Another way to enumerate rational curves with torus actions

Aaron Bertram

1. Introduction. In 1991 Candelas, de la Ossa, Green and Parkes used the “string-theoretic” principle of mirror symmetry to predict the numbers of rational curves of any degree on a quintic three-fold [7]. This prediction took mathematicians by surprise, as the best results at the time only counted rational curves of degree three or less. Since then, exciting new developments have led to mathematical proofs of these predictions and many other related conjectures coming from string theory. While this does not address the deeper problem of constructing a mathematical foundation for string theory, it does represent a major advance in field of enumerative algebraic geometry.

An enumerative question is usually interpreted in terms of intersection theory on a moduli space. Moduli spaces of stable maps were introduced by Kontsevich and Manin [19] on which to calculate Gromov-Witten invariants, including the expected numbers of rational curves on a quintic three-fold. Like the spaces of stable pointed curves (which are stable maps to a point), the boundary of a stable map space has a self-similar property that has been exploited in many of the important recent results in enumerative geometry, including the associativity of the quantum cohomology rings [19, 25] and the reconstruction theorems [8, 19] for genus zero invariants.

Any torus action on the target space is inherited by all spaces of stable maps. Kontsevich applied the Bott residue theorem for such torus actions in his Enumeration of rational curves with torus actions [18] to the problem of computing Gromov-Witten invariants of rational curves, and although his approach would in principle compute the numbers of rational curves of all degrees on the quintic, the combinatorics becomes unmanageable after degree four. Givental seems to have been the first to fully appreciate the significance of the more subtle circle action induced from the domain of a “parametrized” stable map. An application of techniques from equivariant cohomology to both actions led to Givental’s proofs [6, 12, 13, 14, 24] of the mirror conjecture for Fano and Calabi-Yau complete intersections in toric varieties, as well as the subsequent proofs and generalizations of Kim [17] and Lian-Liu-Yau [21, 22]. In particular, these verify the predicted numbers of rational curves on the quintic three-fold.

In this paper we will produce a new and direct computation of the onepoint Gromov-Witten invariants for rational curves on Fano and Calabi-Yau complete interesections in complex projective space. This proof is selfcontained and much simpler than previous proofs. The idea is to use the selfsimilar properties of the boundary of stable map space within the context of equivariant cohomology to find a self-similar decomposition of the relevant equivariant “virtual” classes. This new idea allows us to dispense entirely with torus actions inherited from the target manifold and instead focus on the more important circle action inherited from the parametrization of the genus zero curve. This proof has the additional advantage in that it gives a computation that does not rely on “already knowing the answer,” as is the case in all previous proofs of the mirror conjecture. Besides being a psychological advantage, our proof thereby generalizes without modification to give a relative version of the mirror conjecture for projective bundles.

The mirror conjecture is best expressed in terms of one-point invariants. The space of stable genus-zero one-pointed maps of degree d to P^{n} is denoted by \overline{M}_{0,1}(P^{n}, d) and comes equipped with evaluation and projection maps:

e : \overline{M}_{0,1}(P^{n}, d) → P^{n} and π : \overline{M}_{0,1}(P^{n}, d) → \overline{M}_{0,0}(P^{n}, d)

where \overline{M}_{0,0}(P^{n}, d) is the space of stable zero-pointed maps. These spaces are always projective orbifolds and when d = 1 the maps are the two projections:

e : Fl(1, 2, n + 1) → P^{n} and π : Fl(1, 2, n + 1) → G(2, n + 1)

from the partial flag manifold.

If S ⊂ P^{n} is a transverse zero section of a vector bundle E which is generated by global sections (e.g. a complete intersection), then S defines a “virtual” class:

[S]_{d} := π^{∗}c_{r}(π_{∗}e^{∗}E)

on \overline{M}_{0,1}(P^{n}, d), where r is the rank of π_{∗}e^{∗}E. This measures the expected constraint imposed by requiring the stable map to land in S. Even simpler, given a variety V ⊂ P^{n} with associated cohomology class [V], the pull-back:

e^{∗}[V]

is the constraint imposed by requiring the marked point to land in V. As opposed to the previous two “enumerative” classes, Witten’s cotangent class:

ψ := c_{1}(ω_{π})

(the first chern class of the relative dualizing sheaf for the projection π) is of less enumerative significance, but very useful for computations.

The mirror conjecture for P^{n} (as formulated by Givental and generalized to homogeoneous spaces by Kim [17]) says that if S is a Fano or Calabi-Yau complete intersection in P^{n} of type (l_{1}, ..., l_{m}), then the Laurent polynomials:

e_{∗} (\frac{[S]_{d}}{t(t − ψ)}) := t^{−2}e_{∗}([S]_{d}) + t^{−3}e_{∗}([S]_{d} ∪ ψ) + t^{−4}e_{∗}([S]_{d} ∪ ψ^{2}) + ...

with coefficients in H^{∗}(P^{n}, Q) are closely related to the rational functions (also expressible as Laurent polynomials):

ϕ_{d}(t, h) := \frac{∏_{i=1}^{m} ∏_{k=0}^{dl_{i}}(l_{i}h + kt)}{∏_{k=1}^{d}(h + kt)^{n+1}}

where h ∈ H^{∗}(P^{n}, Z) is the hyperplane class.

More precisely, the relationship depends on the positivity of S as follows:

Fano of Index Two or More: If l_{1} + ... + l_{m} < n, then:

e_{∗} (\frac{[S]_{d}}{t(t − ψ)}) = ϕ_{d}

Fano of Index One: If l_{1} + ... + l_{m} = n, then:

e_{∗} (\frac{[S]_{d}}{t(t − ψ)}) = ∑_{r=0}^{d} \frac{(− ∏ l_{i}!)^{r}ϕ_{d−r}}{r!t^{r}}

Calabi-Yau: If l_{1} + ... + l_{m} = n + 1, then there exist power series:

f(q) = ∑_{d=1}^{∞} a_{d}q^{d} and g(q) = ∑_{d=1}^{∞} b_{d}q^{d}

with constant coefficients such that the two power series:

Φ(q) = ∑_{d=0}^{∞} ϕ_{d}q^{d} and Σ(q) = [S] + ∑_{d=1}^{∞} e_{∗} (\frac{[S]_{d}}{t(t − ψ)}) q^{d}

(with Laurent polynomial coefficients) satisfy:

Σ(q) = e^{\frac{h}{t} f(q)+g(q)}Φ(qe^{f(q)})

If we expand the right side, we get:

e^{\frac{h}{t} f(q)+g(q)}Φ(qe^{f(q)}) = ∑_{d=0}^{∞} ϕ_{d}(q^{d}e^{(d+ \frac{h}{t})f(q)+g(q)})

= ∑_{d} ∑_{0≤d_{1}<d_{2}<...<d_{r+1}=d} \frac{ϕ_{d_{1}} ∏_{i=1}^{r}(a_{d_{i+1}−d_{i}}(d_{1} + \frac{h}{t}) + b_{d_{i+1}−d_{i}})}{r!}q^{d}

so that the mirror conjecture is equivalent to the existence of constants a_{d} and b_{d} for d = 1, ..., ∞ such that:

e_{∗} (\frac{[S]_{d}}{t(t − ψ)}) = ∑_{0≤d_{1}<d_{2}<...<d_{r+1}=d} \frac{ϕ_{d_{1}} ∏_{i=1}^{r}(a_{d_{i+1}−d_{i}}(d_{1} + \frac{h}{t}) + b_{d_{i+1}−d_{i}})}{r!}

All the previous proofs proceed by finding sufficient conditions shared by the power series Σ(q) and the change of variables of the power series Φ(q) to uniquely characterize them, hence to conclude that they are the same. It is in this sense that one needs to “know the answer” beforehand. Our proof is completely different. We will explicitly produce the coefficients of the power series one coefficient at a time. Out of the proof it becomes clear how the self-similar properties of the coefficients of the Σ(q) in the Calabi-Yau and Fano of index one cases are a reflection of the self-similar properties of the boundary of the moduli spaces of stable maps.

Acknowledgements: This paper grew out of a seminar in the summer of 1998 co-organized with Holger Kley during which Chris Hacon and Herb Clemens made many useful suggestions. The use of MacPherson’s graph construction in Lemma 4.4 was suggested by Bill Fulton. Alexander Givental was very helpful when I reported on the Fano version of the result at MSRI in December 1998, and the key step in the Calabi-Yau proof arose out of a conversation with Ravi Vakil. Rahul Pandharipande, Pavel Etinghof and especially Bumsig Kim helped relate my original Calabi-Yau formula to the mirror conjecture and Bumsig Kim and Bong Lian made some useful remarks on an earlier version of the paper. Holger Kley very helpful with a critical revision of the paper and I’d like to thank Michael Thaddeus for explaining the basics of equivariant cohomology one sunny afternoon in Park City.

Note: A very recent preprint of Gathmann [11] explains how one might compute rational Gromov-Witten invariants of hypersurfaces without any torus actions. This seems promising, particularly in the general-type case.

2. Genus Zero Gromov-Witten Invariants on Projective Space.

Definition: If (C; p_{1}, ..., p_{k}) is a pointed genus-zero curve, then f : C → P^{n} is stable if:

(a) C has only nodes for singularities and the p_{i} are nonsingular points.

(b) Every component of C which is collapsed by f has at least three distinguished points (a point is distinguished if it is either a node or marked).

Existence Theorem (see [19],[1]): The moduli problem for isomorphism classes of flat families of genus-zero k-pointed stable maps of degree d to P^{n} is represented by a projective orbifold. This moduli space is denoted by \overline{M}_{0,k}(P^{n}, d) with universal curve C ≅ \overline{M}_{0,k+1}(P^{n}, d) and maps:

\overline{M}_{0,k}(P^{n}, d) ←^{π} C →^{e} P^{n}

For each 1 ≤ i ≤ k, if σ_{i} is the section of π determined p_{i}, then e_{i} := e∘σ_{i} is the evaluation of a stable map at the marked point p_{i}.

The relevant Chern classes on \overline{M}_{0,k}(P^{n}, d) come from three sources.

• The pull-back classes e_{i}^{∗}(h^{b})

• Witten’s cotangent classes ψ_{i} := c_{1}(N_{σ_{i}}^{∗}) for i = 1, ..., k. When k = 1, then ψ := ψ_{1} is the relative canonical class c_{1}(ω_{π}).

• The top chern class c_{r}(π_{∗}e^{∗}(E)) whenever E is a vector bundle on P^{n} which is generated by its global sections. If S ⊂ P^{n} is the zero locus of a general section of E, then this chern class will be denoted by [S]_{d} as before.

The classes [S]_{d} are all pulled back from \overline{M}_{0,0}(P^{n}, d), leading to relations:

Example: If S ⊂ P^{4} is a quintic hypersurface, then the “physical” number of rational curves of degree d on S is given by:

∫_{\overline{M}_{0,0}(P^{4},d)}[S]_{d} = \frac{1}{d} ∫_{\overline{M}_{0,1}(P^{4},d)} e^{∗}(h) ∪ [S]_{d} = \frac{1}{d} ∫_{P^{4}} e_{∗}[S]_{d} ∪ h

so that the cohomology classes e_{∗}[S]_{d} determine all the “physical” numbers of rational curves on the quintic (we will mostly ignore the subtleties of multiple covers in this paper).

The general k-point genus zero Gromov-Witten invariant of a complete intersection S ⊂ P^{n} of type (l_{1}, ..., l_{m}) is an intersection number of the form:

∫_{\overline{M}_{0,k}(P^{n},d)} p(e_{i}^{∗}(h), ψ_{i}) ∪ [S]_{d}

where p(x_{i}, y_{i}) is a polynomial (or power series) in 2k variables.

Notice that in contrast to zero-point invariants, there are interesting onepoint invariants on any complete intersection (because of the ψ’s), including projective space itself.

Definition: Let (C ⊃ P^{1}; p_{1}, ..., p_{k}) be a genus-zero curve with k marked points and a distinguished parametrized component P^{1} ⊂ C. Then a map f : C → P^{n} is stable if:

(a) C has only nodes for singularities and the p_{i} are nonsingular points.

(b) Every component of C which is collapsed by f is either distinguished or else has at least three distinguished points.

The existence theorem applies in the context of stable maps with a parametrized component. Following Givental, we’ll call these moduli spaces the “graph spaces” and denote them by \overline{N}_{0,k}(P^{n}, d). In addition to the maps of the theorem, the universal curve over the graph space admits an evaluation map ϵ : C → P^{1} leading to corresponding evaluation maps at the points. The zero-pointed graph space is rational via an explicit birational morphism:

u : \overline{N}_{0,0}(P^{n}, d) → P_{d}^{n} := P^{(n+1)(d+1)−1}

defined as follows. The general point of \overline{N}_{0,0}(P^{n}, d) is represented by a map f : P^{1} → P^{n} of degree d. Such a map is given by n + 1 polynomials of degree d and thus represents a point in the projective space P_{d}^{n}. Boundary points of the graph space are represented by maps f : C → P^{n} where C has several components, one of which is P^{1}. But if q_{1}, ..., q_{k} ∈ P^{1} are the nodes of C on P^{1}, and if the curve C_{i} growing out of q_{i} is mapped to P^{n} with degree d_{i}, then f maps P^{1} to P^{n} with degree d − ∑ d_{i}. Letting this map be given by polynomials p_{i}, we get a well-defined point of P_{d}^{n} by choosing linear forms q_{i}(x, y) dual to the points q_{i} and sending the stable map f to:

(∏_{i=1}^{k} q_{i}(x, y)^{d_{i}}p_{0}(x, y) : .... : ∏_{i=1}^{k} q_{i}(x, y)^{d_{i}}p_{n}(x, y))

(It is easy to exhibit u as a morphism. See Jun Li’s argument in [21]) There is an inclusion:

i : \overline{M}_{0,1}(P^{n}, d) ↪ \overline{N}_{0,0}(P^{n}, d)

defined as follows. Given a stable map f : C → P^{n} with one marked point p ∈ C, construct a new curve C′ = C ∪ P^{1} by joining P^{1} and C at p ∈ C and 0 ∈ P^{1}. Extend f to a map f′ : C′ → P^{n} by collapsing the P^{1} component to the image point f(p). Then there is a:

Basic Diagram:

\overline{N}_{0,0}(P^{n}, d) →^{u} P_{d}^{n}
i ↑  j ↑
\overline{M}_{0,1}(P^{n}, d) →^{e} P^{n}

where j(a_{0} : ... : a_{n}) = (a_{0}x^{d} : ... : a_{n}x^{d}).

3. Equivariant Basics. Let X be a compact complex manifold (or orbifold) equipped with a C^{∗} action. (We will let T = C^{∗}.) Then:

ET := C^{∞+1} − {0} → CP^{∞} =: BT

is the “universal” principal C^{∗} bundle, which we use to construct:

π_{X} : X_{T} := X ×_{C^{∗}} ET → ET/C^{∗} = BT

Following Borel, we define the equivariant cohomology by setting:

H_{T}^{∗}(X, Q) := H^{∗}(X_{T} , Q)

which is a module via π_{X}^{∗} over H^{∗}(BT, Q) ≅ Q[t]. At one extreme, if the T action were trivial, this would be the tensor product H^{∗}(X, Q)⊗_{Q} Q[t] while at the other extreme if the action were free, it would be the cohomology ring of the quotient H^{∗}(X/T, Q), which is torsion as a module over Q[t].

A vector bundle E over X is linearized if it is equipped with an action of T which is lifted from the action on X so that:

E_{T} → X_{T}

is a vector bundle which pulls back to E → X on each of the fibers of X_{T} over BT. As such, its chern classes represent elements of the equivariant cohomology ring, and one defines:

c_{i}^{T} (E) := c_{i}(E_{T}) ∈ H_{T}^{∗}(X, Q)

If f : X → X′ is a T -equivariant morphism of compact complex manifolds with T actions, let f : X_{T} → X′_{T} also denote the induced map on these spaces (which commutes with the projections to BT). The pull-back on equivariant cohomology is the ordinary pull-back with respect to the induced map f:

f^{∗} : H_{T}^{∗}(X′, Q) → H_{T}^{∗}(X, Q)

and if f is proper, then the equivariant proper push-forward is defined in the same way. We may now state:

The Atiyah-Bott Localization Theorem: Let X be a compact complex manifold with a T -action, let F_{1}, ..., F_{n} ⊂ X be the (necessarily smooth) connected components of the fixed-point locus and let i_{k} : F_{k} ↪ X denote their embeddings. Each normal bundle to F_{k} is canonically linearized, its top equivariant chern class ϵ^{T} (N_{F_{k}/X}) is invertible in H^{∗}(F_{k}, Q)[t, t^{−1}], and every torsion-free element c_{T} ∈ H_{T}^{∗}(X, Q) uniquely decomposes in H_{T}^{∗}(X, Q)⊗Q(t) as a sum of contributions from fixed-point loci:

c_{T} = ∑_{k=1}^{n}(i_{k})_{∗} \frac{i_{k}^{∗}(c_{T})}{ϵ^{T} (N_{F_{k}/X})}

Remark: The localization theorem as stated here and proved in [4] only applies to compact complex manifolds X. Generalizations to the case where X is an orbifold may be found in the papers of Graber-Pandharipande [15] and Kresch [20]. See [2] for a general reference on equivariant cohomology.

Corollary (Correspondence of Residues): Suppose f : X → Y is an equivariant map of compact complex manifolds (or orbifolds) with T actions and suppose i : F ↪ X and j : G ↪ Y are components of the fixed-point loci with the property that F is the unique fixed component to map to G. Then equivariant cohomology classes c_{T} on X satisfy:

(f|_{F})_{∗} (\frac{i^{∗}c_{T}}{ϵ^{T} (N_{F/X})}) = \frac{j^{∗}f_{∗}c_{T}}{ϵ^{T} (N_{G/Y})}

Proof of the Corollary: (See also Lemma 2.1 in [22].) The free part of c_{T} is a push-forward of classes from the fixed-point loci of X, by the theorem. Since F is the only such locus to map to G, only its contribution survives under j^{∗}f_{∗}c_{T} and under i^{∗}c_{T}, so we may as well assume that c_{T} = i_{∗}b_{T} for b_{T} ∈ H^{∗}(F, Q)[t, t^{−1}]. But then:

j^{∗}f_{∗}c_{T} = j^{∗}f_{∗}i_{∗}b_{T} = j^{∗}j_{∗}(f|_{F})_{∗}b_{T} = ϵ^{T} (N_{G/Y})(f|_{F})_{∗}b_{T}

whereas

b_{T} = \frac{i^{∗}i_{∗}b_{T}}{ϵ^{T} (N_{F/X})} = \frac{i^{∗}c_{T}}{ϵ^{T} (N_{F/X})}

completing the proof.

Returning to the basic diagram of the previous section, note that the “standard” linearized action of C^{∗} on P^{1}:

μ · (a, b) ↦ (a, μb)

induces actions on \overline{N}_{0,0}(P^{n}, d) and on P_{d}^{n} and that u : \overline{N}_{0,0}(P^{n}, d) → P_{d}^{n} is C^{∗}-equivariant. The vector bundles O_{P_{d}^{n}}(1) and π_{∗}e^{∗}E are equipped with natural linearizations, so we may define equivariant chern classes:

h^{T} := c_{1}^{T} (O_{P_{d}^{n}}(1)) and [S]_{d}^{T} := c_{r}^{T} (π_{∗}e^{∗}E)

on the spaces P_{d}^{n} and \overline{N}_{0,0}(P^{n}, d) respectively. Then:

Proposition 3.1:

(1a) \overline{M}_{0,1}(P^{n}, d) ⊂ \overline{N}_{0,0}(P^{n}, d) is a component of the fixed-point locus.

(1b) [S]_{d} and e^{∗}(h) extend to the equivariant classes [S]_{d}^{T} and u^{∗}(h^{T}).

(2) \overline{M}_{0,1}(P^{n}, d) is the only fixed component to map to P^{n}.

(3) The equivariant Euler classes are:

ϵ^{T} (N_{\overline{M}_{0,1}/\overline{N}_{0,0}}) = t(t − ψ) and ϵ^{T} (N_{P^{n}/P_{d}^{n}}) = ∏_{k=1}^{d}(h + kt)^{n+1}

Proof: (1a,b) and (2) are immediate. (For a careful treatment, see [6].) As for (3), there are inclusions:

\overline{M}_{0,1}(P^{n}, d) ⊂ \overline{M}_{0,1}(P^{n}, d) × P^{1} ⊂ \overline{N}_{0,0}(P^{n}, d)

The (equivariant!) first chern class of the normal bundle to the first inclusion is clearly t, and the second is c_{1}(T_{P^{1}}) − ψ, which restricts to t − ψ on the fixed-point locus \overline{M}_{0,1}(P^{n}, d). This gives one Euler class computation. The second Euler class computation follows from the Euler sequences for the tangent bundles to P^{n} and P_{d}^{n} (see §7 for a generalization).

The correspondence of residues immediately(!) therefore gives us:

e_{∗} (\frac{[S]_{d}}{t(t − ψ)}) = \frac{j^{∗}u_{∗}[S]_{d}^{T}}{∏_{k=1}^{d}(h + kt)^{n+1}}

Notice that we get every one-point Gromov-Witten invariant associated to S in this way by expanding t(t − ψ) = t^{−2} + t^{−3}ψ + t^{−4}ψ^{2} + ... so that:

∫_{\overline{M}_{0,1}(P^{n},d)} ψ^{a} ∪ e^{∗}(h^{b}) ∪ [S]_{d} = coeff of t^{−2−a} in ∫_{P^{n}} \frac{h^{b} ∪ j^{∗}u_{∗}[S]_{d}^{T}}{∏_{k=1}^{d}(h + kt)^{n+1}}

Thus our challenge is the computation of:

j^{∗}u_{∗}[S]_{d}^{T} ∈ H^{∗}(P^{n}, Q) ⊗ Q[t]

In one case, this is easy. Since j^{∗}u_{∗}1 = 1, we have:

One-Point Invariants of Projective Space:

e_{∗} (\frac{1}{t(t − ψ)}) = \frac{1}{∏_{k=1}^{d}(h + kt)^{n+1}}

4. The Fano Cases: It is difficult to see how to compute j^{∗}u_{∗}[S]_{d}^{T} when E is a non-split bundle. In this section we will develop techniques for making the computation when E is a direct sum of line bundles, and use these techniques to prove the Fano cases of the mirror conjecture.

Let D ⊂ \overline{N}_{0,0}(P^{n}, d) be the exceptional divisor for u : \overline{N}_{0,0}(P^{n}, d) → P_{d}^{n}. This is a divisor with normal crossings which we will eventually describe in detail, but first observe that there is a birational map

\overline{N}_{0,1}(P^{n}, d) → P^{1} × \overline{N}_{0,0}(P^{n}, d)

and that the exceptional divisor for this map lies over D. Then:

Proposition 4.1: There is an equivariant map:

Φ : π_{∗}e^{∗}(O_{P^{n}}(l)) → Sym^{dl}W^{∗} ⊗ u^{∗}O_{P_{d}^{n}}(l)

of vector bundles on \overline{N}_{0,0}(P^{n}, d) which is an isomorphism when restricted to the complement of the boundary divisor D.

Proof: For W ≅ C^{2} and V ≅ C^{n+1}, the map v below:

\overline{N}_{0,1}(P^{n}, d) →^{v} P(W) × P(Sym^{d}W^{∗} ⊗ V) − − > P^{n}

resolves a rational map which is linear in the second factor and has degree d in the first. The composition is the evaluation map e.

Corollary 4.2: Let f : D ↪ \overline{N}_{0,0}(P^{n}, d) be the inclusion. Then:

u_{∗}[S]_{d}^{T} = ∏_{i=1}^{m} ∏_{k=0}^{dl_{i}}(l_{i}h^{T} + kt) + u_{∗}f_{∗}c

for some equivariant class c supported on D.

Proof: This follows from the projection formula, the computation of the top equivariant chern class of Sym^{dl}W^{∗} ⊗ O_{P_{d}^{n}}(l) and an application of the proposition to the sum of vector bundles π_{∗}e^{∗}O_{P^{n}}(l_{i}).

The Fano of Index 2 or More Case: If l_{1} + ... + l_{m} < n then:

e_{∗} (\frac{[S]_{d}}{t(t − ψ)}) = \frac{j^{∗}u_{∗}[S]_{d}^{T}}{∏_{k=1}^{d}(h + kt)^{n+1}} = \frac{∏_{i=1}^{m} ∏_{k=0}^{dl_{i}}(l_{i}h + kt)}{∏_{k=1}^{d}(h + kt)^{n+1}} =: ϕ_{d}

i.e. the difference supported on D contributes nothing.

To prove this, we need to understand D better, as well as the behavior of the map Φ when restricted to D. The self-similar properties of both will lead to this formula and all the others. Our description will roughly follow Fulton-Pandharipande [10]:

Components of the boundary divisor: There are natural maps:

f_{i} : D̃_{i} := \overline{N}_{0,1}(P^{n}, d − i) ×_{P^{n}} \overline{M}_{0,1}(P^{n}, i) → \overline{N}_{0,0}(P^{n}, d)

that desingularize the d components of D. The map u pulls back as follows:

\overline{N}_{0,0}(P^{n}, d)  →^{u}  P_{d}^{n}
f_{i} ↑  j_{i} ↑
\overline{N}_{0,1}(P^{n}, d − i) ×_{P^{n}} \overline{M}_{0,1}(P^{n}, i) →^{π_{i}} P^{1} × P_{d−i}^{n}

Here π_{i} is the projection ρ_{i} : D̃_{i} → \overline{N}_{0,1}(P^{n}, d − i) followed by the map v defined in Proposition 4.1, and j_{i} is the “multi-linear” map:

P(W) × P(Sym^{d−i}W^{∗} ⊗ V) → P(Sym^{d}W^{∗} ⊗ V)

(we identify P(W) with P(W^{∗}) via the canonical W ≅ W^{∗} ⊗ ∧^{2}W).

A stratification of the graph space: We partially stratify the boundary of \overline{N}_{0,0}(P^{n}, d) in terms of “comb” types, i.e. sequences μ of the form:

0 ≤ d_{1} < d_{2} < ... < d_{r} < d_{r+1} = d

We define Δ_{i} := d_{i+1} − d_{i} and

D̃_{μ} := \overline{N}_{0,r}(P^{n}, d_{1}) ×_{(P^{n})^{r}} ∏_{i=1}^{r} \overline{M}_{0,1}(P^{n}, Δ_{i})

and note that the natural finite map f_{μ} : D̃_{μ} → D_{μ} ⊂ \overline{N}_{0,0}(P^{n}, d) is a “desingularization” of its image D_{μ}. The general point of D_{μ} is a stable map of degree d_{1} on the parametrized component and Δ_{i} on r other components (the “teeth” of the comb) each of which meets the parametrized component. Different comb types with the same r and d_{1} and the same sets of Δ_{i}’s will share the same image, and the map:

∐_{μ} D̃_{μ} → D_{μ}

from the union over all comb types with image D_{μ} has degree r!, the order of the permutation group of the set of Δ_{i}’s. The map u pulls back to D̃_{μ} according to:

\overline{N}_{0,0}(P^{n}, d) →^{u}  P_{d}^{n}
f_{μ} ↑  j_{μ} ↑
D̃_{μ}  →^{π_{μ}} (P^{1})^{r} × P_{d_{1}}^{n}

Here ρ_{μ} : D̃_{μ} → \overline{N}_{0,r}(P^{n}, d_{1}) and π_{μ} and j_{μ} are defined as before, with the obvious generalized map v : \overline{N}_{0,r}(P^{n}, d) → (P^{1})^{r} × P_{d}^{n}.

The following two lemmas are the heart of the Fano proof.

Lemma 4.3: The fixed-point locus j : P^{n} ↪ P_{d}^{n} factors uniquely through:

j : P^{n} →^{j′_{μ}} (P^{1})^{r} × P_{d_{1}}^{n} →^{j_{μ}} P_{d}^{n}

and if b is an equivariant cohomology class on (P^{1})^{r} × P_{d_{1}}^{n}, then:

j^{∗}(j_{μ})_{∗}b = (j′_{μ})^{∗}b ∪ \frac{∏_{k=1}^{d}(h + kt)^{n+1}}{t^{r} ∏_{k=1}^{d_{1}}(h + kt)^{n+1}}

Proof: j′_{μ} maps (0, ..., 0) × P^{n} ↪ (P^{1})^{r} × P^{n} ⟶^{(1,j)} (P^{1})^{r} × P_{d_{1}}^{n}

If j_{μ} were an embedding, then the second part of the lemma would be an immediate consequence of the excess intersection formula. Since it isn’t, we will instead use the localization theorem. The components of the fixed-point locus of the C^{∗} action on P_{d}^{n} are all copies of P^{n}, embedded via Segre as:

(e0 + (d − e)∞) × P^{n} ⊂ P^{d} × P^{n} ⊂ P_{d}^{n} = P(Sym^{d}W^{∗} ⊗ V)

and similarly, the components of the fixed-point locus in (P^{1})^{r} × P_{d_{1}}^{n} are all copies of P^{n}, embedded via:

s × (e0 + (d_{1} − e)∞) × P^{n} ⊂ (P^{1})^{r} × P_{d_{1}}^{n}

where s ∈ {0, ∞}^{r}. Of the fixed-point loci in this latter space, there is thus only one that maps to the image of j, namely the image of j′_{μ}. By the localization theorem, an equivariant class on (P^{1})^{r} × P_{d_{1}}^{n} is a sum of its contributions from fixed-point loci. The contribution from the image of j′_{μ} is easily computed to be j′_{μ}_{∗}b′ where:

b′ := \frac{j_{μ}^{′∗}b}{t^{r} ∏_{k=1}^{d_{1}}(h + kt)^{n+1}}

But this is the unique component to map to the image of j, hence:

j^{∗}j_{μ}_{∗}b = j^{∗}j_{∗}b′ = b′ ∪ ∏_{k=1}^{d}(h + kt)^{n+1}

Lemma 4.4: The equivariant virtual class [S]_{d}^{T} decomposes as:

[S]_{d}^{T} = ∑_{μ} \frac{1}{r!}f_{μ}_{∗}(c_{μ} ∪ π_{μ}^{∗} ∏_{i=1}^{m} ∏_{k=0}^{d_{1}l_{i}}(l_{i}h^{T} + kt))

where h^{T} is pulled back from the projection to P_{d_{1}}^{n} and the c_{μ} are equivariant cohomology classes which will be explicitly described in the proof.

Proof: We’ll prove this first when S is a hypersurface of degree l.

MacPherson’s graph construction [9, 23] gives a decomposition of the difference [S]_{d}^{T} − u^{∗} ∏_{k=0}^{ld}(lh^{T} + kt) as follows.

Let E_{d} = π_{∗}e^{∗}O(l), F_{d} = Sym^{dl}W^{∗} ⊗ u^{∗}O_{P_{d}^{n}}(l), and let G(dl + 1, E_{d} ⊕ F_{d}) be the Grassmann bundle. Consider the locus of scaled graphs:

\overline{N}_{0,0}(P^{n}, d) × A^{1} ⊂ G(dl + 1, E_{d} ⊕ F_{d}); (x, λ) ↦ Γ(λΦ_{x})

where Γ(λΦ_{x}) = {(e, λΦ_{x}(e))|e ∈ E_{d}(x)}, and let V ⊂ G(dl+1, E_{d} ⊕F_{d})×P^{1} be the closure of this locus. Let V_{∞} ⊂ G(dl + 1, E_{d} ⊕ F_{d}) be the fiber of V over {∞} ∈ P^{1}. One of the components of V_{∞} is a (reduced) copy of \overline{N}_{0,0}(P^{n}, d) itself, embedded in the Grassmann bundle via the fibers of F_{d}, and all other components map to proper subvarieties of \overline{N}_{0,0}(P^{n}, d). If V_{Z} is such a component, surjecting onto Z ⊂ \overline{N}_{0,0}(P^{n}, d) via the projection map π_{V_{Z}} : V_{Z} → \overline{N}_{0,0}(P^{n}, d), let n_{V_{Z}} be its multiplicity in V_{∞}. Then:

[S]_{d}^{T} − u^{∗} ∏_{k=0}^{ld}(lh^{T} + kt) = ∑_{V_{Z}} n_{V_{Z}}π_{V_{Z}}_{∗}c_{dl+1}(T_{V_{Z}})

where T_{V_{Z}} is the pull-back of the tautological sub-bundle on G(dl+1, E_{d}⊕F_{d}) (see [9], Example 18.1.6).

Now notice that Φ is the push-forward of the inclusion:

e^{∗}O_{P^{n}}(l) ≅ L(− ∑_{i=1}^{d} ilC_{i}) ↪ L ≅ v^{∗}O_{P^{1}×P_{d}^{n}}(dl, l)

where C_{i} ≅ \overline{N}_{0,1}(P^{n}, d−i)×_{P^{n}} \overline{M}_{0,2}(P^{n}, i) is the unparametrized component of the universal curve over D_{i} ⊂ D.

Since the divisors C_{i} are exactly the exceptional divisors for the map \overline{N}_{0,1}(P^{n}, d) → P^{1} × \overline{N}_{0,0}(P^{n}, d) it is clear that the sheaf inclusion holds with some negative coefficients of the C_{i}. The computation of the coefficients follows from the observation that the pull-back f^{∗}O_{P^{n}}(l) under a stable map f : C → P^{n} corresponding to a general point of D_{i} has degree il on the unparametrized component of C.

Alternatively, as in the pointwise description of u, if we are given a stable map f : C → P^{n} such that the parametrized component C_{0} has degree d_{1} and such that r curves “bubble” off this component with degrees Δ_{1}, ..., Δ_{r} at nodes q_{1}, ..., q_{r} ∈ C_{0}, then Φ is the following linear map at the point of \overline{N}_{0,0}(P^{n}, d) corresponding to f:

H^{0}(C, f^{∗}O(l)) → H^{0}(C_{0}, f^{∗}O(l)|_{C_{0}}) ≅ H^{0}(P^{1}, O_{P^{1}}(d_{1}l)) ⊂ H^{0}(P^{1}, O_{P^{1}}(dl))

where the last inclusion comes from multiplying by the Δ_{i}lth powers of the equations of the linear forms dual to the q_{i}. It follows that Φ enjoys the following pleasant properties:

(i) Φ successively drops rank in codimension one, always along the selfintersection strata of the boundary divisor D. Moreover, if such a stratum is not a D_{μ} (I call these other strata “hairy combs”), then it can be embedded in a comb stratum such that the generic corank of Φ is the same along the two strata.

(ii) Φ drops rank “transversally” along comb-type strata. That is, if Φ has generic coranks m and n along two comb-type strata which intersect transversally along a third comb-type stratum, then Φ has generic corank m + n along the intersection.

Precisely, Φ factors as follows when it is pulled back to D̃_{μ}:

f_{μ}^{∗}Φ : f_{μ}^{∗}E_{d} → ρ_{μ}^{∗}E_{d_{1}} ⟶^{ρ_{μ}^{∗}Φ} ρ_{μ}^{∗}F_{d_{1}} ⊂ f_{μ}^{∗}F_{d}

where E_{d_{1}}, F_{d_{1}} and Φ on \overline{N}_{0,r}(P^{n}, d_{1}) are pulled back from \overline{N}_{0,0}(P^{n}, d_{1}), and the map ρ_{μ}^{∗}F_{d_{1}} → f_{μ}^{∗}F_{d} is pulled back via π_{μ}^{∗} from the projection to (P^{1})^{r} × P_{d_{1}}^{n} of the sheaf inclusion given by multiplication by the Δ_{i}l powers of the equations of the appropriate diagonals on P^{1} × (P^{1})^{r} × P_{d_{1}}^{n}:

O(d_{1}l, 0, ..., 0, l) → O(dl, Δ_{1}l, ..., Δ_{r}l, l)

For boundary divisors, the kernel of f_{i}^{∗}Φ is pulled back from the kernel of the evaluation map of bundles on \overline{M}_{0,1}(P^{n}, i):

0 → E_{i}^{1} → π_{∗}e^{∗}O_{P^{n}}(l) → e^{∗}O_{P^{n}}(l) → 0

via the projection τ_{i} : D̃_{i} → \overline{M}_{0,1}(P^{n}, i). If Z ⊂ \overline{M}_{0,1}(P^{n}, i) is the image of the section σ_{1}, then E_{i}^{1} is the first in a filtration of sub-bundles:

0 = E_{i}^{il+1} ⊂ E_{i}^{il} ⊂ ... ⊂ E_{i}^{k} = π_{∗} (e^{∗}O_{P^{n}}(l) ⊗ O(−kZ)) ⊂ ... ⊂ E_{i}^{1} ⊂ E_{i}

which filter ker(f_{i}^{∗}Φ) according to higher-order behavior of Φ along D_{i}:

0 = τ_{i}^{∗}E_{i}^{il+1} ⊂ ⋯ ⊂ τ_{i}^{∗}E_{i}^{k} = f_{i}^{∗}(ker(Φ|_{kD_{i}})) ⊂ ⋯ ⊂ τ_{i}^{∗}E_{i}^{1} ⊂ f_{i}^{∗}E_{d}

There is a filtration of f_{i}^{∗}F_{d} given by the spans of the images of Φ|_{kD_{i}}:

ρ_{i}^{∗}F_{d−i} = π_{i}^{∗}F_{d−i}^{1} ⊂ ⋯ ⊂ π_{i}^{∗}F_{d−i}^{k} ⊂ ⋯ ⊂ f_{i}^{∗}F_{d}

where F_{d−i}^{k} is the push-forward of O_{P^{1}×P^{1}×P_{d−i}^{n}}((d − i)l + k − 1, k − 1, l).

From these descriptions of Φ along boundary divisors (and induction) we can conclude that infinitessimal versions of the pleasant properties (i) and (ii) also hold for Φ. Property (i) tells us immediately that any Z ⊂ \overline{N}_{0,0}(P^{n}, d) in the image of a component V_{Z} must be a comb-type boundary stratum, hence that this gives a decomposition of the form:

[S]_{d}^{T} = ∑_{D_{μ}} γ_{μ}

summed over the images (including \overline{N}_{0,0}(P^{n}, d)) of the comb-types. But thanks to the filtrations of f_{i}^{∗}E_{d} and f_{i}^{∗}F_{d}, we can be much more explicit. Namely, there are il components V_{D_{i}}^{k} over each boundary divisor D_{i}, each of which is a birational image of a P^{1}-bundle V k ; _{D_{i}} on D̃_{i}. By a local coordinate computation, V_{D_{i}}^{k} has multiplicity k in V_{∞} and tautological bundle fitting into an extension of the form:

0 → π_{i}^{∗}F_{d−i}^{k} ⊕ τ_{i}^{∗}E_{i}^{k+1} → T_{V k ; _{D_{i}}} → O(−1) → 0

hence the codimension one contributions γ_{i} have the desired form:

γ_{i} = f_{i}_{∗} (c_{i} ∪ c_{(d−i)l+1}^{T}(π_{i}^{∗}F_{d−i}^{1})) = f_{i}_{∗} (c_{i} ∪ π_{i}^{∗} ∏_{k=0}^{(d−i)l}(lh^{T} + kt))

where c_{i} = ∑_{k=1}^{il}(−k)τ_{i}^{∗}c_{il−k}^{T}(E_{i}^{k+1}) ∪ π_{i}^{∗}c_{k−1}^{T}(F_{d−i}^{k}/F_{d−i}^{1}).

The second property of Φ allows us to construct the γ_{μ} inductively. Namely, if we fix a V_{D_{i}}^{k} (or rather the corresponding P^{1}-bundle over D̃_{i}) and apply the MacPherson construction to the map:

f_{i}^{∗}E_{d}/τ_{i}^{∗}E_{i}^{1} ≅ ρ_{i}^{∗}E_{d−i} →^{ρ_{i}^{∗}Φ} ρ_{i}^{∗}F_{d−i} = π_{i}^{∗}F_{d−i}^{1} (⊂ T_{V k ; _{D_{i}}} )

then the components of this V_{∞} over comb type strata of \overline{N}_{0,1}(P^{n}, d − i) map birationally to the components of the original V_{∞} over comb type strata of \overline{N}_{0,0}(P^{n}, d). It follows by induction that for each μ (which includes an ordering of the Δ_{i}), we will obtain towers of P^{1}-bundles mapping birationally to the components V_{D_{μ}} over D_{μ}, and we obtain the Lemma with the following explicit formula for the c_{μ}:

c_{μ} = ∏_{i=1}^{r} ∑_{k_{i}=1}^{Δ_{i}l}(−k_{i})τ_{Δ_{i}}^{∗}c_{Δ_{i}l−k_{i}}^{T}(E_{Δ_{i}}^{k_{i}+1}) ∪ π_{μ}^{∗}c_{k_{i}−1}^{T}(F_{d_{i}}^{k_{i}}/F_{d_{i}}^{1})

where F_{d_{i}}^{1} ⊂ F_{d_{i}}^{k_{i}} (we are abusing notation slightly in the interest of clarity) is pushed forward from the inclusion of sheaves on P^{1} × (P^{1})^{r} × P_{d_{1}}^{n}:

O(d_{i}l, Δ_{1}l, ..., Δ_{i−1}l, 0, ..., 0, l) ↪ O(d_{i}l+k_{i}−1, Δ_{1}l, ..., Δ_{i−1}l, k_{i}−1, 0, ..., 0, l)

and τ_{Δ_{i}} is induced from the projection (so again we are abusing notation) \overline{N}_{0,1}(P^{n}, d_{i}) ×_{P^{n}} \overline{M}_{0,1}(P^{n}, Δ_{i}) → \overline{M}_{0,1}(P^{n}, Δ_{i}).

In the complete intersection case, the components V k ; _{D_{i}} are only generically P^{1}-bundles, and the explicit form of the c_{i} is thus more difficult to compute, though the existence of the decomposition of the Lemma and the inductive nature of the c_{μ} terms follow from the same argument. Alternatively, one can use the product:

[S]_{d}^{T} = ∏_{i=1}^{m}[S_{l_{i}}]_{d}^{T}

and the decompositions of each of the hypersurface virtual classes [S_{l_{i}}]_{d}^{T} along with excess intersection theory to obtain the desired decomposition of [S]_{d}^{T}. There is one subtlety, in that contributions from hairy combs need to be pushed forward to their underlying (bald) comb strata.

Proof of the Fano of Index ≥ 2 Case: Using the decomposition of [S]_{d}^{T} from Lemma 4.4, we have:

u_{∗}[S]_{d}^{T} = ∑_{μ} \frac{1}{r!}u_{∗}f_{μ}_{∗}(c_{μ} ∪ π_{μ}^{∗}b_{μ}) = ∑_{μ} \frac{1}{r!}j_{μ}_{∗} (π_{μ}_{∗}c_{μ} ∪ b_{μ})

where b_{μ} = ∏_{i=1}^{m} ∏_{k=0}^{d_{1}l_{i}}(l_{i}h^{T} + kt). On the other hand, the codimension of the image of j_{μ} : (P^{1})^{r} × P_{d_{1}}^{n} → P_{d}^{n} is

(n + 1)(d − d_{1}) − r ≥ n(d − d_{1})

because of the obvious inequality r ≤ d − d_{1}. Thus the codimension of j_{μ}_{∗}b_{μ} is already at least:

(∑_{i=1}^{m} l_{i})d_{1} + n(d − d_{1}) > codim(u_{∗}[S]_{d}^{T}) = (∑_{i=1}^{m} l_{i})d

since (∑_{i=1}^{m} l_{i}) < n by assumption. This implies that π_{μ}_{∗}c_{μ} = 0 on all strata of positive codimension, hence u_{∗}[S]_{d}^{T} = ∏_{i=1}^{m} ∏_{k=0}^{dl_{i}}(l_{i}h^{T} + kt). Together with the argument from §3, this proves the desired result:

e_{∗} (\frac{[S]_{d}}{t(t − ψ)}) = ϕ_{d}

The Fano of Index One Case: Suppose l_{1} + ... + l_{m} = n. Then:

e_{∗} (\frac{[S]_{d}}{t(t − ψ)}) = ∑_{r=0}^{d} \frac{(− ∏ l_{i}!)^{r}ϕ_{d−r}}{r!t^{r}}

Proof: By Lemmas 4.3 and 4.4, the contribution of the stratum indexed by μ to the ratio:

\frac{j^{∗}u_{∗}[S]_{d}^{T}}{∏_{k=1}^{d}(h + kt)^{n+1}}

is given by:

\frac{j^{∗}j_{μ}_{∗}(π_{μ}_{∗}c_{μ} ∪ b_{μ})}{r! ∏_{k=1}^{d}(h + kt)^{n+1}} = \frac{j_{μ}^{′∗}π_{μ}_{∗}c_{μ} ∪ j_{μ}^{′∗}b_{μ}}{r!t^{r} ∏_{k=1}^{d_{1}}(h + kt)^{n+1}} = \frac{j_{μ}^{′∗}π_{μ}_{∗}c_{μ} ∪ ϕ_{d_{1}}}{r!t^{r}}

where b_{μ} is defined as in the higher index case.

When (∑ l_{i})d_{1} + (n + 1)(d − d_{1}) − r > (∑ l_{i})d, then as we’ve already seen, π_{μ}_{∗}c_{μ} = 0 for dimension reasons. Under the Fano index one assumption, this inequality holds unless r = d − d_{1}, in which case equality holds. This occurs exactly once for each r from 0 to d, namely when μ is the comb type:

0 ≤ d_{1} < d_{1} + 1 < ... < d_{1} + r = d

For these comb strata, π_{μ}_{∗}c_{μ} is a codimension zero class, so that the formula here amounts to the identity: π_{μ}_{∗}c_{μ} = (− ∏_{i=1}^{m} l_{i}!)^{r} · 1.

In the hypersurface case, we note that the positive chern classes of the π_{1}^{∗}(F_{d−1}^{k}/F_{d−1}^{1}) cannot contribute to the push-forward, hence the only term which does contribute to π_{1}_{∗}c_{1} is −τ_{i}^{∗}c_{l−1}(E_{1}^{2}). The classes c_{1}(E_{1}^{k}/E_{1}^{k+1}) are easily computed to be e^{∗}(lh) + kψ. The e^{∗}(lh) do not contribute to the push-forward, and we are left with the term −l!ψ^{l−1} which pushes forward to −l!. Using the inductive description of c_{μ}, we readily obtain the desired computation π_{μ}_{∗}c_{μ} = (−l!)^{r}.

In the complete intersection case, from the product of decompositions [S]_{d}^{T} = ∏_{i=1}^{m}[S_{l_{i}}]_{d}^{T} and excess intersection, we can conclude that modulo terms pulled back under π_{1}^{∗}, the term c_{1} agrees with f_{1}^{∗}(D_{1})^{m−1} ∏_{i=1}^{m} −τ_{1}^{∗}c_{l_{i}−1}(E_{l_{i},1}^{2}) where E_{l_{i},1} is the bundle π_{∗}e^{∗}O_{P^{n}}(l_{i}) on \overline{M}_{0,1}(P^{n}, 1). For all i, we have f_{i}^{∗}(D_{i}) = −τ_{i}^{∗}ψ − ρ_{i}^{∗}ψ, the second term of which does not contribute to the push-forward, and treating the rest of the expression as in the previous paragraph, we obtain

π_{1}_{∗}c_{1} = π_{1}_{∗}τ_{i}^{∗} ((−ψ)^{m−1} ∏_{i=1}^{m}(−l_{i}!ψ^{l_{i}−1})) = − ∏_{i=1}^{m} l_{i}!

As in the hypersurface case, the desired form of π_{μ}_{∗}c_{μ} follows from the inductive description of the c_{μ}.

5. The Calabi-Yau Case. The decomposition of Lemma 4.4 together with the push-pull formula of Lemma 4.3 always yield:

e_{∗} (\frac{[S]_{d}}{t(t − ψ)}) = ∑_{μ} \frac{ϕ_{d_{1}} ∪ j′_{μ}^{∗}π_{μ}_{∗}c_{μ}}{r!t^{r}}

so that the challenge is to find a method for computing the j′_{μ}^{∗}π_{μ}_{∗}c_{μ}. This seems to be very difficult in the general-type case, but in the Calabi-Yau case there is a wonderful simplification:

Lemma 5.1: When S is Calabi-Yau, the decomposition of Lemma 4.4 has the following additional properties:

(a) The equivariant chern classes λ_{μ}(h, t) := j_{μ}^{′∗}π_{μ}_{∗}c_{μ} satisfy

λ_{μ}(h, t) = λ_{d_{1},d_{2}}(h, t)λ_{d_{2},d_{3}}(h, t) ⋯ λ_{d_{r},d}(h, t)

where λ_{d−i,d}(h, t) = j_{i}^{′∗}π_{i}_{∗}c_{i}.

(b) The “simple” classes λ_{d−i,d}(h, t) are linear and satisfy: λ_{d−i,d}(h, t) = λ_{0,i}(h + (d − i)t, t)

We immediately obtain the following: A Formula for the Calabi-Yau Case: If l_{1} + ... + l_{m} = n + 1, then there are linear forms λ_{i}(h, t) only depending upon (l_{1}, ..., l_{m}) such that:

e_{∗} (\frac{[S]_{d}}{t(t − ψ)}) = ∑_{μ} \frac{ϕ_{d_{1}} ∪ ∏_{i=1}^{r} λ_{Δ_{i}}(h + d_{i}t, t)}{r!t^{r}}

Proof: Set λ_{i}(h, t) = λ_{0,i}(h, t) and apply the lemma.

Proof of Lemma 5.1: Recall that Lemma 4.4 gave us:

c_{μ} = ∏_{i=1}^{r} ∑_{k_{i}=1}^{Δ_{i}l}(−k_{i})τ_{Δ_{i}}^{∗}c_{Δ_{i}l−k_{i}}(E_{Δ_{i}}^{k_{i}+1}) ∪ π_{μ}^{∗}c_{k_{i}−1}(F_{d_{i}}^{k_{i}}/F_{d_{i}}^{1})

(again we will start with the hypersurface case) and in particular,

c_{i} = ∑_{k=1}^{il}(−k)τ_{i}^{∗}c_{il−k}(E_{i}^{k+1}) ∪ π_{i}^{∗}c_{k−1}(F_{d−i}^{k}/F_{d−i}^{1})

Recall also the construction of the map π_{i}:

π_{i} : \overline{N}_{0,1}(P^{n}, d − i) ×_{P^{n}} \overline{M}_{0,1}(P^{n}, i) →^{ρ_{i}} \overline{N}_{0,1}(P^{n}, d − i) →^{v} P^{1} × P_{d−i}^{n}

Let ξ^{T} be the equivariant hyperplane class on P^{1}. Then one computes:

c_{k−1}^{T}(F_{d−i}^{k}/F_{d−i}^{1}) = ∏_{j=1}^{k−1}(l(h^{T} + (d − i)t) + jξ^{T})

On the other hand, by the Künneth decomposition of in Δ ⊂ P^{n} × P^{n}, the class ρ_{i}_{∗}τ_{i}^{∗}c_{il−k}^{T}(E_{i}^{k+1}) is of the form κ_{a}^{i}e^{∗}(h^{a}) for a rational number κ_{a}^{i} that is clearly independent of d. Putting these together, we see that if S is Calabi-Yau, then π_{i}_{∗}c_{i} is a codimension one class, hence a = 1 or a = 0, and:

π_{i}_{∗}c_{i} = −v_{∗}κ_{1}^{i}e^{∗}(h) − 2κ_{0}^{i}(l(h^{T} + (d − i)t) + ξ^{T})

But e^{∗}(h) = v^{∗}(h^{T} + (d − i)ξ^{T}) − ϵ on \overline{N}_{0,1}(P^{n}, d − i) for a v-exceptional divisor ϵ (from the explicit description of Φ in Lemma 4.4). Hence:

j′_{i}^{∗}π_{i}_{∗}c_{i} = −κ_{1}^{i}(h + (d − i)t) − 2κ_{0}^{i}(l(h + (d − i)t) + t)

proving part (b) of the lemma with λ_{0,i}(h, t) = (−κ_{1}^{i} − 2lκ_{0}^{i})h − 2κ_{0}^{i}t.

Part (a) is proved similarly. It follows from the projection formula that

ρ_{μ}_{∗}c_{μ} = ∏_{i=1}^{r} ∑_{k_{i}}(−k_{i})ρ_{μ}_{∗}τ_{Δ_{i}}^{∗}c_{Δ_{i}l−k_{i}}^{T}(E_{Δ_{i}}^{k_{i}+1}) ∪ v^{∗}c_{k_{i}−1}^{T}(F_{d_{i}}^{k_{i}}/F_{d_{i}}^{1})

If we let ξ_{i}^{T} be the hyperplane class of the ith copy of P^{1}, then:

c_{k_{i}−1}^{T}(F_{d_{i}}^{k_{i}}/F_{d_{i}}^{1}) = ∏_{j=1}^{k_{i}−1}(l(h^{T} + d_{1}t + Δ_{1}ξ_{1}^{T} + ... + Δ_{i−1}ξ_{i−1}^{T}) + jξ_{i}^{T})

and as in the proof of (b), ρ_{μ}_{∗}τ_{Δ_{i}}^{∗}c_{Δ_{i}l+1}(E_{Δ_{i}}^{2}) = κ_{1}^{i}(v^{∗}(h^{T} + d_{i}t) − ϵ_{i}) and ρ_{μ}_{∗}τ_{Δ_{i}}^{∗}c_{Δ_{i}l+2}(E_{Δ_{i}}^{3}) = κ_{0}^{i}. Different ϵ_{i} divisors multiply together to yield vexceptional classes, so putting all this together, we get:

j′_{μ}^{∗}π_{μ}_{∗}c_{μ} = ∏_{i=1}^{r}(−κ_{1}^{i}(h + d_{i}t) − 2κ_{0}^{i}(l(h + d_{i}t) + t))

proving the lemma in the hypersurface case.

The complete intersection case is proved similarly. The only wrinkle in that case is the presence of normal classes to the strata in the decomposition. But f_{i}^{∗}(D_{i}) = τ_{i}^{∗}ψ + ψ_{P^{1}} + ϵ where ϵ is exceptional for the v map, and thus (modulo ϵ) does not depend upon d. Once again, the lemma now follows from the linearity of the ρ_{i}_{∗}c_{i} and the inductive description of the c_{μ}.

6. Using the Formula. The real beauty of the Calabi-Yau formula:

e_{∗} (\frac{[S]_{d}}{t(t − ψ)}) = ∑_{μ} \frac{ϕ_{d_{1}} ∪ ∏_{i=1}^{r} λ_{Δ_{i}}(h + d_{i}t, t)}{r!t^{r}}

is that it recursively computes itself! This is well-known, but I include the computation here for the reader’s enjoyment.

Namely, consider the simple comb type {0 ≤ 0 < d} which contributes:

\frac{ϕ_{0} ∪ λ_{d}(h, t)}{t} = \frac{∏_{i=1}^{m} l_{i}h ∪ λ_{d}(h, t)}{t}

to right side of the formula. This term is irrelevant to the left side of the formula, which only involves the powers t^{−2}, t^{−3}, ... so the “error” coefficients of t^{−1} and t^{0} (there are no more when S is Calabi-Yau) in the rest of the right side of the formula determine this term. Since these only depend upon λ_{1}, ..., λ_{d−1}, we therefore have an inductive construction of the λ_{d}.

We may read off the coefficients of λ_{d}(h, t) = α_{d}h + β_{d}t via:

α_{d}t^{−1} = \frac{1}{∏_{i=1}^{m} l_{i}} ∫_{P^{n}} \frac{h^{n−m−1} ∪ ϕ_{0} ∪ λ_{d}(h, t)}{t}

β_{d} = \frac{1}{∏_{i=1}^{m} l_{i}} ∫_{P^{n}} \frac{h^{n−m} ∪ ϕ_{0} ∪ λ_{d}(h, t)}{t}

In other words, not only does the formula in degree d compute the onepoint Gromov-Witten invariants, but via the “error” coefficients, it computes the form λ_{d} which is to be used in higher degrees!

Let S be the quintic threefold in P^{4}. Then

e_{∗} (\frac{[S]_{d}}{t(t − ψ)}) = n_{d}h^{3}t^{−2} + m_{d}h^{4}t^{−3}

and it follows from the projection formula that the “actual” physical number of rational curves of degree d on S is n_{d}/d = −m_{d}/2. For the reader who is unused to this and nervous about the fact that these are not typically integers, the Aspinwall-Morrison formula (see [3]) translates these numbers into the expected numbers of immersed rational curves of degree d. The formula now produces:

n_{1} = 2875, λ_{1} = −(770)h − (120)t
n_{2} = 4876875/4, λ_{2} = −(421375)h − (60000)t
n_{3} = 8564575000/9, λ_{3} = −(436236875)h − (59937500)t

n_{4} = 15517926796875/16, λ_{4} = −(17351562078125/6)h − (390555125000)t

Via the Aspinwall-Morrison formula, this translates into:

\frac{n_{d}}{d} = ∑_{e|d} \frac{N_{e}}{e^{3}}

where N_{e} is the expected number of immersed curves of degree e. This gives:

N_{1} = 2875, N_{2} = 609250, N_{3} = 317206375 and N_{4} = 242467530000

the well-known numbers of rational curves of degree ≤ 4 on the quintic.

7. A Relative Version. Given a projectivized vector bundle:

π : P(V) → X

over a projective manifold X, there are relative moduli spaces:

π_{M} : \overline{M}_{0,k}(P(V), d) → X and π_{N} : \overline{N}_{0,k}(P(V), d) → X

for stable maps to the fibers of π. The fibers of π_{M} and π_{N} are the moduli spaces we studied previously, and these moduli spaces are equipped with evaluation maps (to P(V)) and forgetful maps with the usual properties. The definition of P_{d}^{n} is readily generalized to:

P(V)_{d} := P(Sym^{d}(W^{∗}) ⊗ V)

and the basic diagram in the relative setting is:

\overline{N}_{0,0}(P(V), d) →^{u} P(V)_{d}
i ↑  j ↑
\overline{M}_{0,1}(P(V), d) →^{e} P(V)

All the results of the paper carry through unchanged, with one exception. Proposition 3.1(3) now needs to take into account the chern classes of V. Namely, if α_{1}, ..., α_{n+1} are the chern roots of π^{∗}V, then:

ϵ^{T} (N_{P(V)/P(V)_{d}}) = ∏_{k=1}^{d} ∏_{j=1}^{n+1}(h + α_{j} + kt)

which follows from the diagram:

0  0
↓  ↓
O_{P(V)} =  O_{P(V)}
↓  ↓
0 → π^{∗}V (1) → Sym^{d}(W^{∗}) ⊗ π^{∗}V (1)|_{P(V)} → N_{P(V)/P(V)_{d}} → 0
↓  ↓  ∥
0 → T_{P(V)/X} →  T_{P(V)_{d}/X}|_{P(V)}  → N_{P(V)/P(V)_{d}} → 0
↓  ↓
0  0

We get the following interesting formula already in degree one:

Relative Schubert Calculus: Chern classes on the relative flag bundle Fl(1, 2, V) over X push forward to P(V) via:

e_{∗} (\frac{σ(q_{1}, q_{2})}{t(t − ψ)}) = \frac{σ(h, h + t)}{∏_{j=1}^{n+1}(h + α_{j} + t)} + O(t^{−1})

where σ(q_{1}, q_{2}) is any chern class pulled back from the relative Grassmann bundle G(2, V) and expressed as a symmetic polynomial in the chern roots of the (dual of the) universal sub-bundle S^{∗}.

Proof: S^{∗} pulls back to π_{∗}e^{∗}O_{P(V)}(1) on Fl(1, 2, V) = \overline{M}_{0,1}(P(V), 1). By Proposition 4.1, the map Φ : π_{∗}e^{∗}O_{P(V)}(1) → W^{∗} ⊗ u^{∗}O_{P(V)_{d}}(1) of bundles over \overline{N}_{0,0}(P(V), 1) is an isomorphism off the unique boundary stratum D_{1}. It follows from the argument of §3 (and Lemma 4.3) that:

e_{∗} (\frac{σ(q_{1}, q_{2})}{t(t − ψ)}) = \frac{σ(h, h + t)}{∏_{j=1}^{n+1}(h + α_{j} + t)} + \frac{j′_{1}^{∗}π_{1}_{∗}c_{1}}{t}

for some equivariant cohomology class c_{1} supported on D_{1}.

We can express the denominator in terms of Segre classes:

\frac{1}{∏_{j=1}^{n+1}(h + α_{j} + t)} = \frac{1}{(h + t)^{n+1}} s_{(h+t)^{−1}}(π^{∗}V)

where s_{(h+t)^{−1}}(π^{∗}V) = 1 + (h + t)^{−1}s_{1}(π^{∗}V) + (h + t)^{−2}s_{2}(π^{∗}V) + ....

In the case of a “linear” complete intersection S ⊂ P(V) defined by m sections of O_{P(V)}(1), then [S]_{1} = (q_{1}q_{2})^{m} and we get a generalized “Porteous” formula for lines:

e_{∗} (\frac{[S]_{1}}{t(t − ψ)}) = h^{m}(h + t)^{m−n−1}s_{(h+t)^{−1}}(π^{∗}V) + O(t^{−1})

The t^{−2} coefficient is h^{m}(s_{m−n+1}(π^{∗}V)−s_{m−n}(π^{∗}V)h). When we multiply this by h and push forward to X, we get the formula:

s_{m−n+1}^{2}(V) − s_{m−n}(V)s_{m−n+2}(V)

for the push-forward to X of the class [S]_{1} on the Grassmann bundle. This agrees with the classical Porteous formula for lines (see e.g. [16]).

For general degree and a general complete intersection S ⊂ P(V), let:

ϕ_{d} = \frac{∏_{i=1}^{m} ∏_{k=0}^{dl_{i}}(l_{i}h + kt)}{∏_{k=1}^{d} ∏_{j=1}^{n+1}(h + α_{j} + kt)}

Then the exact analogues of the Fano formulas hold with this ϕ_{d}, and for Calabi-Yau’s we have the analogous:

Relative Calabi-Yau Formula: If ∑_{i=1}^{m} l_{i} ≤ n + 1, then there are linear equivariant classes λ_{e}(h, t) ∈ H^{∗}(P(V), Q)[t] such that:

e_{∗} (\frac{[S]_{d}}{t(t − ψ)}) = ∑_{μ} \frac{ϕ_{d_{1}} ∪ ∏_{i=1}^{r} λ_{Δ_{i}}(h + d_{i}t, t)}{r!t^{r}}

As an application, consider the “linear relative Calabi-Yau’s”, i.e. the S ⊂ P(V) that are cut out by n + 1 transverse sections of O_{P(V)}(1). The following was first proved in [5] in the context of symmetric products of a smooth curve C, where the g − 1st symmetric product C_{g−1} is an example of a linear relative Calabi-Yau over the Jacobian of C:

Proposition 7.1: The three-point Gromov-Witten invariants:

∫_{\overline{M}_{0,3}(P(V),d)} e_{1}^{∗}h ∪ e_{2}^{∗}h ∪ e_{3}^{∗}c ∪ [S]_{d}

of linear relative Calabi-Yau’s are independent of d ≥ 1. Remark: When X admits no rational curves, this says that the quantum product of h with itself in the quantum cohomology ring of S is of the form:

h ∗ h = h^{2} + bq + bq^{2} + bq^{3} + ...

Proof: By the projection formula and the relative Schubert calculus, the proposition is equivalent to e_{∗}([S]_{d}) = (1/d^{2})e_{∗}([S]_{1}) = (1/d^{2})h^{n+1}(s_{2} − s_{1}h).

In this case, ϕ_{d} = h^{n+1}s_{(h+t)^{−1}}(π^{∗}V)s_{(h+2t)^{−1}}(π^{∗}V)...s_{(h+dt)^{−1}}(π^{∗}V)

= h^{n+1} (1 + \frac{s_{1}}{t} ∑_{k=1}^{d} \frac{1}{k} + \frac{s_{1}^{2}}{t^{2}} ∑_{1≤j<k≤d} \frac{1}{jk} + \frac{s_{2} − s_{1}h}{t^{2}} ∑_{k=1}^{d} \frac{1}{k^{2}} + ...)

We make the following (only valid for linear Calabi-Yau’s): Assumption: The λ_{e}(h, t) of the Calabi-Yau formula are independent of h.

We can separate variables in the formula:

e_{∗} (\frac{[S]_{d}}{t(t − ψ)}) = ∑_{d_{1}} ϕ_{d_{1}} ∑_{0<e_{1}<...<e_{r}=d−d_{1}} \frac{∏_{i=1}^{r}(λ_{e_{i}−e_{i−1}}(t)/t)}{r!}

and if we express this in terms of generating functions, we get:

h^{n+1} + ∑_{d>0} q^{d}e_{∗} (\frac{[S]_{d}}{t(t − ψ)}) = (∑_{d≥0} q^{d}ϕ_{d})exp(∑_{e>0} q^{e}λ_{e}(t)/t)

Let λ_{e}(t) = a_{e}t + b_{e} where a_{e} ∈ Q and b_{e} is a cohomology class of degree one coming from X. Then equating coefficients of t^{0} gives:

1 = (1 + q + q^{2} + ...)exp(∑_{e} a_{e}q^{e})

so that ∑_{e} a_{e}q^{e} = log(1 − q) and hence a_{e} = −\frac{1}{e}.

Similarly, the t^{−1} coefficients give 0 = \frac{s_{1}}{t} ∑_{k=1}^{d} \frac{1}{k} + ∑_{k=1}^{d} \frac{b_{k}}{t} for each d, so that by induction, b_{e} = −\frac{s_{1}}{e}, and we get λ_{e}(t) = −\frac{1}{e}(t + s_{1}) which is indeed independent of h. Plug these λ_{e}(t) in the generating function:

h^{n+1} + ∑ q^{d}e_{∗}[S]_{d}t^{−2} + ... =

(∑_{e} q^{e}ϕ_{e})(1 − q)(1 + \frac{s_{1}}{t} log(1 − q) + \frac{s_{1}^{2}}{2!t^{2}} log(1 − q)^{2} + ...)

and the t^{−2} term on the right is (1/d^{2})h^{n+1}(s_{2} − s_{1}h), as desired.

8. The Relationship with the Mirror Conjecture. The astute reader will have noticed that the Calabi-Yau formula in §6 does not quite match with the mirror conjecture for Calabi-Yau complete intersections from the introduction! We will establish the latter with the aid of the following:

Proposition 8.1: Let

F (q) = ∑_{d=0}^{∞} ∑_{0<d_{1}<...<d_{r}=d} \frac{∏_{i=1}^{r}(y_{d_{i}−d_{i−1}} + x_{d_{i}−d_{i−1}}d_{i−1})}{r!}q^{d}

Then log(F (q)) is a linear function of y_{1}, y_{2}, ....

Proof: (Pavel Etinghof) Let E be the Euler vector field:

E = ∑_{k=1}^{∞} kx_{k} \frac{∂}{∂x_{k}} + ky_{k} \frac{∂}{∂y_{k}}

Then we may rewrite F (q) as:

F (q) = exp(∑_{d=1}^{∞} q^{d}y_{d} + q^{d}x_{d}E) · 1 = ∑_{r=0}^{∞} \frac{(∑ q^{d}y_{d} + q^{d}x_{d}E)^{r}}{r!} · 1

Consider the function G(t, q) = exp(t∑_{d=1}^{∞} q^{d}y_{d} + q^{d}x_{d}E) · 1 satisfying G(1, q) = F, G(0, q) = 1 and q \frac{∂G}{∂q} = E · G (because the Euler vector field is homogeneous of degree zero). Thus:

\frac{∂G}{∂t} = ∑_{d=1}^{∞} q^{d}y_{d}G + ∑_{d=1}^{∞} q^{d}x_{d}E · G

hence

\frac{∂log(G)}{∂t} = \frac{1}{G} \frac{∂G}{∂t} = ∑_{d=1}^{∞} q^{d}y_{d} + ∑_{d=1}^{∞} q^{d+1}x_{d} \frac{∂log(G)}{∂q}.

By the fundamental theorem of calculus(!) this gives:

log(G)(t, q) = ∑_{d=1}^{∞} tq^{d}y_{d} + ∑_{d=1}^{∞} q^{d+1}x_{d} ∫_{0}^{t} \frac{∂log(G(s, q))}{∂q} ds

which proves the desired linearity of log(F) = log(G)(1, q) in the y variables by induction on the power of q.

Corollary: If we let F (q) = exp(∑_{d=1}^{∞} y′_{d}q^{d}) then

y′_{d} = ∑_{0<d_{1}<...<d_{r}=d} \frac{y_{d_{1}} ∏_{i=2}^{r}(x_{d_{i}−d_{i−1}}d_{i−1})}{r!}

Proof: Cast out the non-linear terms (in the y_{k}) from the identity:

∑_{0<d_{1}<...<d_{r}=d} \frac{∏_{i=1}^{r} y′_{d_{i}−d_{i−1}}}{r!} = ∑_{0<d_{1}<...<d_{r}=d} \frac{∏_{i=1}^{r}(y_{d_{i}−d_{i−1}} + x_{d_{i}−d_{i−1}}d_{i−1})}{r!}

Finally, suppose S is Calabi-Yau and let λ_{d} = α_{d}h+β_{d}t. Then our formula may be written as follows:

Σ(q) = ∑_{d} ϕ_{d} ∑_{0<e_{1}<...<e_{r}=e} \frac{∏_{i=1}^{r}(α_{e_{i}−e_{i−1}}(d + \frac{h}{t}) + β_{e_{i}−e_{i−1}}) + α_{e_{i}−e_{i−1}}e_{i−1}}{r!} q^{d+e}

The proposition applies to give us the new coordinates y′_{e} which are linear in α_{e}(d + \frac{h}{t}) + β_{e} (and polynomial in the α_{k}’s), hence of the form:

y′_{e} = a_{e}(d + \frac{h}{t}) + b_{e}

where a_{e} and b_{e} are independent of d. This gives:

Σ(q) = ∑_{d} ϕ_{d} ∑_{0<e_{1}<...<e_{r}=e} \frac{∏_{i=1}^{r}(a_{e_{i}−e_{i−1}}(d + \frac{h}{t}) + b_{e_{i}−e_{i−1}})}{r!}q^{d+e}

which proves the mirror conjecture.

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University of Utah, Salt Lake City, UT 84112

bertram@math.utah.edu