) of holomorphic maps f : P^{1} → X of degree d. There is another compactification, the Uhlenbeck compactification, which is easy to describe set-theoretically:

\overline{U}(X, d) = Map(X, d) ∪ Map(X, d − 1) × P^{1} ∪ ... ∪ X × (P^{1})^{d}

and the diagram for projective space generalizes to:

\overline{G}_{0,1}(X, d) →^{f} \overline{U}(X, d)
i ↑  j ↑
\overline{M}_{0,1}(X, d) →^{ev}  X

where j is the inclusion of X = X × 0^{d}.

The fundamental problem with this approach is that in all cases other than X = P^{m}, the Uhlenbeck compactification is not smooth!

This brings up an important point. The spaces \overline{G}_{0,n}(X, β) are not smooth either, but a theorem of Graber and Pandharipande states that they are “virtually smooth” in the sense that the functoriality of residues holds with “c” replaced by “c ∩ [\overline{G}_{0,n}(X, β)]^{vir}” in an equivariant Chow group. The Uhlenbeck compactification has no such virtual smoothness property.

Nevertheless, a variation of this approach works for Fano toric varieties.

Product Formula: There is a diagram:

\overline{G}_{0,1}(X × Y, (α, β)) →^{f_{G}}  \overline{G}_{0,1}(X, α) × \overline{G}_{0,1}(Y, β)
i_{(X,Y)} ↑  (i_{X}, i_{Y}) ↑
\overline{M}_{0,1}(X × Y, (α, β)) →^{f_{M}} \overline{M}_{0,1}(X, α) × \overline{M}_{0,1}(Y, β) →^{ev} X × Y

where f_{G} and f_{M} are pairs of forgetful maps. One expects for dimension reasons that f_{G} (but not f_{M}) is birational, and a theorem of Behrend states that in fact, the virtual class pushes forward to the product of virtual classes, from which it follows that:

J_{(α,β)}(t) = ev_{∗} (\frac{[\overline{G}_{0,1}(X × Y, (α, β))]^{vir}}{t(t − ψ)} )

= ev_{∗} (\frac{[\overline{G}_{0,1}(X, α)]^{vir} × [\overline{G}_{0,1}(Y, β)]^{vir}}{ϵ_{T} (\overline{M}_{0,1}(X, α)) ⊗ ϵ_{T} (\overline{M}_{0,1}(Y, β))}) = J_{α}(t) ⊗ J_{β}(t)

Hypersurfaces: Let i : X ↪ P^{m} be a hypersurface of degree l. Consider the diagram:

\overline{G}_{0,1}(X, d) →^{f_{G}} \overline{G}_{0,1}(P^{m}, d)
↑  ↑
\overline{M}_{0,1}(X, d) →^{f_{M}} \overline{M}_{0,1}(P^{m}, d)

One of the basic properties of the virtual class is:

f_{M}_{∗}([\overline{M}_{0,1}(X, d)]^{vir}) = c_{dl+1}E

where E is the rank dl + 1 bundle π_{∗}e^{∗}O_{P^{m}}(l), and the same is true of the pushforward under f_{G} of the virtual class, which we will denote c.

It follows that:

i_{∗}J_{d}^{X}(t) = ev_{∗} (\frac{c}{t(t − ψ)}) ∈ H^{∗}(P^{m}, Q)[t^{−1}]

and one can extract the J-function itself from this pushforward.

So we need a method for computing such things. By the functoriality of residues and the diagram for projective space, we have:

ev_{∗} (\frac{c}{t(t − ψ)}) = \frac{j^{∗}f_{∗}c}{∏_{k=1}^{d}(H + kt)^{m+1}}

and we can thus reduce the problem to one of computing j^{∗}f_{∗}c.

There are several ways to do this. Using the additional action of the “big” torus (C^{∗})^{m} on P^{m} and considering residues for the total action of C^{∗} × (C^{∗})^{m}, Givental found his original proof.

The proof of Lian-Liu-Yau is similar, and in their series of papers Mirror Principle I-IV, they develop a machine for computing pushforwards of residues of any class c satisfying “nice” properties. This also applies, for example, to classes of the form c_{dl−1}F where F is the bundle R^{1}π_{∗}e^{∗}O_{P^{m}}(−l). These latter computations are useful for “local” mirror symmetry.

For my approach, recall that f : \overline{G}_{0,1}(P^{m}, d) → P^{(m+1)(d+1)−1} is birational. If c were the pullback of an equivariant cohomology class c′ on P^{(m+1)(d+1)−1}, then we’d have j^{∗}f_{∗}c = j^{∗}c′ and the computation would be easy. This doesn’t happen, but c does decompose as a sum of pullbacks for a sequence of maps from boundary strata starting with the map f. The point is that there is a map:

Φ : E = π_{∗}e^{∗}O_{P^{m}}(l) → f^{∗}O(l)^{⊕dl+1}

of vector bundles on \overline{G}_{0,1}(P^{m}, d). Given ϕ : P^{1} → P^{m} of degree d, then ϕ^{∗}O_{P^{m}}(l) ≅ O_{P^{1}}(dl) and the fiber of E over ϕ is thus H^{0}(P^{1}, O_{P^{1}}(dl)). This globalizes to the map Φ, which is an isomorphism over the points corresponding to maps ϕ. The precise relationship between the top equivariant Chern classes:

c_{dl+1}^{T}(E) and ∏_{k=0}^{dl}(lH + kt)

is determined by means of MacPherson’s graph construction and the “self-similar” nature of the boundary leads to the proofs.

WDVV-Like Relations: Assume that X is embedded in P^{m}, for simplicity and for n ≥ 1, consider the diagram:

P^{(m+1)(d+1)−1} ←^{f_{n+1}} \overline{G}_{0,n+1}(X, d) →^{f_{G}} \overline{G}_{0,n}(X, d) × P^{3}
↑  ↑
F  →^{f_{M}}  \overline{M}_{0,n}(X, d) →^{ev} X^{n}

where f is made up of two forgetful maps to:

\overline{G}_{0,n}(X, d) and \overline{M}_{0,0}(P^{1} × P^{1}, (1, 1)) ≅ P^{3}

The fixed loci can be described explicitly, yielding:

ev_{∗}f_{M}_{∗} ∑_{F} (\frac{1}{ϵ_{T} (F)}) = ∑_{1∈S} ∑_{e} P (Ĵ_{d−e}(t_{S}, t) ⊗ J_{e}(−t, t_{S^{c}}))

and more generally, for positive integers b, we have:

ev_{∗}f_{M}_{∗} ∑_{F} (\frac{i^{∗}f_{n+1}^{∗}H_{T}^{b}}{ϵ_{T} (F)} ) = ∑_{1∈S} ∑_{e} P_{b,e} (Ĵ_{d−e}(t_{S}, t) ⊗ J_{e}(−t, t_{S^{c}}))

with the conventions on Poincaré duality from the previous section.

On the other hand, by functoriality of residues, we have:

ev_{∗}f_{M}_{∗} ∑_{F} (\frac{i^{∗}f_{n+1}^{∗}H_{T}^{b}}{ϵ_{T} (F)} ) = \frac{j^{∗}f_{G}_{∗}f_{n+1}^{∗}H_{T}^{b}}{∏ t_{i}(t_{i} − ψ_{i})} · \frac{1}{t_{1}t(t_{1} + t)}

(the extra factor is the equivariant Euler class of a fixed point in P^{3})

Here’s the punchline.

The numerator: j^{∗}f_{G}_{∗}f_{n+1}^{∗}H_{T}^{b} is an equivariant cohomology class, so without calculating it, we know that it is polynomial in t. Hence the relation of the previous section. In a couple of cases, it can be computed. When b = 0, it is 1 since f_{G} is birational. When b = 1, it is e_{1}^{∗}H + dt_{1}. This requires more work. (It may follow from the divisor relation) And for b > 1, it seems to be hard to calculate. But as we’ve noted, the calculation is not necessary for reconstruction!