arXiv:1405.2045v2  [math.AG]  12 May 2014
Absolute vs. Relative Gromov-Witten Invariants
Mohammad F. Tehrani and Aleksey Zinger∗
May 3, 2014. Updated: May 13, 2014
Abstract
In light of recent attempts to extend the Cieliebak-Mohnke approach for constructing Gromov-
Witten invariants to positive genera, we compare the absolute and relative Gromov-Witten
invariants of compact symplectic manifolds when the symplectic hypersurface contains no rele-
vant holomorphic curves. We show that these invariants are then the same, except in a narrow
range of dimensions of the target and genera of the domains, and provide examples when they
fail to be the same.
Contents
1
Introduction
1
2
Review of GW-invariants
6
3
Proof of Theorem 1
12
3.1
By direct comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
3.2
Via the symplectic sum formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
3.3
Extension to virtual cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
4
Details on the counter-examples
22
4.1
Genus 1 degree 0 invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
4.2
Genus 2 degree 1 invariants of P1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
4.3
Genus 3 degree 1 primary invariants of P4 . . . . . . . . . . . . . . . . . . . . . . . .
32
4.4
The δ =0, 1 numbers in Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
5
The Cieliebak-Mohnke approach to GW-invariants
42
1
Introduction
Gromov-Witten invariants of a compact symplectic manifold (X, ω) are certain, often delicate,
counts of J-holomorphic curves in X; they play prominent roles in symplectic topology, alge-
braic geometry, and string theory. For a symplectic hypersurface V in (X, ω), i.e. a closed sym-
plectic submanifold of real codimension 2, relative Gromov-Witten invariants of (X, ω, V ) count
∗Partially supported by NSF grant 0846978
1

J-holomorphic curves in X with speciﬁed contacts with V . If V contains no (non-constant) J-
holomorphic curves that could possibly contribute to a speciﬁc absolute invariant of X, one could
hope that such an absolute invariant equals the corresponding relative invariant with the basic
contact condition, divided by the number of orderings of the contact points. We show that this is
indeed the case, except in a narrow range of dimensions of the target and genera of the domains;
see Theorem 1 and Remarks 1.2-1.4. Examples 1-3 illustrate the three cases when the absolute and
relative invariants can fail to be equal. We also overview the geometric approach to constructing
genus 0 Gromov-Witten invariants suggested in [2] and the attempts to extend this approach to
higher genera in [9] and [18].
For g, k ∈Z≥0, we denote by Mg,k the Deligne-Mumford moduli space of stable k-marked genus g
connected nodal curves. If 2g+k<3, Mg,k is empty with this deﬁnition, though it is often convenient
to formally take it to be a point in these cases, as done when we set up notation for GW-invariants
below. If g, k ∈Z≥0, A ∈H2(X; Z), and J is an an almost complex structure on X compatible
with (or tamed by) ω, let Mg,k(X, A) denote the moduli spaces of stable J-holomorphic k-marked
maps from connected nodal curves of genus g. If in addition V ⊂X is a symplectic hypersurface,
s≡(s1, . . . , sℓ) is an ℓ-tuple of positive integers such that
s1 + . . . + sℓ= A · V,
(1.1)
and J is compatible with V in a suitable sense, let M
V
g,k;s(X, A) denote the moduli spaces of stable
J-holomorphic (k +ℓ)-marked maps from connected nodal curves of genus g that have contact
with V at the last ℓmarked points of orders s1, . . . , sℓ, respectively.
These moduli spaces are
introduced in [20, 16, 21] under certain assumptions on J and reviewed in Section 2. The expected
dimensions of these two moduli spaces are given by
dimvir Mg,k(X, A) = 2
 ⟨c1(X), A⟩+ (n−3)(1−g) + k

,
dimvir M
V
g,k;s(X, A) = 2
 ⟨c1(X), A⟩+ (n−3)(1−g) + k + ℓ(s)−|s|

,
(1.2)
where ℓ(s)≡ℓand |s|≡s1+. . .+sℓ. In particular, these dimensions are the same if
s = 1ℓ≡
 1, . . . , 1
| {z }
ℓ

,
i.e. the tuple s imposes no contact conditions on degree A J-holomorphic curves, beyond what a
generic such curve can be expected to satisfy.
For each i=1, . . . , k, let
evi : Mg,k(X, A), M
V
g,k;s(X, A) −→X
(1.3)
be the i-th evaluation map. It sends the equivalence class of a J-holomorphic map u : Σ −→X
from a genus g nodal curve Σ to u(xi)∈X, where xi ∈Σ is the i-th marked point. Let
st: Mg,k(X, A), M
V
g,k;s(X, A) −→Mg,k
(1.4)
denote the forgetful morphism to the Deligne-Mumford space. If 2g+k≥3, it sends the equivalence
class of a J-holomorphic map u: Σ−→X from a marked genus g nodal curve Σ to the equivalence
class of the stable k-marked genus g nodal curve Σ′ obtained from (Σ, x1, . . . , xk) by contracting the
2

u
st
x1
x2
u
st
x1
x2
Figure 1: Examples of the stabilization morphism (1.4).
unstable components (spheres with one or two special, i.e. nodal or marked, points); see Figure 1.
Along with the virtual class for Mg,k(X, A), constructed in [34] in “semi-positive” cases, in [1] in
the algebraic case, and in [8, 24] in the general case, the ﬁrst morphisms in (1.3) and (1.4) give rise
to the (absolute) GW-invariants of (X, ω):
GWX
g,A(κ; α1, . . . , αk) ≡

st∗κ
k
Y
i=1
ev∗
i αi, [Mg,k(X, A)]vir

∀κ∈H∗(Mg,k), αi ∈H∗(X),
(1.5)
where H∗denotes the cohomology with Q-coeﬃcients. The number above vanishes unless
deg κ +
k
X
i=1
deg αi = 2
 ⟨c1(X), A⟩+ (n−3)(1−g) + k

.
(1.6)
Along with the virtual class for M
V
g,k;s(X, A), the second morphisms in (1.3) and (1.4) give rise to
the relative GW-invariants of (X, ω, V ):
GWX,V
g,A;s(κ; α1, . . . , αk) ≡

st∗κ
k
Y
i=1
ev∗
i αi, [M
V
g,k;s(X, A)]vir

∀κ∈H∗(Mg,k), αi ∈H∗(X).
(1.7)
Such a virtual class is constructed in [16] in “semi-positive” cases and in [21] in the algebraic case
and is used in [20] in the general case; see Section 2 for more details. The number in (1.7) vanishes
unless
deg κ +
k
X
i=1
deg αi = 2
 ⟨c1(X), A⟩+ (n−3)(1−g) + k + ℓ(s)−|s|

.
The numbers in (1.5) and (1.7) are (graded-) symmetric and linear in the inputs αi. By the latter
property, they give rise to well-deﬁned numbers
GWX
g,A(κ; α), GWX,V
g,A;s(κ; α) ∈Q
∀κ∈H∗(Mg,k), α∈H∗(X)⊗k .
The numbers
GWX
g,A(α) ≡GWX
g,A(1; α)
and
GWX,V
g,A;s(α) = GWX,V
g,A;s(1; α)
3

are called primary GW-invariants or GW-invariants with primary insertions. In some cases, the num-
bers (1.5) and (1.7) can be described as signed counts of concrete geometric objects, J-holomorphic
or (J, ν)-holomorphic maps; see Sections 2 and 5.
Remark 1.1. The numbers (1.5) and (1.7) do not cover GW-invariants that arise from natural
classes on Mg,k(X, A) and M
V
g,k;s(X, A), such as ψ-classes (which are generally diﬀerent from the
ψ-classes on Mg,k pulled back by the morphism (1.4)) and the euler classes of obstruction bundles
of various kinds; both types of classes are central to GW-theory. The geometric constructions of
the numbers (1.5) and (1.7) reviewed in Sections 2 and 5 are not compatible with such classes.
Deﬁnition 1. Let (X, ω) be a compact symplectic manifold, g ∈Z≥0, and A∈H2(X; Z). A sym-
plectic hypersurface V ⊂X is (g, A)-hollow if there exists an ω|V -tame almost complex structure JV
on V such that every non-constant JV -holomorphic map u: Σ−→V from a smooth connected Rie-
mann surface Σ satisﬁes
g(Σ) > g,
or
⟨u∗ω, Σ⟩>ω(A),
or
⟨u∗ω, Σ⟩= ω(A), u∗[Σ]̸=A.
Theorem 1. Suppose (X, ω) is a compact symplectic manifold of real dimension 2n, g, k ∈Z≥0,
A∈H2(X; Z), and V ⊂X is a (g, A)-hollow symplectic hypersurface such that A·V ≥0. If
(g, A) ̸= (1, 0)
and
(n−5)g(g−1) ≥0 ,
(1.8)
then the absolute GW-invariants (1.5) and the basic corresponding relative GW-invariants (1.7)
agree:
GWX
g,A(κ; α) =
1
(A · V )!GWX,V
g,A;1A·V (κ; α)
∀κ ∈H∗(Mg,k), α ∈H∗(X)⊗k .
(1.9)
This identity also holds if κ=1, A̸=0, and either g=2 or n̸=4.
Remark 1.2. By Theorem 1, the absolute GW-invariants with primary insertions, i.e. κ=1, and
the corresponding relative invariants in degree A ̸= 0 may fail to be equal only if n = 4 and g ≥3
at the same time; the possibility of such a failure is illustrated by Example 3. With non-trivial
constraints κ, the two invariants in degree A̸=0 may fail to be equal only if 1≤n≤4 and g ≥2 at
the same time; the possibility of such a failure is illustrated by Example 2. Example 1, which is
motivated by [17, Example 12.5], illustrates the possibility of failure of (1.9) with A=0.
Remark 1.3. In Section 3, we give two versions of essentially the same proof of Theorem 1.
The ﬁrst version is a direct comparison of the two invariants. It is particularly suitable for con-
sidering the independence of the geometrically constructed curve counts of the chosen Donaldson
divisor in [2, 9, 18]; see Section 5. The argument involves several cases; in all, but one of them, the
conclusion is established by a dimension-counting argument. In the exceptional case, when κ=1,
n = 3, and g ≥3, we also use the fact that λ2
g = 0 on Mg; see [30, (5.3)]. The second version of
the proof is a formal application of the symplectic sum formula for GW-invariants, as in the setup
introduced in [26, Section 2.2], successfully applied in the genus 0 case in [14], and used in the
attempted proof of [18, Theorem 11.1]. As indicated by [18], establishing (1.9) in this way leads
to the analogue of (1.9) for virtual classes, at least in the algebraic category; see Section 3.3. The
crucial step in the proof is that we start by taking a generic regularization for the maps to V , i.e. a
horizontal deformation of the parameters (J, ν) along V , before deforming the parameter ν in the
4

normal direction to V . The order of the deformations is reversed in [18, Section 12], which makes
the horizontal directions not even deﬁned and crucially misses out the opportunity to quickly settle
most cases of Theorem 1.2. The argument in [18, Section 11] instead misinterprets [4, (9)] to arrive
at the conclusion of Lemma 3.2 in Section 3.3 without the restrictions in (3.16) and the conclusion
of Corollary 3.3 without the restrictions in (3.17) or the projective assumptions on X and A.
Remark 1.4. It is suﬃcient to verify the condition of Deﬁnition 1 for J-holomorphic maps
u: Σ−→V that are simple in the sense of [29, Section 2.5]. By [29, Section 3.2], moduli spaces of
such maps have the expected dimensions for a generic ωV -tame (or compatible) almost complex
structure JV on V . Thus, by the ﬁrst equation in (1.2), V is (g, A)-hollow if
A′ · V > ⟨c1(X), A′⟩+ (n−4)(1−g′)
(1.10)
for all g′ ∈Z≥0 with g′ ≤g and A′ ∈H2(X; Z) with ω(A′)≤ω(A) such that A′ can be represented by
a J0-holomorphic curve for some ﬁxed ω-tame almost complex structure J0 on X. By Gromov’s
Compactness Theorem, the number of such classes A′ is ﬁnite. Since ω(A′)>0 for all such classes,
(1.10) can be achieved by taking V to be Poincare dual to a suﬃciently high multiple of a rational
symplectic form close to ω. Such V , called Donaldson hypersurfaces, always exist by [3] and are
central to the construction of genus 0 curve counts in [2] and its attempted extensions to positive
genera in [9] and [18]; see Section 5.
Remark 1.5. For the purposes of the direct proof of Theorem 1 in Section 3.1, it is suﬃcient to
assume that there exist an almost complex structure JV on V and an arbitrarily small perturba-
tion ν on V as in Section 2 so that every (JV , ν)-holomorphic map u : Σ −→V from a smooth
connected Riemann surface Σ satisﬁes
u∗[Σ] = 0,
or
g(Σ) > g,
or
⟨u∗ω, Σ⟩>ω(A),
or
⟨u∗ω, Σ⟩= ω(A), u∗[Σ]̸=A.
For the purposes of the proof of Theorem 1 via the symplectic sum formula in Section 3.2, it is
suﬃcient to assume the GW-invariants of V of genus g′ and in the class A′ vanish whenever A′ ̸=0,
g′ ≤g, and ω(A′)≤ω(A).
The next three examples illustrate diﬀerent cases when (1.9) fails to hold. They are justiﬁed in
Section 4.
Example 1. Suppose (X, ω) is a compact symplectic manifold of real dimension 2n and V ⊂X is a
symplectic hypersurface. Let j∈H2(M1,1) be the Poincare dual of a generic point and α∈H2(X).
The genus 1 degree 0 GW-invariants of (X, ω) and (X, ω, V ) satisfy
GWX
1,0(j; 1) = χ(X)
2
= 1
0!GWX;V
1,0;()(j; 1) + χ(V )
2
,
(1.11)
GWX
1,0(α) = −⟨α cn−1(X), X⟩
24
= 1
0!GWX,V
1,0;()(α) −⟨α|V cn−2(V ), V ⟩
24
,
(1.12)
where χ(·) is the euler characteristic and () in the subscript is the length 0 contact vector (and
thus gives 0! in the denominators).
Example 2. Denote by P1 the one-dimensional complex projective space with the standard sym-
plectic form and by Vδ ⊂P1 the symplectic hypersurface consisting of δ ∈Z≥0 distinct points. Let
pt∈H2(P1) be the Poincare dual of a point and κ∈H2(M2,2) be the Poincare dual of the divisor
5

whose generic element consists of two components, one of genus 2 and the other of genus 0; see the
bottom right diagram in Figure 1. The genus 2 degree 1 GW-invariants of P1 and (P1, Vδ) satisfy
1
240 = GWP1
2,1(κ4; pt, pt) = 1
δ!GWP1,Vδ
2,1;1δ(κ4; pt, pt) +
δ
1, 152 .
(1.13)
Example 3. Denote by P4 the four-dimensional complex projective space with the standard sym-
plectic form and by Vδ ⊂P4 a smooth complex hypersurface of degree δ. Let pt ∈H8(P4) be the
Poincare dual of a point. The genus 3 degree 1 primary GW-invariants of P4 and (P4, Vδ) satisfy
−
37
82, 944 = GWP4
3,1(pt) = 1
δ!GWP4,Vδ
3,1;1δ(pt) + δ(δ2−5δ+8)
72, 576
.
(1.14)
Remark 1.6. The proof in Section 4.3 of the second equality in (1.14) applies to primary GW-
invariants of P4 and (P4, Vδ) in degree d as long as Vδ contains no curves of genus at most 3 and
degree at most d that pass through the constraints. In these cases, the last term in (1.14) should
be multiplied by the genus 0 degree d absolute GW-invariant with an extra point insertion. The
condition on Vδ in particular excludes the d=1 GW-invariants with primary insertions (P1, P2) if
δ =1.
We review the deﬁnitions of absolute and relative invariants in Section 2, focusing on the geo-
metric diﬀerences for the requirements on generic parameters (J, ν) determining the two types
of invariants. These diﬀerences are fundamental to establishing Theorem 1 in Section 3 and the
claims of Examples 1-3 in Section 4. In Section 5, we review the Cieliebak-Mohnke approach to
constructing GW-invariants and relate a key issue in this approach to Theorem 1 and Examples 1-3.
This note was inspired by the discussions regarding [18, Theorem 11.1] and the related aspects
of [9] at and following the SCGP workshop on constructing the virtual cycle in GW-theory. We
would like to thank the SCGP for organizing and hosting this very enlightening workshop and the
authors of [18] and [9] for bringing up important questions concerning relative GW-invariants. We
are also grateful to C.-C. Liu and D. Maulik for sharing invaluable insights on [18, Theorem 11.1]
and C. Faber for providing intersection numbers for Deligne-Mumford moduli spaces.
2
Review of GW-invariants
Let g, k∈Z≥0 be such that 2g+k≥3,
}
Mg,k −→Mg,k
(2.1)
be the branched cover of the Deligne-Mumford space of stable k-marked genus g curves by the
associated moduli space of Prym structures constructed in [25], and
πg,k : qUg,k −→}
Mg,k
be the corresponding universal curve. A k-marked genus g nodal curve with a Prym structure is a
connected compact nodal k-marked Riemann surface (Σ, z1, . . . , zk) of arithmetic genus g together
with a holomorphic map stΣ : Σ −→qUg,k which surjects on a ﬁber of πg,k and takes the marked
points of Σ to the corresponding marked points of the ﬁber.
6

If J is an almost complex structure on a smooth manifold X, A∈H2(X; Z), and
ν ∈Γg,k(X, J) ≡Γ
  qUg,k×X, π∗
1(T ∗qUg,k)0,1⊗Cπ∗
2(TX, J)

,
(2.2)
a k-marked genus g degree A (J, ν)-map is a tuple (Σ, z1, . . . , zk, stΣ, u) such that (Σ, z1, . . . , zk, stΣ)
is a genus g k-marked nodal curve with a Prym structure and u: Σ−→X is a smooth (or Lp
1, with
p> 2) map such that
u∗[Σ] = A
and
¯∂J,ju

z ≡1
2
 du + J ◦du ◦j

= ν(stΣ(z), u(z))
∀z ∈Σ,
where j is the complex structure on Σ. Two such tuples are equivalent if they diﬀer by a reparametriza-
tion of the domain commuting with the maps to qUg,k.
Suppose (X, ω) is a compact symplectic manifold and J is an ω-tame almost complex structure. By
[34, Corollary 3.9], the space Mg,k(X, A; J, ν) of equivalence classes of k-marked genus g degree A
(J, ν)-maps is Hausdorﬀand compact in Gromov’s convergence topology. By [34, Theorem 3.16],
for a generic ν each stratum of Mg,k(X, A; J, ν) consisting of simple (not multiply covered) maps
of a ﬁxed combinatorial type is a smooth manifold of the expected even dimension, which is less
than the expected dimension of the subspace of simple maps with smooth domains (except for
this subspace itself). By [34, Theorem 3.11], the last stratum has a canonical orientation. By [34,
Proposition 3.21], the images of the strata of Mg,k(X, A; J, ν) consisting of multiply covered maps
under the morphism
st×ev1×. . .×evk : Mg,k(X, A; J, ν) −→Mg,k×Xk
(2.3)
are contained in images of maps from smooth even-dimensional manifolds of dimension less than
this stratum if ν is generic and (X, ω) is semi-positive in the sense of [29, Deﬁnition 6.4.1]. Thus,
(2.3) is a pseudocycle. Intersecting it with generic representatives for the Poincare duals of the
classes κ and αi and dividing by the order of the covering (2.1), we obtain the (absolute) GW-
invariants (1.5) of a semi-positive symplectic manifold (X, ω) in the stable range, i.e. with (g, k)
such that 2g+k≥3. If g=0, the same reasoning applies with ν =0 and yields the same conclusion if
(X, ω) satisﬁes a slightly stronger condition (c1(A)>0 instead of c1(A)≥0 in [29, Deﬁnition 6.4.1]).
For general symplectic manifolds (X, ω), the GW-invariants (1.5) are deﬁned in [8, 24] using Ku-
ranishi structures (or ﬁnite-dimensional approximations) and local perturbations ν as in (2.2).
Suppose in addition V ⊂X is a closed symplectic hypersurface and J(TV )=TV . Thus, J induces
a complex structure iX,V on (the ﬁbers of) the normal bundle
πX,V : NXV ≡TX|V

TV −→V.
A connection ∇NXV in (NXV, iX,V ) induces a splitting of the exact sequence
0 −→π∗
X,V NXV −→T(NXV )
dπX,V
−→π∗
X,V TV −→0
(2.4)
of vector bundles over NXV which restricts to the canonical splitting over the zero section and is
preserved by the multiplication by C∗; see [37, Lemma 1.1]. For each trivialization
NXV |U ≈U ×C
7

over an open subset U of V , there exists α ∈Γ(U; T ∗V ⊗R C) such that the image of π∗
X,V TV
corresponding to this splitting is given by
T hor
(x,w)(NXV ) =

(v, −αx(v)w): v∈TxV
	
∀(x, w)∈U ×C.
The isomorphism (x, w)−→(x, w−1) of U ×C∗maps this vector space to
T hor
(x,w−1)
 (NXV )∗
=

(v, w−2αx(v)w): v∈TxV
	
=

(v, αx(v)w−1): v∈TxV
	
∀(x, w)∈U ×C∗.
Thus, the splitting of (2.4) induced by a connection in (NXV, iX,V ) extends to a splitting of the
exact sequence
0 −→T vrt(PXV ) −→T(PXV )
dπX,V
−→π∗
X,V TV −→0,
where
πX,V : PXV ≡P
 NXV ⊕V ×C

−→V ;
(2.5)
this splitting restricts to the canonical splittings over
PX,∞V ≡P(NXV ⊕0)
and
PX,0V ≡P(0 ⊕X×C)
(2.6)
and is preserved by the multiplication by C∗. Via this splitting, the almost complex structure
JV ≡JX|V and the complex structure iX,V in the ﬁbers of πX,V induce an almost complex struc-
ture JX,V on PXV which restricts to almost complex structures on PX,∞V and PX,0V and is pre-
served by the C∗-action. Furthermore, the projection πX,V : PXV −→V is (JV , JX,V )-holomorphic.
By [37, Lemma 2.2], ξ ∈Γ(V, NXV ) is (JX,V , J|V )-holomorphic if and only if ξ lies in the kernel of
the ¯∂-operator on (NXV, iX,V ) corresponding to the connection used above.
For each m∈Z≥0, let
XV
m =
 X ⊔{1}×PXV ⊔. . . ⊔{m}×PXV

/∼,
where
(2.7)
x ∼1×PX,∞V |x ,
r×PX,0V |x ∼(r+1)×PX,∞V |x
∀x∈V, r = 1, . . . , m−1;
see Figure 2. Deﬁne
qm: XV
m −→X
by
qm(x) =
(
x,
if x∈X;
πX,V ([v, w]),
if x=(r, [v, w])∈r×PXV .
We denote by Jm the almost complex structure on XV
m so that
Jm|X = J
and
Jm|{r}×PXV = JX,V
∀r = 1, . . . , m.
For each (c1, . . . , cm)∈C∗, deﬁne
Θc1,...,cm : XV
m −→XV
m
by
Θc1,...,cm(x) =
(
x,
if x∈X;
(r, [crv, w]),
if x=(r, [v, w])∈r×PXV.
(2.8)
This diﬀeomorphism is biholomorphic with respect to Jm and preserves the ﬁbers of the projection
PXV −→V and the sections PX,0V and PX,∞V .
8

Suppose J(TV )=TV and J is ω-tame. We denote by ∇the Levi-Civita connection of the metric gJ
on X determined by (ω, J) as in [29, (2.1.1)], by e∇the corresponding JX-linear connection, as above
[29, (3.1.3)], and by b∇the connection given by
b∇vζ = e∇vζ −1
4

∇JζJ + J∇ζJ
	
(v)
∀ζ ∈Γ(X; TX), v ∈TX.
By the next paragraph, the ¯∂-operator
b∇0,1 : Γ(X; TX) −→Γ
 X; T ∗X0,1⊗CTX

,
ζ −→1
2
 ∇·ζ + J∇J·ζ

restricts to an operator
b∇0,1 : Γ(V ; TV ) −→Γ
 V ; T ∗V 0,1⊗CTV

,
and thus descends to a ¯∂-operator
Γ(V ; NXV ) −→Γ
 V ; T ∗V 0,1⊗CNXV

corresponding to some connection ∇NXV in (NXV, iX,V ); see [37, Section 2.3]. Let JX,V denote
the complex structure on PXV induced by JV and ∇NXV as in the paragraph above the previous
one; it depends only on the above ¯∂-operator and not on the connection ∇NXV realizing it.
If in addition u : (Σ, j) −→(X, J) is (J, j)-holomorphic, i.e. ¯∂J,ju = 0, the linearization of the
¯∂J,j-operator at u is given by
Du : Γ(Σ, u∗TX) −→Γ0,1
J,j (Σ; u∗TX) ≡Γ
 Σ, (T ∗Σ, j)0,1⊗Cu∗(TX, J)

,
Duξ = 1
2
 b∇uξ + {u∗J} ◦b∇uξ ◦j) + 1
4N u
J (ξ, du),
(2.9)
where b∇u and N u
J are the pull-backs of the connection b∇and of the Nijenhuis tensor NJ of J
normalized as in [29, p18], respectively, by u; see [29, (3.1.7)]. If in addition u(Σ)⊂V ,
Du
 Γ(Σ, u∗TV )

⊂Γ0,1
J,j (Σ, u∗TV ),
because the restriction of Du to Γ(Σ; u∗TV ) is the linearization of the ¯∂J,j-operator at u for the
space of maps to V . Thus, Du descends to a ﬁrst-order diﬀerential operator
DNXV
u
: Γ(Σ, u∗NXV ) −→Γ0,1
J,j (Σ, u∗NXV ).
(2.10)
By (2.9), this operator is C-linear if
NJ(v, w) ∈TxV
∀v, w∈TxX, x∈V.
(2.11)
Under this assumption, ξ ∈Γ(Σ, u∗NXV ) is a (JX,V , j)-holomorphic map if and only if ξ ∈ker DNXV
u
.
If J(TV )⊂V , Σ is a smooth connected Riemann surface, and u: Σ−→X is a J-holomorphic map
such that u(Σ)̸⊂V , then u−1(V ) is an isolated set of points zi; see the beginning of [5, Section 5.1].
Furthermore, u has a well-deﬁned order of contact with V at each zi ∈u−1(V ), ordV
ziu∈Z+; if Σ is
compact,
X
zi∈u−1(V )
ordV
ziu = u∗[Σ] · V .
9

V
PX,∞V
PX,0V
PX,∞V
PX,0V
X
1×PXV
2×PXV
z1
z2
z3
z4
Figure 2: The image of a relative map with k=1 and s=(2, 2, 2) to the space XV
2 .
If A∈H2(X; Z), g, k, ℓ∈Z≥0, and s=(s1, . . . , sℓ)∈(Z+)ℓis a tuple satisfying (1.1), let
MV
g,k;s(X, A) ⊂Mg,k+ℓ(X, A)
(2.12)
denote the subset of equivalence of stable J-holomorphic maps u from marked genus g nodal curves
(Σ, z1, . . . , zk+ℓ) such that
u−1(V ) =

zk+1, . . . , zk+ℓ
	
and
ordV
zk+iu = si
∀i = 1, . . . , ℓ.
If J satisﬁes (2.11), we denote by
M
V
g,k;s(X, A) ⊃MV
g,k;s(X, A)
(2.13)
the space of equivalence classes of stable JX,V -holomorphic maps u : Σ −→XV
m, with m ∈Z≥0,
from connected marked genus g nodal curves (Σ, z1, . . . , zk+ℓ) such that the restriction of u to each
irreducible component of XV
m is contained in either X or in {r}×PXV for some r =1, . . . , m, but
not in V or {r}×PX,0V for any r,
qm∗u∗[Σ] = A,
ord{m}×PX,0V
zk+i
u = si
∀i = 1, . . . , ℓ,
and the orders of contacts of the two branches at each node on V , {r} × PX,0V , or {r} × PX,0V
agree; see Figure 2. Two maps u as above are equivalent if they diﬀerent by an isomorphism of
marked domains and a composition with an isomorphism (2.8); see [5, Section 4.2] for more details.
The relative moduli spaces M
V
g,k;s(X, A) are introduced in [20] in a somewhat diﬀerent formulation
and under a stronger assumption on J than (2.11), which essentially requires it to be given via the
Symplectic Neighborhood Theorem [28, Theorem 3.30] and makes the setup very amenable for the
gluing needed to construct a virtual class. In [16], the relative moduli spaces are re-introduced,
again in a somewhat diﬀerent formulation from the previous paragraph, with ω-compatible J sat-
isfying (2.11). The relative non-amenability of this setup with the gluing is not material in cases
when the relative invariants (1.7) can be deﬁned geometrically, as in the next paragraph. By [20,
Section 3.2] and [16, Section 6], the spaces M
V
g,k;s(X, A) are compact; they are also Hausdorﬀ.
With notation as in (2.2) and J as in the previous two paragraphs, let
ΓV
g,k(X, J) ⊂Γg,k(X, J)
10

denote the subspace of elements ν such that
ν| qUg,k×V ∈Γg,k(V, J|V ),
e∇wν + J e∇Jwν ∈(T ∗qUg,k)0,1⊗CTxV
∀w∈TxX, x∈V.
(2.14)
The ﬁrst condition in (2.14) insures that every (J, ν)-holomorphic map u: Σ−→X has well-deﬁned
order of contact with V at all points of u−1(V ) not contained in an irreducible component of Σ
mapped into V . The second condition in (2.14) implies that the linearization of the ¯∂J,j−ν operator
at u: Σ−→V induces a C-linear map
DNXV
u
: Γ(Σ, u∗NXV ) −→Γ0,1
J,j (Σ, u∗NXV )
for every (J, ν)-holomorphic map u: Σ−→V . The moduli spaces
MV
g,k;s(X, A; J, ν) ⊂M
V
g,k;s(X, A; J, ν)
can then be deﬁned analogously to (2.12) and (2.13). The component maps into the rubber layers
{r}×PXV are then (JX,V , ν′)-holomorphic, with
ν′ ∈Γg′,k′(PXV, J),
{ν′|w}(v) =
 {e∇wν}(v), ν(v)

∈T vrt
w NXV ⊕T hor
w NXV
∀w ∈NXV, v ∈T qUg′,k′ .
By the same reasoning as for JX,V , ν′ given by the second line above extends over PX,∞V , is
C∗-equivariant, and satisﬁes (2.14) with (X, V ) replaced by (PXV, PX,0V ) and (PXV, PX,∞V ).
By [16, Proposition 7.3], the space M
V
g,k;s(X, A; J, ν) is compact.
By [16, Lemma 7.5], if ν is
generic each stratum of M
V
g,k;s(X, A; J, ν) consisting of simple maps of a ﬁxed combinatorial type
is a smooth manifold of the expected even dimension, which is less than the expected dimension
of the subspace of simple maps with smooth domains (except for this subspace itself). By [16,
Theorem 7.4], the last stratum has a canonical orientation. As explained in [5, Section 4.3], the
images of the strata of M
V
g,k;s(X, A) consisting of multiply covered maps under the morphism
st×ev1. . .×evk×evk+1. . .×evk+ℓ: M
V
g,k;s(X, A; J, ν) −→Mg,k+ℓ×Xk×V ℓ
(2.15)
are contained in images of maps from smooth even-dimensional manifolds of dimension less than the
main stratum if ν is generic, subject to the conditions (2.11) and (2.14), (V, ω|V ) is semi-positive,
and (X, ω, V ) is semi-positive in the sense of [5, Deﬁnition 4.7(1)]. Such strata do not even exist
if the domains of all elements of M
V
g,k;s(X, A) possibly contributing to the number (1.7) are stable
for some J, as happens in Section 5. By the proof of [18, Proposition 8.2], all relevant domains are
stable for a generic J if
A′ · V ≥⟨c1(X), A′⟩+ 1
2 dimR X + 2g
(2.16)
for all A′ ∈H2(X; Z) with ω(A′)≤ω(A) such that A′ can be represented by a J-holomorphic curve.
In the above cases, (2.15) is thus a pseudocycle. Intersecting it with generic representatives for the
Poincare duals of the cohomology classes κ on Mg,k+ℓ, α1, . . . , αk on X, and αk+1, . . . , αk+ℓon V
and dividing by the order of the covering (2.1), we obtain the relative GW-invariant
GWX,V
g,k;s
 κ; α1, . . . , αk; αk+1, . . . , αk+ℓ

= GWX,V
g,k;s
 κ; α1⊗. . .⊗αk; αk+1⊗. . .⊗αk+ℓ

.
11

The relative GW-invariant (1.7) is the above invariant with κ pulled back from Mg,k by the forgetful
morphism from Mg,k+ℓand αk+i = 1 for all i = 1, . . . , ℓ.
If g = 0, the same reasoning applies
with ν = 0 and yields the same conclusion if (X, ω, V ) satisﬁes the slightly stronger condition of
[5, Deﬁnition 4.7(2)]. For general triple (X, ω, V ), the relative GW-invariants (1.7) are deﬁned
similarly to [8, 24] using Kuranishi structures (or ﬁnite-dimensional approximations) and local
perturbations ν as in (2.14).
3
Proof of Theorem 1
A generic (J, ν)-holomorphic map contributing to the absolute GW-invariant (1.5) has intersection
number A·V with V . One would thus expect it to meet V at A·V distinct points. The diﬀerent
orderings of these points would ideally give rise to (A·V )! distinct relative maps contributing to the
relative GW-invariant (1.7). However, a regular pair (J, ν) determining the number (1.5) may not
satisfy the conditions (2.11) and (2.14) required of the pairs (J, ν) determining the number (1.7),
while a generic pair satisfying (2.11) and (2.14) may not be regular for the purposes of determining
the number (1.5). Thus, there is no `a priori reason for the identity (1.9) to hold in general. Below
we give two versions of nearly the same proof of Theorem 1: ﬁrst by a direct comparison and then
by formally applying the symplectic sum formula.
3.1
By direct comparison
The restriction (2.11) on J (or even the stronger one in [20]) is not material, as we can simply ﬁx one
admissible J and then choose a suitable ν to compute the GW-invariants (1.5) and (1.7). We start
by choosing a generic ν|V ∈Γg′,k′(V, J) and then extend it to X so that it satisﬁes the second con-
dition in (2.14). A generic such extension ν determines the relative GW-invariant (1.7). It counts
the (J, ν)-maps that pass through generic representatives of the Poincare duals of κ and αi have
images in X with no components mapped into V . Dropping the contact marked points, we obtain a
regular element of Mg,k(X, A; J, ν) which contributes to the absolute GW-invariant (1.5). However,
because ν may not be generic as far as the absolute invariants are concerned, Mg,k(X, A; J, ν) may
contain other elements u which meet generic representatives of the Poincare duals of κ and αi.
Any such u must have at least some components mapped into V , as all other components can be
regularized with ν subject to the condition (2.14).
Spaces MΓ of maps as at the end of the previous paragraph can be represented by decorated
connected bipartite graphs Γ with vertices v
• alternating between those representing the topological components Σv of the domain of the maps
into V and into X (without being contained in V ),
• labeled by pairs indicating the genus gv of and the degree Av of the map on Σv, and
• decorated by disjoint subsets of {1, . . . , k}, indicating the marked points carried by Σv;
see Figure 3. Since MΓ is contained in Mg,k(X, A; J, ν),
gΓ +
X
v∈Γ
gv = g,
X
v∈Γ
Av = A ∈H2(X; Z) ,
and
X
v∈Γ
kv = k,
12

X
V
1
2
(g, A)
X
V
1
2
(g4, A4) (g5, A5)
(g1, A1)
(g3, A3)
(g2, A2)
Figure 3: Bipartite graphs Γ representing elements of Mg,2(X, A; J, ν).
where v ∈Γ means that v is a vertex in Γ, gΓ is the genus of the graph Γ (number of edges minus
the number vertices plus 1), and kv is the number of original marked points attached to a vertex
v ∈Γ (the number of the original marked points carried by the topological component Σv of Σ).
We denote by ΓV the set of vertices of Γ corresponding to the components mapped into V and by
ΓX the set of remaining vertices. For each v∈Γ, let ℓv ∈Z≥0 denote the number of edges leaving v
(the number of nodes joining Σv to other topological components of Σ). The stability condition
on the elements of Mg,k(X, A) implies that kv+ℓv ≥3 for each v∈Γ with (gv, Av)=(0, 0).
If the domains of all relevant elements of Mg,k(X, A) are stable, as is the case in Section 5, the
above perturbations ν can be chosen globally as elements of ΓV
g,k(X, J). Otherwise, the same general
principle applies by using compatible Kuranishi structures for maps to X and to V . Theorem 1 is
established by showing that the subspace
MΓ(κ; α) ⊂MΓ ⊂Mg,k(X, A; J, ν)
of the elements that are of type Γ and meet generic representatives of the Poincare duals of κ and α
is empty for a generic ν satisfying (2.14) unless Γ is the one-vertex graph of maps to X, as in the
ﬁrst diagram in Figure 3. We can assume that κ and α satisfy (1.6).
Since V is assumed to be (g, A)-hollow in Theorem 1, we can use the Symplectic Neighborhood The-
orem [28, Theorem 3.30] to choose an ω-tame almost complex structure J on X so that J(TV )⊂TV ,
JV ≡J|V satisﬁes the conditions of Deﬁnition 1, and J satisﬁes (2.11). Thus, the degree Av of the
restriction of any element of MΓ to a topological component Σv of the domain mapped into V is
zero. If the genus gv of such Σv is zero, the restriction of any element u of MΓ to Σv is regular
as a map into X and stays so after a small generic deformation ν as in the previous paragraph.
If gv = 0 for all v ∈ΓV , MΓ consists of regular maps into X for a generic ν satisfying (2.14) and
thus has the expected dimension. Since this dimension is smaller than the virtual dimension of
Mg,k(X, A), unless ΓV = ∅, MΓ(κ; α) = ∅. In particular, if g = 0, all (J, ν)-maps for a generic ν
satisfying (2.14) are regular as maps to X and transverse to V . Thus, the sets of stable maps
contributing to the numbers on the two sides of (1.9) are the same in this case, up to the orderings
of the A·V intersection points with V . This establishes the g=0 case of (1.9).
If n≥5,
dimvir Mg′,0(V, 0) = 2(n−4)(1−g′) < 0
∀g′ ≥2.
In these cases, we can choose deformations ν satisfying (2.14) so that MΓ = ∅if gv ≥2 for any
v ∈ΓV . For the purposes of establishing the g ≥1 cases of (1.9), it thus remains to consider the
13

spaces MΓ so that gv ∈{0, 1} for all v ∈ΓV . Denote by Γv;1 ⊂ΓV the subset of vertices so that
gv =1. In the next paragraph, we show that
dim MΓ ≤dimvir Mg,k(X, A) −2
X
v∈Γv;1
ℓv
(3.1)
for a generic ν satisfying (2.14), if either n ≥5 or gv ≤1 for all v ∈ΓV (in particular, if g = 1).
Thus, MΓ(κ; α)=∅in these cases if Γ is not the basic one-vertex graph as in the ﬁrst diagram in
Figure 3, and so (1.9) again holds.
Removing the vertices of Γv;1 from Γ and replacing the edges leading to them by the marked points
on the remaining vertices, we obtain graphs Γi, with i = 1, . . . , N for some N ∈Z+, representing
subspaces MΓi of the moduli spaces Mgi,ki+ℓi(X, Ai) with
N
X
i=1
(gi−1) +
X
v∈Γv;1
ℓv = g−1,
N
X
i=1
Ai = A,
N
X
i=1
ki +
X
v∈Γv;1
kv = k ,
N
X
i=1
ℓi =
X
v∈Γv;1
ℓv ,
where ki ∈Z≥0 is the number of the original marked points carried by the component Γi. The
moduli spaces M1,kv+ℓv(V, 0; J, ν) corresponding to v ∈Γv;1 are of dimension 2(kv +ℓv) ∈Z+ for a
generic choice of ν|V . Since MΓi contains no component of positive genus mapped into V , it has
the expected dimension for a generic extension of ν|V satisfying (2.14). Taking into account the
matching conditions at the nodes joining elements of MΓi to elements of M1,kv+ℓv(V, 0; J, ν), we
ﬁnd that
dim MΓ ≤
N
X
i=1
dim MΓi +
X
v∈Γv;1
dim M1,kv+ℓv(V, 0; J, ν) −2n
X
v∈Γv;1
ℓv
≤2
N
X
i=1
 ⟨c1(X), Ai⟩+(n−3)(1−gi)+ki+ℓi

+ 2
X
v∈Γv;1
(kv+ℓv) −2n
X
v∈Γv;1
ℓv
= 2
 ⟨c1(X), A⟩+(n−3)(1−g)+k

+ 2(n−3+1+1−n)
X
v∈Γv;1
ℓv .
Along with the ﬁrst equation in (1.2), this establishes (3.1) and concludes the proof of the ﬁrst
claim of Theorem 1.
Remark 3.1. A regular genus 1 degree 0 (J, ν)-map into V may not be regular as a (J, ν)-map
into X. However, the space of such maps has the expected dimension for the target X because
this dimension is the same as the expected dimension for the target V in the g = 1 case. Thus,
a boundary stratum of (J, ν)-maps with only g = 0, 1 components contained in V is of smaller
dimension than the main stratum of maps into X. However, the space of (J, ν)-maps from smooth
genus 1 domains into V has the same dimension as the main stratum; this is precisely what makes
Example 1 possible.
Suppose next that κ=1 and g≥2 in (1.9), i.e. only the primary insertions are considered. Given a
bipartite graph Γ describing a subspace MΓ of Mg,k(X, A) as in Figure 3, let Γ0 be the decorated
bipartite graph obtained by replacing the genus labels of all vertices v ∈ΓV with 0. Thus, MΓ0 is
14

a subspace of Mg0,k(X, A) for some g0 < g, unless gv = 0 for all v ∈ΓV (in which case Γ0 = Γ and
thus g0 =g). If n=1, 2 and g0 <g,
dimvir Mg0,k(X, A) < dimvir Mg,k(X, A)
by the ﬁrst equation in (1.2). Thus, for a generic ν ∈ΓV
g0,k(X, J), MΓ0(1; α)=∅in this case, and so
ν ∈ΓV
g,k(X, J) can be chosen so that MΓ(1; α)=∅whenever g′
v >0 for any v ∈ΓV . This establishes
the n=1, 2 cases of the last claim of Theorem 1.
If g≥2 in (1.9) and n=3,
dimvir Mg0,k(X, A) = dimvir Mg,k(X, A) .
(3.2)
For any v∈ΓV with gv ≥1,
Mgv,kv+ℓv(V, 0) = Mgv,kv+ℓv × V ;
the obstruction bundle for this moduli space is
π∗
1E∗⊗π∗
2TV −→Mgv,kv+ℓv ×V ,
(3.3)
where E−→Mgv,kv+ℓv is the rank gv Hodge vector bundle of holomorphic diﬀerentials; it has chern
classes λi ≡ci(E). For gv ≥2, it is the pull-back of the Hodge vector bundle over Mg by the
forgetful morphism; if gv =1, it is the pull-back of the Hodge line bundle over M1,1. By [30, (5.3)]
in the ﬁrst case and for dimensional reasons in the second case,
λ2
gv = 0 ∈H4gv Mgv,kv+ℓv

.
(3.4)
Since the obstruction bundle is given by (3.3),

Mgv,kv+ℓv(V, 0; J, ν)

= e
 π∗
1E∗⊗π∗
2TV

∩

Mgv,kv+ℓv ×V

(3.5)
for a generic ν ∈ΓV
gv,kv
 X, J). By (3.2), MΓ0(1; α) consists of isolated maps meeting V transver-
sality at ﬁnitely many points pj for such a choice of ν (if MΓ0(1; α) is not empty). These points
include the nodes where irreducible components of elements of MΓ0(1; α) meet the elements of
Mgv,kv+ℓv(V, 0; J, ν) with v ∈ΓV . By (3.5) and (3.4), the homology class represented by the sub-
space of the latter passing through pj is
e
 E∗⊗TpjV

∩

Mgv,kv+ℓv

= λ2
gv ∩

Mgv,kv+ℓv

= 0.
Thus, the contribution of MΓ(1; α) to the left-hand side of (1.9) is the degree of a zero-cycle which
vanishes in the homology and thus is 0, if gv ≥1 for any v ∈ΓV . This establishes the κ=1, n=3,
and g≥2 case of (1.9).
The remaining case of Theorem 1 is κ=1, n=4, g=2, and A̸=0 (otherwise both sides of (1.9) vanish
for dimensional reasons). By the previous discussion, it is suﬃcient to show that MΓ(1; α)=∅for
a generic ν satisfying (2.14) if gv =2 for some v ∈ΓV . This assumption implies that gv′ =0 for all
v′ ∈ΓV −v and
dimvir Mg0,k(X, A) = dimvir Mg,k(X, A) + 4 .
(3.6)
By the ﬁrst equation in (1.2), the virtual dimension of M2,0(V, 0) is 0. Thus, we can choose a
deformation ν satisfying (2.14) so that the image of all elements of Mg0,kv+ℓv(X, A; J, ν) is contained
in arbitrary small neighborhoods of ﬁnitely many points of V . By (3.6), for a generic such ν there
are no elements of MΓ0(1; α) that pass through these images, since each point in V ⊂X imposes a
condition of real codimension 6 on maps to X. Thus, MΓ(1; α)=∅for a generic ν satisfying (2.14)
in this case as well.
15

3.2
Via the symplectic sum formula
We next give a proof of Theorem 1 by applying the symplectic sum formula to the symplectic
decomposition
X = X
#
V =PX,∞V
PXV ,
(3.7)
with PX,∞V ⊂PXV as in (2.5) and (2.6). The P1-bundle PXV −→V carries a symplectic form
induced from ω|V in a way well-deﬁned up to symplectic deformation equivalence; see the beginning
of Section 3.3.
According to the symplectic sum formula, the left-hand side of (1.9) is a weighted count of k-marked
genus g degree A (J, ν)-maps u into
XV
1 ≡X
∪
V =PX,∞V PXV
(3.8)
that have the same contact order with the common hypersurface V at the two branches of each
node, take no smooth point of the domain to V , and meet generic representatives of the Poincare
duals of κ and αi. The degree of such u is the class in X represented by the composition of u with
the natural projection
q: X
∪
V =PX,∞V PXV −→X ;
(3.9)
its weight is the product of the contacts with the common hypersurface (counted once for each pair
of contacts from the two sides).
Spaces MΓ(κ; α) of such maps to XV
1 can be represented by the same kind of connected bipartite
graphs Γ as in Section 3.1 with an additional decoration de ∈Z+ for each edge e; see Figure 4, where
edge labels 1 are not explicitly indicated. The subset ΓV of vertices now describes the topological
components Σv of the domain Σ that are mapped to PXV ; the additional decorations de specify
the orders of contacts with V of the branches of the nodes associated with the edges. The stability
condition on Γ described before now applies only to the vertices v ∈ΓX. The composition of an
element u in such a space MΓ(κ; α) with q produces an element of the space M¯Γ(κ; α) considered
above with ¯Γ obtained from Γ by dropping the edge labels and contracting oﬀthe unstable vertices
v∈ΓV and the edges leaving from them.
Breaking a graph Γ as in the previous paragraph at the mid-point of each edge, we obtain the
relative moduli spaces
M
V
gv,kv;sv(X, Av)
and
M
PX,∞V
gv,kv;sv(PXV, Av(sv))
with v ∈ΓX and v ∈ΓV , respectively, where sv is the tuple given by the labels on the edges and
Av(sv) is the sum of the push-forward of Av under the inclusion PX,0V −→PXV and |sv| ﬁber
classes. The left-hand side of (1.9) is the sum over all admissible graphs Γ of the weighted products
of the corresponding relative invariants with the relative primary insertions given by the usual
Kunneth decomposition of the diagonal in V 2 at each node; see the second-to-last equation on
page 201 in [22] and equations (5.4), (5.7), and (5.8) in [20]. Since the intersection points of ele-
ments of MΓ(κ; α) are unordered, while the contact points of the corresponding relative invariants
are ordered, the contribution from each graph Γ should be divided by the number of orderings of
16

X
PXV
V
PX,∞V
1
2
(g, A)
(0, 0)
X
PXV
V
PX,∞V
1
2
(g4, A4) (g5, A5)
(g1, A1)
(g3, A3)
(0, 0)
(g2, A2)
2
Figure 4: Bipartite graphs Γ representing elements of Mg,2(XV
1 , A; J, ν) with A·V =7.
the intersection points.
Some care is needed in translating the constraints κ and αi in (1.5) into constraints for the relative
invariants of (X, V ) and (PXV, PX,∞V ). If v ∈ΓX, the corresponding relative invariant of (X, V )
keeps the insertion αi at the absolute marked point corresponding to i, if it is carried by Σv. If
v ∈ΓV , the corresponding relative invariant of (PXV, PX,∞V ) gets the insertion π∗
X,V (αi|V ) at
the absolute marked point corresponding to i, where πX,V : PXV −→V is the projection map.
Denote by
stv : M
V
gv,kv;sv(X, Av) −→Mgv,kv+ℓ(sv)
or
stv : M
PX,∞V
gv,kv;sv(PXV, Av(sv)) −→Mgv,kv+ℓ(sv)
the stabilization map, depending on whether v∈ΓX or v∈ΓV , respectively; in the unstable range,
the target of this map is one point. Let
glΓ :
Y
v∈Γ
Mgv,kv+ℓ(sv) −→Mg,k
be the morphism given by identifying pairs of points corresponding to the same edge in Γ. In
particular,
glΓ ◦
Y
v∈Γ
stv = st◦ιΓ : MΓ −→Mg,k ,
where ιΓ : MΓ −→Mg,k(XV
1 , A) is the inclusion map. By the Kunneth formula,
gl∗
Γκ =
X
j
O
v∈Γ
κj;v ∈
O
v∈Γ
H∗ Mgv,kv+ℓ(sv)

= H∗
 Y
v∈Γ
Mgv,kv+ℓ(sv)

for some κj;v ∈H∗(Mgv,kv+ℓ(sv)). In the Γ-summand in the symplectic sum decomposition for
the absolute GW-invariant (1.5), the insertion κ is replaced by the insertion κv;j in the relative
invariant corresponding to the vertex v and the resulting products are summed over all j. This is
carried out in a speciﬁc case in Section 4.2.
Since V is assumed to be (g, A)-hollow in Theorem 1, we can choose an almost complex structure JV
on V so that it satisﬁes the conditions of Deﬁnition 1. Using a connection in NXV as in Section 2,
we can extend JV to an almost complex structure J on PXV so that the condition (2.11) is satisﬁed
17

and the projection πX,V : PXV −→V is (JV , J)-holomorphic. Using the same connection, we can
extend any ν ∈Γg,k(V, JV ) to
π∗
X,V ν ∈ΓPX,∞V
g,k
(PXV, J)
so that πX,V ◦u: Σ−→V is (JV , ν)-holomorphic whenever u: Σ−→PXV is (J, π∗
X,V ν)-holomorphic.
By the previous paragraph, we can assume that the degree Av of the composition of the restriction
of any element of MΓ to a topological component Σv of the domain mapped into PXV with πX,V is
zero, i.e. all relevant relative invariants of (PXV, PX,∞V ) lie in the ﬁber classes dvF with dv ∈Z≥0.
A key point of the paragraph above the previous one is that the class integrated over the relative
moduli space M
PX,∞V
gv,kv;sv(PXV, dF) corresponding to the vertex v is pulled back by the projection
map
ϕ ≡st×πX,V : M
PX,∞V
gv,kv;sv(PXV, dvF) −→Mgv,kv+ℓ(sv)×V .
(3.10)
In particular, if
dimvir M
PX,∞V
gv,kv;sv(PXV, dvF) > dim
 Mgv,kv+ℓ(sv)×V

,
(3.11)
then the relative invariant corresponding to the vertex v∈ΓV vanishes and such bipartite graph Γ
does not contribute to the left-hand side of (1.9).
By the second equation in (1.2) and the condition |sv|=dv, (3.11) is equivalent to
dv + (n−3)(1−gv) + kv + ℓ(sv) > n−1 +
(
0,
if gv =0, kv+ℓ(sv)≤2;
3gv−3+kv+ℓ(sv),
otherwise.
If gv = 0, either dv ∈Z+ (and thus ℓ(sv) ∈Z+) or kv ≥3 for stability reason. Thus, the relative
invariant corresponding to a vertex v∈ΓV with gv =0 is zero unless dv =1, kv =0, and sv =(1). In
this remaining case, the only nonzero relative invariant is
GWPXV,PX,∞V
0,F ;(1)
 1, 1; PDV ([pt])

= 1.
In particular, the contribution to the left-hand side of (1.9) from the simplest graph, i.e. as in the
ﬁrst diagram in Figure 4, is
1
(A · V )!GWX,V
g,A;1A·V
 κ; α; 1A·V 
≡
1
(A · V )!GWX,V
g,A;1A·V
 κ; α

.
(3.12)
All other nonzero contributions to the left-hand side of (1.9) can come only from graphs Γ such
that (dv, kv, sv)=(1, 0, (1)) for all v ∈ΓV with gv =0 and gv ∈Z+ for some v ∈ΓV . Since there are
no such graphs if g=0, this concludes the proof of the g=0 case of (1.9).
We next show that the relative invariants corresponding to v ∈ΓV with gv ∈Z+ also vanish under
the assumptions of (1.8). If gv ≥2 and ν ∈Γgv,0(V, JV ), the composition with the projection πX,V
induces a continuous map
πX,V : M
PX,∞V
gv,kv;sv
 PXV, dvF; J, π∗
X,V ν

−→Mgv,0
 V, 0; JV , ν

.
(3.13)
Since
dim Mgv,0
 V, 0; JV , ν

= dimvir Mgv,0
 V, 0

= (n−4)(1−gv)
∀gv ≥2
18

for a generic ν ∈Γgv,0(V, JV ), the moduli spaces in (3.13) are empty if gv ≥2 and n≥5. In particu-
lar, the relative invariants vanish in these cases.
If gv =1, then dv, ℓ(sv)∈Z+ by the ﬁrst assumption in (1.8). For a generic ν ∈Γ1,1(V, JV ),
πX,V : M
PX,∞V
1,kv;sv
 PXV, dvF; J, π∗
X,V ν

−→M1,1
 V, 0; JV , ν

is then a ﬁbration with typical ﬁber M
pt
1,kv;sv(P1, d)j, where the subscript j denotes the moduli space
with j ﬁxed on M1,1. Since the obstruction bundle for M1,1(V, 0) is given by (3.3),

M1,1
 V, 0; JV , ν)

= e
 π∗
1E∗⊗π∗
2TV

∩

M1,1×V

= {j}×V1 + M1,1×V0,
(3.14)
where V0, V1 ⊂V are some cycles of R-dimensions 0 and 2, respectively, and j is a ﬁxed element
of M1,1. Since
dimvir M
pt
1,kv;sv(P1, dv) = dv + kv+ℓ(sv) > kv + ℓ(sv) = dim M1,kv+ℓ(sv),
the integral of the pull-back of any class by (3.10) vanishes on the last term in (3.14). Since
dimvir M
pt
1,kv;sv(P1, dv)j = dv−1 + kv+ℓ(sv) > kv+ℓ(sv) −1 = dim M1,kv+ℓ(sv);j ,
the integral of the pull-back of any class by (3.10) vanishes on the ﬁrst term on the RHS of (3.14)
as well.
In summary, the only graph Γ that contributes to the left-hand side of (1.9) via the symplectic
sum formula applied to the decomposition (3.7) under the assumptions (1.8) is the graph with
|ΓX| = 1
and
(gv, dv, kv, sv) = (0, 1, 0, (1))
∀v∈ΓV ;
see the ﬁrst diagram in Figure 4. Since its contribution is given by (3.12), we have established the
ﬁrst claim of Theorem 1.
Suppose next that κ = 1 and g ≥2 in (1.9), i.e. only the primary insertions are considered. The
relative invariants of (X, V ) that enter into the symplectic sum formula then count curves that
meet generic Poincare duals of all the constraints αi. If n=1, 2 and g0 <g,
dimvir M
V
g0,k;s0(X, A) < dimvir Mg,k(X, A)
(3.15)
by (1.2). Thus, these relative invariants vanish if n=1, 2 and the total genus of the vertices in ΓX
is less than g. This happens in particular if gv > 0 for any v ∈ΓV . Along with the paragraph
containing (3.12), this establishes the n=1, 2 cases of the last claim of Theorem 1.
Suppose gv ≥2 for some v ∈ΓV and n=3. The dimensions of the two moduli spaces in (3.15) are
then the same. The relative invariants of (X, V ) that enter into the symplectic sum formula thus
count curves that meet V at ﬁnitely many distinct points {pj}. Since the obstruction bundle for
Mgv,0(V, 0) is given by (3.3), the homology class of the subspace of elements of Mgv,0(V, 0; JV , ν)
that pass through pj

Mgv,0(V, 0; JV , ν)|pj

= e
 E∗⊗TpjV

∩

Mgv,0×{pj}

= λ2
gv ∩

Mgv,0

= 0;
19

see (3.4). Thus, by (3.13), the genus gv relative invariants of (PXV, PX,∞V ) with a relative point
insertion vanish in this case as well.
The remaining case of Theorem 1 is κ=1, n=4, and g=2. Since A̸=0 in this case, dv, ℓ(sv)∈Z+.
For a generic ν ∈Γ2,0(V, JV ), the target in (3.13) is a ﬁnite set of points, while the dimension of
the ﬁber is
dv + 1 −gv + kv + ℓ(sv) ≥1 −1 + 0 + 1 = 1.
Thus, the genus 2 relative invariants of (PXV, PX,∞V ) with only primary insertions from V vanish.
This concludes the proof of the last claim of Theorem 1.
3.3
Extension to virtual cycles
In the process of establishing the ﬁrst claim of Theorem 1 above, we showed that the relative invari-
ants in the ﬁber classes of P1-bundles often vanish. This, more technical, conclusion is summarized,
in Lemma 3.2 below. It leads to a version of Theorem 1 for virtual moduli cycles; see Corollary 3.3.
Let (V, ω) be a compact symplectic manifold, πL : L−→V be a complex line bundle, and
πL,V : PL ≡P(L⊕V ×C) −→V
be the bundle projection map. Given a Hermitian metric ρ (square of the norm) and a ρ-compatible
connection ∇in L, let α denote the connection 1-form on the ρ-circle bundle in L and its extension
to L−V via the retraction given by v−→v/|v|. The closed 2-form
eω ≡π∗
X,V ω −ǫ d

α
1+ρ2

on L−V extends to a closed 2-form on PL, which is symplectic if ǫ>0 is suﬃciently small; we will
take the symplectic deformation equivalence class of this form to be the default one. Let
PL,∞= P(L⊕0) ⊂PL .
The projection map
ϕ ≡st×πL,V : M
PL,∞
g,k;s (PL, dF) −→Mg,k+ℓ(s)×V,
where F ∈H2(PL; Z) is the ﬁber class, induces a push-forward on the virtual class:
ϕ∗

M
PL,∞
g,k;s (PL, dF)
vir ∈H∗
 Mg,k+ℓ(s)×V

.
By the Poincare duality applied on Mg,k+ℓ(s)×V , this push-forward is determined by the evaluation
of cohomology classes pulled back from Mg,k+ℓ(s)×V by ϕ on the virtual class of M
PL,∞
g,k;s (PL, dF).
Thus, Section 3.2 establishes the following statement.
Lemma 3.2. Let (V, ω) be a compact symplectic manifold of real dimension 2(n−1) and L−→V
be a complex line bundle. If g, d, k∈Z≥0 and s∈(Z+)ℓare such that
(g, d) ̸= (1, 0),
and
(n−5)g(g−1) ≥0 ,
(3.16)
then
ϕ∗

M
PL,∞
g,k;s (PL, dF)
vir =
(
[V ],
if (g, d, k, s)=(0, 1, 0, (1));
0,
otherwise.
20

Corollary 3.3 (D. Maulik). Suppose (X, ω) is a projective manifold of real dimension 2n, g, k∈Z≥0,
A∈H2(X; Z), and V ⊂X is a (g, A)-hollow projective hypersurface such that A·V ≥0. If
(g, A) ̸= (1, 0)
and
(n−5)g(g−1) ≥0 ,
(3.17)
then

Mg,k(X, A)
vir =
1
(A · V )! f∗

Mg,k;1A·V (X, A)
vir ,
(3.18)
where f is the morphism between the moduli spaces dropping the relative marked points.
Proof. Let ∆⊂C denote a small disk around the origin, Z be the blowup of ∆×X along 0×V ,
and π: Z −→∆be the projection map. Thus,
Zλ = X
∀λ ∈∆∗≡∆−0
and
Z0 ≡π−1(0) = XV
1 ,
with notation as in (3.8).
As summarized in [22, Section 0], there are moduli stacks Mg,k(XV
1 , A) and Mg,k(Z, A).
The
former carries a virtual class so that

Mg,k(XV
1 , A)
vir =

Mg,k(Zλ, A)
vir =

Mg,k(X, A)
vir
(3.19)
under the inclusion into Mg,k(Z, A).
In the case of the given family Z −→∆, (3.19) can be
written as
q∗

Mg,k(XV
1 , A)
vir =

Mg,k(X, A)
vir ,
(3.20)
with q as in (3.9). By the last formula on page 201 in [22],

Mg,k(XV
1 , A)
vir =
X
Γ
m(Γ)
ℓ(Γ)! ΦΓ∗∆! 
M(X, ΓX)
vir×

M(PXV, ΓV )vir
.
(3.21)
This sum is taken over the same bipartite graphs Γ as in Section 3.2. For such a graph Γ, m(Γ)
is the product of the edge labels (of contacts with the common divisor V ) and ℓ(Γ) is the number
of edges (of contacts with V ). In the notation of Section 3.2, the two moduli spaces appearing on
the right-hand side of (3.21) are
Y
v∈ΓX
M
V
gv,kv;sv(X, Av)
and
Y
v∈ΓV
M
PX,∞V
gv,kv;sv
 PXV, Av(sv)

,
respectively.
The symbol ∆! indicates the cap product with the product over the edges of Γ
of the pull-back of the diagonal ∆V ⊂V 2 by the evaluation maps at the relative marked points
corresponding to the same edge, while ΦΓ is the morphism given by identifying these marked points.
Since V is (g, A)-hollow, the only possible nonzero summands in (3.21) correspond to Γ with Av =0
for all v ∈ΓV . For such Γ, the relative evaluation maps are given by the composition with the
projection map πX,V : PXV −→V , while q◦ΦΓ factors through id×ϕ. Combining (3.20) and (3.21),
we thus obtain

Mg,k(X, A)
vir =
X
Γ
m(Γ)
ℓ(Γ)! ΦΓ∗∆! 
M(X, ΓX)
vir×ϕ∗

M(PXV, ΓV )vir
,
(3.22)
21

with the sum now taken over bipartite graphs Γ as in Section 3.2 with Av = 0 for all v ∈ΓV . For
graphs Γ, the restrictions (3.17) imply the restrictions (3.16) for all (g, d) = (gv, dv) with v ∈ΓV .
By Lemma 3.2, the last term in (3.22) thus vanishes except for the basic graph Γ with |ΓX| = 1,
(gv, Av, kv) = (0, 0, 0) for all v ∈ΓV , and all edge labels equal to 1, i.e. as in the ﬁrst diagram in
Figure 4. The summand in (3.22) corresponding to this basic graph gives (3.18).
Remark 3.4. The equality (3.19) is established in [22] for a general ﬂat degeneration π: Z −→∆,
with Z0 consisting of two smooth varieties joined along a smooth hypersurface, only after summing
over all classes A of the same degree with respect to an ample line bundle over Z. However, in the
given case, the relevant summands on the two sides of (3.19) lie in diﬀerent spaces and thus must
be equal pairwise.
Remark 3.5. The conclusion of Corollary 3.3 should apply to any compact symplectic mani-
fold (X, ω) and (g, A)-hollow symplectic hypersurface V . Unfortunately, the above proof of Corol-
lary 3.3 makes use of the symplectic sum (degeneration) formula for virtual fundamental cycles
(not just numbers) in GW-theory, which is not even claimed in the symplectic category in any
work we are aware of. In particular, [17] is concerned only with equating GW-invariants (viewed
as numbers), contrary to the claim just above [18, (11.4)].
4
Details on the counter-examples
In Sections 4.1-4.3, we establish the claims made in Examples 1-3, respectively; see Section 1. In
the case of Example 1, we give two computations of the relative invariants. In the cases of Exam-
ples 2 and 3, we include localization computations of the δ = 0, 1 numbers as consistency checks;
the localization computations for Example 3 are separated oﬀinto Section 4.4.
In Sections 4.2-4.4, we use some degree 1 relative GW-invariants of (P1, ∞) and rubber relative
invariants of (P1, ∞, 0) with respect to the standard C∗-action. In principle, all such invariants are
computed in [7, 31]. As it is not completely trivial to extract actual numbers from the generating
series in [7, 31], we include alternative computations for the few numbers relevant to our purposes.
4.1
Genus 1 degree 0 invariants
Let (X, ω, V ) be as in Example 1.
Fix an ω-tame almost complex structure J on X so that
J(TV )=TV and the Nijenhuis condition (2.11) holds.
For a complex structure j on a smooth 1-marked genus 1 Riemann surface (Σ, x1), the space
of degree 0 holomorphic maps u : Σ −→X consists of the constant maps and so is canonically
isomorphic to X. The obstruction bundle (i.e. the bundle of the cokernels of the linearizations DJ,u
of the ¯∂J,j-operator at u) is isomorphic to H0,1
j
⊗CTX, where H0,1
j
is the complex one-dimensional
space of anti-holomorphic one-forms on Σ. Thus,
TX ≈Obs −→Holj(X, 0) ≈X.
(4.1)
By deﬁnition, the 1-marked genus 1 degree 0 ﬁxed j absolute GW-invariant with primary insertion
1∈H∗(X) is the (signed) number of solutions u: Σ−→X of
¯∂J,ju

z = ν
 z, u(z)

∀z ∈Σ,
u∗[Σ] = 0 ∈H2(X; Z),
(4.2)
22

for a generic element
ν ∈Γj(X, J) ≡Γ
 Σ×X, (T ∗Σ)0,1⊗CTX

.
The projection ¯ν of this element to the cokernel of DJ,u for each u∈Holj(X, 0) induces a transverse
section of the obstruction bundle (4.1). The solutions of (4.2) correspond to the zeros of ¯ν, as the
obstruction to solving (4.2) vanishes at these points. Thus, the number of solutions of (4.2) is

e(Obs), Holj(X, 0)

=

c1(TX), X

= χ(X).
If j ∈H2(M1,1) is the Poincare dual of a generic element, the absolute GW-invariant GWX
1,0(j; 1)
is half this number, because the group of automorphisms of a generic element of M1,1 is Z2. This
establishes the ﬁrst equality in (1.11).
For maps of degree A=0, A·V =0 and so the only compatible contact vector is the length 0 vector,
which we denote by (). By deﬁnition, the 1-marked genus 1 degree 0 ﬁxed j GW-invariant relative
to V with contact vector () and primary insertion 1∈H∗(X) is the number of solutions u: Σ−→X
of
¯∂J,ju

z = ν
 z, u(z)

∀z ∈Σ,
u∗[Σ] = 0 ∈H2(X; Z),
u(Σ) ̸⊂V,
(4.3)
for a generic ν ∈ΓV
j (X, J), where
ΓV
j (X, J) ⊂Γj(X, J)
is the subspace of elements ν such that
ν|Σ×V ∈Γj(V, J|V ),
e∇wν + J e∇Jwν ∈(T ∗Σ)0,1⊗CTxV
∀w∈TxX, x∈V.
By the ﬁrst assumption above, the number of maps u : Σ −→V ⊂X that satisfy the ﬁrst two
conditions in (4.3) and fail the third is χ(V ). The total number of maps u : Σ −→X that sat-
isfy the ﬁrst two conditions in (4.3) is χ(X), as in the previous paragraph. Thus, the number of
solutions of (4.3) is χ(X)−χ(V ). Similarly to the previous paragraph, the relative GW-invariant
GWX,V
1,0;()(j; 1) is half this number. This establishes the second equality in (1.11).
With α as in (1.12), let Y ⊂X be a generic representative of the Poincare dual of α. Since every
degree 0 J-holomorphic map is constant,
M1,1(X, 0) = M1,1×X.
Similarly to the previously case, the obstruction bundle in this case is isomorphic to
Obs = π∗
1E∗⊗π∗
2TX −→M1,1×X,
(4.4)
where E −→M1,1 is the Hodge line bundle of holomorphic diﬀerentials.
Its ﬁrst chern class,
λ≡c1(E), satisﬁes

λ, M1,1

= 1
24 .
(4.5)
By deﬁnition, the 1-marked genus 1 degree 0 absolute GW-invariant with primary insertion α is
the (signed) number of solutions u: (Σ, j)−→X of
¯∂J,ju

z = ν
 z, u(z)

∀z ∈Σ,
u∗[Σ] = 0 ∈H2(X; Z),
u(x1) ∈Y,
(4.6)
23

where Σ is a smooth torus, x1 ∈Σ is the marked point, ν is a generic element of
Γ1,1(X, J) ≡Γ
 U1,1×X, (T ∗U1,1)0,1⊗CTX

,
and U1,1 −→M1,1 is the universal curve.
Similarly to the case considered above, ν induces a
transverse section ¯ν of the obstruction bundle (4.4). The solutions of (4.6) correspond to the zeros
of ¯ν with u(x1)∈Y . Thus,
GWX
1,0(α) =

e(Obs), M1,1×Y

= −

λ, M1,1

cn−1(X), Y

= −1
24

α cn−1(X), X

.
(4.7)
This establishes the ﬁrst equality in (1.12).
By deﬁnition, the 1-marked genus 1 degree 0 GW-invariant relative to V with contact vector ()
and primary insertion α is the number of solutions u: (Σ, j)−→X of
¯∂J,ju

z = ν
 z, u(z)

∀z ∈Σ,
u∗[Σ] = 0 ∈H2(X; Z),
u(x1) ∈Y,
u(Σ) ̸⊂V,
(4.8)
for a generic ν ∈ΓV
1,1(X, J), where ΓV
1,1(X, J) ⊂Γ1,1(X, J) is the subspace of elements ν sat-
isfying (2.14). By the ﬁrst assumption in (2.14) and previous paragraph, the number of maps
u: Σ−→V ⊂X that satisfy the ﬁrst three conditions in (4.8) and fail the fourth is
GWV
1,0(PDV (V ∩Y )) = −1
24

PDV (V ∩Y ) cn−2(Y ), Y

= −1
24

α|V cn−2(Y ), Y

.
(4.9)
Since the total number of maps u : Σ −→X that satisfy the ﬁrst three conditions in (4.8) is
GWX
1,0(α), GWX,V
1,0;()(α) is the diﬀerence of (4.7) and (4.9), as claimed in the second equality in (1.12).
Remark 4.1. Strictly speaking, the arguments in the last two paragraphs should be applied to the
universal curve qU1,1 over the moduli space }
M1,1 of 1-marked genus 1 curves with Prym structures
in place of U1,1 and the resulting numbers should then be divided by the order of the covering (2.1)
with (g, k)=(1, 1). This nuance is taken into account by E−→M1,1 being a line orbi-bundle over
an orbifold with the chern class given by (4.5).
The absolute invariant GWX
1,0(j; 1) can also be computed using the same framework as GWX
1,0(α).
If σ∈M1,1 represents the Poincare dual of j, (4.7) becomes
GWX
1,0(j; 1) =

e(Obs), [σ]×X

=

1, [σ]

cn(X), X

= 1
2χ(X).
Below we recall a similar framework for computing the relative invariants in the algebraic category,
based on [23, Section 8], and note that it applies equally well in the symplectic category.
If X is a complex manifold and V ⊂X is a complex submanifold, the sheaf OX(TX) of holomorphic
vector ﬁelds contains the subsheaf OX(TX(−log V )) of vector ﬁelds with values in TV along V .
If (z1, . . . , zn) is a coordinate chart on U ⊂X such that U∩V is the slice zn =0, OU(TX(−log V ))
is freely generated by the vector ﬁelds
∂
∂z1
, . . . ,
∂
∂zn−1
, zn
∂
∂zn
.
24

Thus, OX(TX(−log V )) is a locally free sheaf of rank n, i.e. the sheaf of a holomorphic sections
of a holomorphic vector bundle TX(−log V ) of rank n. In the symplectic category, such a vector
bundle can be constructed using the Symplectic Neighborhood Theorem [28, Theorem 3.30]; the
resulting complex vector bundle is well-deﬁned up to equivalence by the deformation equivalence
class of ω as a symplectic form on X ⊃V .
Lemma 4.2. Suppose (X, ω) is a compact symplectic manifold of real dimension 2n and V ⊂X is
a compact symplectic manifold. If α∈H2(n−k)(X), then

α ck(TX(−log V )), X

=

α ck(X), X

−

α ck−1(V ), V

.
Proof. By deﬁnition, there is a short exact sequence of sheaves
0 −→OX(TX(−log V )) −→OX(TX) −→OV (V ) −→0,
(4.10)
where the second non-trivial homomorphism is the restriction to V followed by the projection to
the normal bundle NXV , which equals to the restriction of the line bundle OX(V ) to V . Combin-
ing (4.10) with the short exact sequence
0 −→OX −→OX(V ) −→OV (V ) −→0,
we ﬁnd that
c
 OX(TX(−log V ))

= c
 OX(TX)

c
 OV (V )
−1 = c
 OX(TX)
  c(OX(V ))c(OX)−1−1
= c(X)
 1+PDX(V )
−1 .
Thus,

α ck(TX(−log V )), X

=

α ck(X), X

−
k−1
X
i=0
(−1)i
α ck−1−i(X) (PDXV )1+i, X

=

α ck(X), X

−
k−1
X
i=0
(−1)i
α ck−1−i(X) c1(NXV )i, V

.
(4.11)
Since c(V ) = c(TX)|V c(NXV )−1, the claim follows from (4.11).
In the projective setting, the analogue of the obstruction bundle (4.4) for the relative moduli
space is
ObsV = π∗
1E∗⊗π∗
2TX(−log V ) −→M1,1×X ;
see [23, Section 8].
In the symplectic setting, the substance of the ﬁrst restriction in (2.14) is
that ν induces a section of ObsV . If ν is generic, subject to the conditions in (2.14), this section
is transverse to the zero set everywhere and when restricted to M1,1×V . Thus, it has no zeros
along V and the two relative invariants in Example 1 are given by
GWX,V
1,0;()(j; 1) =

e(ObsV ), [σ]×X

=

1, [σ]

cn(TX(−log V )), X

GWX,V
1,0;()(α) =

e(ObsV ), M1,1×Y

= −

λ, M1,1

cn−1(TX(−log V )), Y

.
The second equalities in (1.11) and (1.12) now follow from Lemma 4.2 and (4.5).
25

X
PXV
V
PX,0V
(1, 0)
X
PXV
V
PX,0V
(1, 0)
Figure 5: The two bipartite graphs Γ contributing to the genus 1 degree 0 GW-invariants of X via
the symplectic sum formula applied to (3.7).
Finally, we note that the two pairs of the GW-invariants in Example 1 are consistent with the
symplectic sum formula as stated in the second-to-last equation on page 201 in [22] and applied to
the decomposition (3.7). Since the degree A=0 in this case, there are only two types of bipartite
graphs Γ as in Section 3.2 to sum over: the two possible one-vertex graphs; see Figure 5. The
symplectic sum formula in these cases gives
GWX
1,0(j; 1) = GWX,V
1,0;()(j; 1) + GWPXV,PX,∞V
1,0;()
(j; 1),
(4.12)
GWX
1,0(α) = GWX,V
1,0;()(α) + GWPXV,PX,∞V
1,0;()
 π∗
X,V (α|V )

,
(4.13)
with PX,∞V ⊂PXV and πX,V : PX,∞V −→V as in (2.5) and (2.6). According to the second equality
in (1.11),
GWX,V
1,0;()(j; 1) = χ(X) −χ(V )
2
,
GWPXV,PX,∞V
1,0;()
(j; 1) = χ(PXV ) −χ(PX,∞V )
2
= 2χ(V ) −χ(V )
2
.
Thus, (4.12) is consistent with the ﬁrst equality in (1.11). According to the second equality in (1.12),
GWX,V
1,0;()(α) = −⟨α cn−1(X), X⟩−⟨α|V cn−2(V ), V ⟩
24
,
GWPXV,PX,∞V
1,0;()
 π∗
X,V (α|V )

= −
⟨π∗
X,V (α|V ) cn−1(PXV ), PXV ⟩−⟨α|V cn−2(V ), V ⟩
24
= 2 ⟨α|V cn−2(V ), V ⟩−⟨α|V cn−2(V ), V ⟩
24
.
Thus, (4.13) is consistent with the ﬁrst equality in (1.12).
4.2
Genus 2 degree 1 invariants of P1
We establish the second equality in (1.13) by applying the symplectic sum formula, as stated in
the second-to-last equation on page 201 in [22], to the absolute GW-invariant in (1.13) via the
decomposition (3.7) with
X = P1,
V = Vδ ≡{p1, . . . , pδ},
PXV = {1, . . . , d} × P1 .
We will make use of some top intersection numbers on the Deligne-Mumford spaces M2 and M2,1,
as summarized in Tables 1 and 2. The numbers for M3 and M3,1, appearing in Tables 1 and 3,
26

P1
{1} ×P1
{7} ×P1
p1
p7
1
2
(2, 1)
(0, 0)
P1
{1} ×P1
{7} ×P1
p1
p7
1
2
(0, 1)
(0, 0)
(2, 0)
Figure 6: Bipartite graphs Γ that contribute to the absolute GW-invariant in (1.13) via the sym-
plectic sum decomposition with respect to V7 ={p1, . . . , p7}.
will be used in Section 4.3. These numbers can be obtained from C. Faber’s computer program,
which implements the formula described in [6].
Since the Poincare duals of the two primary insertions in (1.13) vanish on the divisor Vδ (the
constraining points can be chosen to be distinct from the δ points in Vδ), kv =0 for all v∈ΓV (the
marked points stay on the X-side) if Γ is a bipartite graph as in Section 3.2 contributing to the
absolute GW-invariant in (1.13). Furthermore, Av = 1 for the unique vertex v ∈ΓX and all edge
labels are 1 in this case (because the curve on the X-side is of degree 1 and so meets each point in
the divisor with order 1). By Section 3.2 or Lemma 3.2 (separately), there are thus only two types
of graphs Γ contributing to the absolute GW-invariant in (1.13):
(1) (gv, Av, kv)=(0, 0, 0) for all v∈ΓV and
(2) the δ graphs with (gv, Av, kv) = (2, 0, 0) for one element v ∈ΓV and (gv, Av, kv) = (0, 0, 0) for
the remaining δ−1 elements v∈ΓV ;
see Figure 6. There are other bipartite graphs Γ, but they all contain a vertex v ∈ΓV with gv =1;
by Section 3.2 or Lemma 3.2, such a graph does not contribute to an absolute GW-invariant via the
symplectic sum formula. In the setup of Section 3.1, such graphs correspond to conﬁgurations with
genus 1 components sinking into the divisor; since there are no higher-genus components sinking
into the divisor in this case, the argument in Section 3.1 also implies that such a conﬁguration does
not contribute.
Thus, by the symplectic sum formula,
GWP1
2,1(κ4; pt, pt) = 1
δ!GWP1,V
2,1;1δ(κ4; pt, pt) + δ
δ!
X
i
GWP1,Vδ
0,1;1δ(κi; pt, pt)GWP1,pt
2,1;(1)(κ′
i; 1) ,
(4.14)
with κi ∈H∗(M0,2+δ) and κ′
i ∈H∗(M2,1) given by
gl∗κ4 =
X
i
κi ⊗κ′
i ∈H∗(M0,2+δ) ⊗H∗(M2,1) = H∗(M0,2+δ×M2,1),
where
gl: M0,2+δ×M2,1 −→M2,2 ,
27

λ3
1
λ1λ2
λ6
1
λ4
1λ2
λ3
1λ3
λ2
1λ2
2
λ1λ2λ3
λ3
2
λ2
3
1
2880
1
5760
1
90720
1
181440
1
725760
1
362880
1
1451520
1
725760
0
Table 1: The top intersections of λ-classes on M2 and M3.
is the morphism obtained by forgetting the last δ−1 points on the genus 0 curve and identifying the
marked point of the genus 2 curve with the third marked point on the genus 0 curve. Since κ is the
Poincare dual of the divisor represented by the bottom right diagram in Figure 1, it follows that
X
i
κi ⊗κ′
i ≡gl∗κ4 = 1 ⊗ψ4
1,
(4.15)
where ψ1 ∈H∗(M2,1) is the chern class of the universal cotangent line bundle. By Theorem 1,
1
δ!GWP1,Vδ
0,1;1δ(1; pt, pt) ≡1
δ!GWP1,Vδ
0,1;1δ(pt, pt) = GWP1
0,1(pt, pt) = 1.
(4.16)
Combining (4.14) with (4.15), (4.16), and Lemma 4.3 below, we conclude that
GWP1
2,1(κ4; pt, pt) = 1
δ!GWP1,Vδ
2,1;1δ(κ4; pt, pt) + δ

ψ4
1, M2,1

.
The second equality in (1.13) now follows from the ﬁrst column in Table 2.
Lemma 4.3 (C.-C. Liu). If st : M
pt
2,0;(1)(P1, 1) −→M2,1 is the forgetful morphism dropping the
map to P1, then
st∗

M
pt
2,0;(1)(P1, 1)
vir =

M2,1

.
(4.17)
Since M2,1 is smooth (as an orbifold) and irreducible, (4.17) is equivalent to

st∗σ, [M
pt
2,0;(1)(P1, 1)]vir
= 1,
(4.18)
where σ∈H8(M2,1) is the Poincare dual of a generic element (Σ, x1) of M2,1. We give two proofs
of (4.18) below. The ﬁrst argument applies the virtual localization theorems of [11, 12] as in [13,
Chapter 27]. The second proof applies the obstruction analysis of [35] as in [36, Section 4].
Proof 1 of (4.18). We use the standard (C∗)-action on P1. It has two ﬁxed points,
p1 = [1, 0]
and
p2 = [0, 1],
ψ4
1
ψ3
1λ1
ψ2
1λ2
1
ψ2
1λ2
ψ1λ3
1
ψ1λ1λ2
1
1152
1
480
7
2880
7
5760
1
1440
1
2880
Table 2: The top intersections of λ-classes and ψ1 on M2,1.
28

(1, 2)
2
1
(1, 1)
(2, 1)
1
1
(2, 2)
1
Figure 7: The three graphs describing the (C∗)2-ﬁxed loci of M
p2
2,0;(1)(P1, 1)
and lifts linearly to an action on OP1(1)−→P1. As in [13, Chapter 27], we let
αi = c1
 OP1(1)

pi ∈H∗
(C∗)2 ≡H∗ B((C∗)2)

= H∗(P∞×P∞) = C[α1, α2] .
The ﬁxed loci of the induced action on M
p2
2,0;(1)(P1, 1) consist of maps sending components of pos-
itive genus to either the ﬁxed point p1 or the rubber P1 attached to the ﬁxed point p2. The three
graphs describing these ﬁxed loci in the notation of [13, Chapter 27] are shown in Figure 7. In
these diagrams, the ﬁrst vertex label indicates the corresponding ﬁxed point of P1, while the sec-
ond indicates the genus of the component taken there, if any. The edge degree is 1 in all cases,
corresponding to the degree 1 cover from P1 −→P1.
The morphism st takes the ﬁxed locus represented by the middle diagram in Figure 7 to the closure
in M2,1 of the locus consisting of two-component maps. Thus, st∗σ vanishes on this locus and the
middle diagram does not contribute to (4.18) via the virtual localization theorem of [11].
The locus represented by the ﬁrst diagram in Figure 7 is isomorphic to M2,1 and is cut down by
st∗σ to a single point. The space of deformations of this locus consists of moving the node and of
smoothing the node; after restricting to the cut-down space, the equivariant chern class of both of
these line bundles equal to the equivariant chern class of TP1 at p1, which is α1−α2 in this case;
see [13, Exercise 27.1.3]. The obstruction bundle after cutting down by st∗σ is
H1(Σ; Tp1P1) =
 H0(Σ; T ∗Σ⊗T ∗
p1P1)
∗≈Tp1P1 ⊕Tp1P1 ;
its equivariant euler class is (α1 −α2)2. Thus, the contribution of the ﬁrst diagram in Figure 7
to (4.18) is
Z
M2,1
st∗σ
(α1−α2)2
(α1−α2) · (α1−α2) = 1 ;
see [11, (7)] or [12, Theorem 3.6].
The locus represented by the last diagram in Figure 7 is isomorphic to the (rubber) moduli space
M
0,∞
2,0;(1),(1)(P1, 1)∼of relative morphisms to the non-rigid target (P1, 0, ∞) with the standard C∗-
action; see [12, Section].
Since the virtual dimension of this moduli space is 3, the restriction
of st∗σ to this ﬁxed locus vanishes. By [12, Theorem 3.6], the last diagram in Figure 7 thus does
not contribute to (4.18). Combining this with the conclusion of the two previous paragraphs, we
obtain (4.18).
Proof 2 of (4.18). Let (Σ, j, x1) be a generic element of M2,1 as before. The number (4.18) is
the number of solutions u: Σ−→P1 of
¯∂J,ju

z = ν
 z, u(z)

∀z ∈Σ,
u∗[Σ] = [P1] ∈H2(P1; Z),
(4.19)
29

for a generic ν ∈Γpt
j (P1, J), where
Γpt
j (P1, J) ⊂Γ
 Σ×P1, (T ∗Σ)0,1⊗CTP1
is the subspace of elements ν such that
ν|Σ×pt = 0,
∇wν + J∇Jwν = 0
∀w∈TptP1.
(4.20)
The moduli space of degree 1 holomorphic maps (Σ, j)−→P1 and its obstruction bundle are given by
H0,1
Σ ⊗TP1 ≈Obs −→Holj(P1, 1) ≈P1 ,
(4.21)
where H0,1
Σ
is the space of harmonic (0, 1)-forms on Σ.
The space of deformations of the domain of the elements in Holj(P1, 1) is the product of the two
tangent bundles at the node, i.e.
Tx1Σ ⊗TP1 ≈TP1 −→P1 .
(4.22)
Each smoothing parameter υ in this line bundle determines an approximately (J, j)-holomorphic
map uυ : Σ−→P1; see [35, Section 3.3]. The ﬁrst-order term of the projection π0,1
υ,−¯∂J,juυ of ¯∂J,juυ
to Obs is given by

L(υ1⊗υ2)
	
(ψ) = ψx1(υ1)

dx2u
	
(υ2) ∈Tu(x2)P1
∀ψ ∈H1,0
Σ ,
where x2 ∈P1 is the node of the rational component of the domain of the map; see [36, Lemma 4.5].
Since L is injective in this case, the solutions of (4.19) correspond to the zeros of the section of
Obs/Im L −→Holj(P1, 1)
(4.23)
induced by a generic ν, excluding the one with u(x2) = pt; see the proof of [36, Corollary 4.7].
Thus, the number of solutions of (4.19) is

e(Obs/Im L), Holj(P1, 1)

−1 =

c1(H0,1
Σ ⊗TP1) −c1(Tx1Σ⊗TP1), P1
−1 = 1.
This establishes (4.18).
We next use the virtual localization theorem of [11] to compute the absolute invariant and the δ =1
case of the relative invariant in Example 2. We continue with the localization setup of the ﬁrst
proof of (4.18) and compute
Z
[M2,2(P1,1)]virst∗κ4 ev∗
1OP1(1−α2) ev∗
2O(1−α2)
and
Z
[M
p2
2,2;(1)(P1,1)]virst∗κ4 ev∗
1OP1(1−α2) ev∗
2OP1(1−α2) .
(4.24)
The (C∗)2-ﬁxed loci consist of maps sending the positive-genus components and the absolute marked
points to the ﬁxed points p1 and p2. Since the equivariant chern class of OP1(1−α2) vanishes at p2,
the only graphs possibly contributing to the integrals in (4.24) must have both absolute marked
points sent to p1. Since the morphism st takes ﬁxed loci with a positive-genus component at both
30

(1, 2)
2
1
2
(1, 0)
(2, 2)
1
2
(1, 2)
2
1
2
3
(1, 0)
(2, 2)
1
2
3
Figure 8: The two pairs of graphs possibly contributing to the two integrals in (4.24)
ﬁxed points to the closure in M2,2 of the locus consisting of two genus 1 curves, st∗κ4 vanishes on
such ﬁxed loci as well. The two remaining graphs possibly contributing to each of the integrals
in (4.24) are shown in Figure 8.
The locus represented by the ﬁrst diagram in Figure 8 is isomorphic to M2,3.
The space of
deformations of this locus consists of moving the node and of smoothing the node; the equivariant
chern classes of these line bundles are α1−α2 and α1−α2−ψ3, respectively. The euler class of the
obstruction bundle is given by
e
 E∗⊗Tp1P1
= λ2 −(α1−α2)λ1 + (α1−α2)2 .
By [11, (7)], the contribution of the ﬁrst diagram in Figure 8 to the ﬁrst integral in (4.24) is thus
Z
M2,3
st∗κ4 (α1−α2)2 λ2 −(α1−α2)λ1 + (α1−α2)2
(α1−α2)(α1−α2−ψ3)
=
Z
M2,3
st∗κ4 λ2−λ1ψ3+ψ2
3)
= 0 + 4⟨ψ3
1λ1, M2,1⟩−⟨(f ∗ψ1)3ψ2
2, M2,2⟩= 4⟨ψ3
1λ1, M2,1⟩−⟨ψ3
1ψ2
2, M2,2⟩,
(4.25)
where f : M2,2 −→M2,1 is the forgetful morphism. The second equality above applies the dila-
ton equation [13, Exercise 25.2.7] to the middle term, while the last equality follows from [13,
Lemma 25.2.3].
The locus represented by the second diagram in Figure 8 is isomorphic to M2,1. The space of
deformations of this locus consists of moving and smoothing the two nodes; the total equivariant
euler class of the deformations is
(α1−α2)(α2−α1)(α1−α2)(α2−α1−ψ1).
The euler class of the obstruction bundle is given by
e
 E∗⊗Tp2P1
= λ2 −(α2−α1)λ1 + (α2−α1)2 .
By [11, (7)], the contribution of the second diagram in Figure 8 to the ﬁrst integral in (4.24) is thus
Z
M2,1
st∗κ4 (α1−α2)2 λ2 −(α2−α1)λ1 + (α1−α2)2
(α1−α2)3(α1−α2+ψ1)
=
Z
M2,1
ψ4
1 =
1
1152 .
(4.26)
Combining (4.25) and (4.26) with
⟨ψ3
1ψ2
2, M2,2⟩=
29
5760
from C. Faber’s program, we obtain the ﬁrst equality in (1.13).
31

P4
PP4V7
V7
PP4,∞V7
(3, 1)
1
(0, 0)
P4
PP4V7
V7
PP4,∞V7
(0, 1)
1
(0, 0)
(3, 0)
Figure 9: Bipartite graphs Γ that contribute to the absolute GW-invariant in (1.14) via the sym-
plectic sum decomposition with respect to V7.
The contribution of the ﬁxed locus of M
p2
2,2;(1)(P1, 1)]vir represented by third diagram in Figure 8
to the second integral in (4.24) is the same as of the ﬁrst diagram to the the ﬁrst integral in (4.24).
The locus represented by the fourth diagram in Figure 8 is isomorphic to M2,2. The space of
deformations of this locus consists of moving node at p1 and smoothing both nodes; the total
equivariant euler class of the deformations is
(α1−α2)(α1−α2)(α2−α1−ψ1).
The euler class of the obstruction bundle in this case is given by
e
 E∗⊗Tp2P1(−p2)

= λ2 .
By [11, (7)], the contribution of the fourth diagram in Figure 8 to the second integral in (4.24)
is thus
Z
M2,2
st∗κ4 (α1−α2)2
λ2
(α1−α2)2(α2−α1−ψ1) = 0.
Along with the two previous paragraphs, this provides a direct check of the d=1 case of the second
equality in (1.13).
4.3
Genus 3 degree 1 primary invariants of P4
We establish the second equality in (1.14) by applying the symplectic sum formula, as stated in
the second-to-last equation on page 201 in [22], to the absolute GW-invariant in (1.14) via the
decomposition (3.7) with X =P4 and V =Vδ, where Vδ ⊂P4 is a smooth degree δ hypersurface.
Since the Poincare dual of the primary insertion in (1.14) vanishes on the hypersurface Vδ (the
constraining point can be chosen outside of Vδ), kv = 0 for all v ∈ΓV (the marked point stays on
the X-side) if Γ is a bipartite graph as in Section 3.2 contributing to the absolute GW-invariant
in (1.14).
Furthermore, Av = 1 for the unique vertex v ∈ΓX.
By Section 3.2 or Lemma 3.2
(separately), the only graphs Γ that may contribute to the absolute GW-invariant in (1.14) satisfy
(1) (gv, Av, kv)=(0, 0, 0) for all v∈ΓV or
(2) (gv, Av, kv) = (3, 0, 0) for one element v ∈ΓV and (gv, Av, kv) = (0, 0, 0) for the remaining
elements v∈ΓV .
32

There are other bipartite graphs Γ, but they all contain a vertex v ∈ΓV with gv ∈{1, 2}; by Sec-
tion 3.2, such a graph does not contribute to an absolute GW-invariant with primary insertions via
the symplectic sum formula. By Section 3.2 or Lemma 3.2, the label of the edge leaving a vertex
v∈ΓV with gv =0 in a contributing graph Γ is 1 (and thus omitted in our diagrams). By the proof
of Lemma 3.2 in Section 3.2, the same is the case if gv = 3; otherwise, the ﬁber of the projection
in (3.13) would have positive dimension, while too many conditions would be imposed on the curve
on the X-side. The same conclusions can be drawn from Section 3.1.
In summary, there are only two graphs that may contribute to the absolute GW-invariant in (1.14)
via the symplectic sum formula; they are shown in Figure 9. Thus,
GWP4
3,1(pt) = 1
δ!GWP4,Vδ
3,1;1δ(pt) + δ
δ! GWP4,Vδ
0,1;1δ(1; pt; 1δ−1, pt) GW
PP4Vδ,PP4,∞Vδ
3,F ;(1)
(1; 1; 1) ,
(4.27)
where F ∈H2(PP4Vδ; Z) is the ﬁber class. The ﬁrst insertion 1 in the last two relative invariants
in (4.27) indicates that no constraint is imposed on the domain of the maps by pulling back a
class κ from a Deligne-Mumford space of curves. The relative insertions (1δ−1, pt) and 1 in these
invariants (shown after the second semi-column in each case) arise from the Kunneth decomposition
of the diagonal ∆V in V 2; the point insertion on the ﬁrst of these invariants corresponds to the
pairing with the second invariant, which arises from a zero-dimensional relative moduli space. It is
immediate from the g=0 part of the argument in Section 3.1 that
δ
δ! GWP4,Vδ
0,1;1δ(1; pt; 1δ−1, pt) = GWP4
0,1(pt, pt) = 1.
(4.28)
Combining (4.27) with (4.28) and Lemma 4.4 below, we conclude that
GWP4
3,1(pt) = 1
δ!GWP4,Vδ
3,1;1δ(pt) + ⟨c1(Vδ)c2(Vδ)−c3(Vδ), Vδ⟩
362880
.
The second equality in (1.14) now follows from
c(Vδ) =
 (1 + x)5(1+δx)−1
Vδ ∈H∗(Vδ; Z),
where x=c1(OP4(1)) ∈H2(P4; Z) is the standard generator.
Lemma 4.4. Let (V, ω) be a compact symplectic manifold of real dimension 6 and L −→V be a
complex line bundle. With notation as at the beginning of Section 3.3, the virtual dimension of the
genus 3 relative moduli space M
PL,∞
3,0;(1)(PL, F) is 0 and
deg

M
PL,∞
3,0;(1)(PL, F)
vir = ⟨c1(V )c2(V )−c3(V ), V ⟩
362880
.
Proof. The ﬁrst claim is immediate from the second equation in (1.2). In order to establish the
second claim, we proceed as in Section 3.2 by ﬁrst choosing a generic deformation ν ∈Γ3,0(V, JV ).
Lifting JV and ν to PL−→V as in Section 3.2, we obtain a ﬁbration
πL,V : M
PL,∞
3,0;(1)
 PL, F; J, π∗
X,V ν

−→M3,0
 V, 0; JV , ν

(4.29)
33

(1, 3)
2
1
(1, 2)
(2, 1)
1
(1, 1)
(2, 2)
1
1
(2, 3)
1
Figure 10: The four graphs describing the (C∗)2-ﬁxed loci of M
p2
3,0;(1)(P1, 1)
as in (3.13). In this case, the base is zero-dimensional. Since the obstruction bundle for M3,0(V, 0)
is given by (3.3) with (gv, kv+ℓv)=(3, 0), the degree of this base is

e(π∗
1E∗⊗π∗
2TV ), M3×V

= ⟨λ1λ2λ3, M3⟩

c1(V )c2(V )−3c3(V ), V

+ ⟨λ3
2, M3⟩

c3(V ), V

+ ⟨λ2
3, M3⟩

c1(V )3−3c1(V )c2(V )+3c3(V ), V

.
The three intersection numbers on M3 above are provided by Table 1 and (3.4). The second claim
of Lemma 4.4 now follows from Lemma 4.5 below.
Lemma 4.5. If st : M
pt
3,0;(1)(P1, 1) −→M3 is the forgetful morphism dropping the map to P1 and
the marked point, then
st∗

M
pt
3,0;(1)(P1, 1)
vir = 4

M3

.
(4.30)
Since M3 is smooth (as an orbifold) and irreducible, (4.30) is equivalent to

st∗σ, [M
pt
3,0;(1)(P1, 1)]vir
= 4,
(4.31)
where σ∈H12(M3) is the Poincare dual of a generic element Σ of M3. We give two proofs of (4.31)
below, which are similar to the two proofs of (4.18).
Proof 1 of (4.31). We continue with the localization setup in the ﬁrst proof of (4.18). The ﬁxed
loci of the induced action on M
p2
3,0;(1)(P1, 1) again consist of maps sending components of positive
genus to either the ﬁxed point p1 or the rubber P1 attached to the ﬁxed point p2. The four graphs
describing these ﬁxed loci, in the notation of [13, Chapter 27] and Figure 7, are shown in Figure 10.
The morphism st takes the ﬁxed loci represented by the two middle diagrams in Figure 10 to the
closure in M3 of the locus consisting of two-component maps. Thus, st∗σ vanishes on these loci
and the two middle diagrams do not contribute to (4.31) via the virtual localization theorem of [11].
The locus represented by the ﬁrst diagram in Figure 10 is isomorphic to M3,1 and is cut down by
st∗σ to the curve Σ (which encodes the position of the node). The space of deformations of this
locus consists of moving the node and of smoothing the node; its euler class equals
(α1−α2)(α1−α2+c1(TΣ))
after restricting to the cut-down space. The obstruction bundle after cutting down by st∗σ is
H1(Σ; Tp1P1) =
 H0(Σ; T ∗Σ⊗T ∗
p1P1)
∗≈Tp1P1 ⊕Tp1P1 ⊕Tp1P1 ;
34

its equivariant euler class is (α1 −α2)3. Thus, the contribution of the ﬁrst diagram in Figure 10
to (4.31) is
Z
M3,1
st∗σ
(α1−α2)3
(α1−α2)(α1−α2+c1(TΣ)) = −
Z
Σ
c1(TΣ) = 4 ;
see [11, (7)] or [12, Theorem 3.6].
The locus represented by the last diagram in Figure 10 is isomorphic to M
0,∞
3,0;(1),(1)(P1, 1)∼. Since
the virtual dimension of this moduli space is 5, the restriction of st∗σ to this ﬁxed locus vanishes.
By [12, Theorem 3.6], the last diagram in Figure 10 thus does not contribute to (4.31). Combining
this with the conclusion of the two previous paragraphs, we obtain (4.31).
Proof 2 of (4.31). Let (Σ, j) be a generic element of M3 as before. The ﬁrst paragraph of the
second proof of (4.18) applies to the present situation; the only change is that the base in (4.21) is
replaced by
Σ×Holj(P1, 1) ≈Σ×P1 .
The line bundle of smoothing parameters (4.22) now becomes
TΣ ⊗TP1 −→Σ×P1 .
Analogously to the sentence containing (4.23), the solutions of the analogue of (4.19) in this situ-
ation correspond to the zeros of the section of
Obs/Im L −→Σ×Holj(P1, 1),
with L as before, induced by a generic admissible ν, excluding the ones with u(x2)=pt. Without
the ﬁrst restriction on ν in (4.20), the number of such zeros would have been

e(Obs/Im L), Σ×Holj(P1, 1)

=

c1(C3/TΣ), Σ

c1(TP1), P1
= 8.
The contribution to this number from the vanishing of ¯ν along Σ×pt is the number of zeros of an
aﬃne bundle map
C ⊕TΣ⊗TptP1 −→H0,1
Σ ⊗TptP1
with an injective linear part. Thus, the latter number is

e(C3/(C⊕TΣ)), Σ

= 4 .
The number in (4.31) is the diﬀerence of the two numbers above.
4.4
The δ=0, 1 numbers in Example 3
We now use the virtual localization theorem of [11] to compute the absolute invariant and the
virtual localization theorem of [12] to compute the δ =1 case of the relative invariant in Example 3.
We apply [11, (7)] with the C∗-action on P4 given by
c · [Z1, Z2, Z3, Z4, Z5] = [Z1, cZ2, cZ3, c−1Z4, c−1Z5]
35

(1, 3)
i
1
(1, 2)
(i, 1)
1
(1, 1)
(i, 2)
1
1
(i, 3)
1
(1, 3)
i
1
2
(1, 2)
(i, 1)
1
2
(1, 1)
(i, 2)
1
2
1
(i, 3)
1
2
Figure 11: The two sets of graphs possibly contributing to the integrals (4.32) and (4.37), with
i∈{+, −} in the ﬁrst case and i∈{2, 3, 4, 5} in the second case.
and its linear lift to OP4(1) deﬁned in the same way. The ﬁxed locus of this action consists of
p1 ≡[1, 0, 0, 0, 0],
P1
23 ≡

[0, Z2, Z3, 0, 0] ∈P4	
,
P1
45 ≡

[0, 0, 0, Z4, Z5] ∈P4	
.
Let
α = c1
 OP4(1)

p2 ∈H∗
C∗.
We denote by
M3,1(P4, 1)p1 ⊂M3,1(P4, 1)
the preimage of p1 under the evaluation morphism ev1. We will compute
Z
[M3,1(P4,1)p1]vir1.
(4.32)
The C∗-ﬁxed loci of this moduli space consist of maps sending the positive-genus components to p1
or a point on P1
23 or P1
45 with the image of a degree 1 rational component running between p1 and
a point on either P1
23 or P1
45. The four types of graphs possibly contributing the integral in (4.32)
are shown in the left half of Figure 11, where ± on the bottom vertex indicates whether it lies
on P1
23 or P1
45, respectively. In the computations below, we ﬁrst assume that i=+.
The locus represented by the ﬁrst diagram in Figure 11 is isomorphic to M3,2 ×P1. The space
of deformations of this locus consists of smoothing the node and turning the line around it away
from P1
23; the equivariant euler class of the space of deformations is thus
(−α−x−ψ2)(2α+x)2 = −(α+x+ψ2)(2α+x)2 ,
where x = c1(OP1(1)) ∈H1(P1; Z) is the standard generator. The euler class of the obstruction
bundle is given by
e
 E∗
3⊗Tp1P4
=
 λ3 −αλ2 + α2λ1 −α32 λ3 + αλ2 + α2λ1 + α32 .
By [30, (5.3)],
 λ3 −αλ2 + α2λ1 −α3 λ3 + αλ2 + α2λ1 + α3
= −α6 .
The contribution of the ﬁrst diagram in Figure 11 to (4.32) is thus
−
Z
M3,2×P1
α12
(α+x+ψ2)(2α+x)2 = 5
2⟨x, P1⟩

ψ8
2, M3,2

= 5
2

ψ7
1, M3,1

=
5
165888 .
(4.33)
The second equality above applies the dilaton equation [13, Exercise 25.2.7]; the last follows from
the ﬁrst column in Table 3.
36

ψ7
1
ψ6
1λ1
ψ2
5λ2
1
ψ5
1λ2
ψ4
1λ3
1
ψ4
1λ1λ2
ψ4
1λ3
1
82944
7
138240
41
290304
41
580608
23
96768
23
193536
31
967680
ψ3
1λ4
1
ψ3
1λ2
1λ2
ψ3
1λ1λ3
ψ3
1λ2
2
ψ2
1λ5
1
ψ2
1λ3
1λ2
ψ2
1λ2
1λ3
ψ2
1λ1λ2
2
ψ2
1λ2λ3
41
181440
41
362880
41
1451520
41
725760
1
7560
1
15120
1
60480
1
30240
1
120960
Table 3: The top intersections of λ-classes and ψi
1 with i ≥2 on M3,1; the intersections with ψ1
1
are obtained by multiplying the corresponding numbers in Table 1 by 4.
The locus represented by the second diagram in Figure 11 is isomorphic to M2,2×M1,1×P1. The
space of deformations of this locus consists of smoothing the two nodes and moving the bottom
node away from P1
23; the equivariant euler class of the space of deformations is thus
(−α−x−ψt)(α+ x−ψb)(α+x)(2α+x)2 = −(α+x+ψt)(α+ x−ψb)(α+x)(2α+x)2 ,
where ψt ∈H∗(M2,2) and ψb ∈H∗(M1,1). The euler class of the obstruction bundle is given by
e
 E∗
2⊗Tp1P4
e
 E∗
1⊗TP4|P1
23

=
 λ2−αλ1+α22 λ2+αλ1+α22 λb−2x
 λb−(α+x)
 λb−(2α+x)
2
= 4α
 λ2−αλ1+α22 λ2+αλ1+α22(3λbx −2αx+αλb)
 λb−(α+x)

,
where λb ∈H∗(M1,1) By [30, (5.3)],
 λ2−αλ1+α2 λ2+αλ1+α2
= α4 .
Since ψ1 =λ on M1,1, the contribution of the second diagram in Figure 11 to (4.32) is thus
Z
M2,2×M1,1×P1
4α9(3λbx −2αx+αλb)
(α+x+ψt)(α+x)(2α+x)2 = 5⟨λ, M1,1⟩⟨x, P1⟩

ψ5
2, M2,2

= 5
24

ψ4
1, M2,1

=
5
27648 .
(4.34)
The second equality above applies the dilaton equation [13, Exercise 25.2.7]; the last follows from
the ﬁrst column in Table 2.
The locus represented by the third diagram in Figure 11 is isomorphic to M1,2×M2,1×P1. The euler
class of its deformation space is as in the previous paragraph. The euler class of the obstruction
bundle is given by
e
 E∗
1⊗Tp1P4
e
 E∗
2⊗TP4|P1
23

=
 λt−α
2 λt+α
2 λ2−2xλ1
 λ2−(α+x)λ1+(α+x)2 λ2−(2α+x)λ1+(2α+x)22,
where λt ∈H∗(M1,2). Since λ2 = 0 on M1,2 and λ2
2, 2λ2−λ2
1 = 0 on M2, the contribution of the
third diagram in Figure 11 to (4.32) is thus
−
Z
M1,2×M2,1×P1
4α8λ1(2α2λ1−4αλ2
1 −(8α2−24αλ1+29λ2
1)x)
(α+x+ψ1)(α+ x−ψ2)(α+x)(2α+x)2
=

ψ2
1, M1,2

⟨x, P1⟩

λ3
1ψ1−8λ2
1ψ2
1+8λ1ψ3
1, M2,1

= −
1
11520 .
(4.35)
37

The second equality above applies the dilaton equation [13, Exercise 25.2.7] and uses the ﬁrst col-
umn in Table 1 and the third in Table 2.
The locus represented by the fourth diagram in Figure 11 is isomorphic to M3,1×P1. The space
of deformations of this locus consists of smoothing the (bottom) node and moving it from P1
23; the
equivariant euler class of the space of deformations is thus
(α+ x−ψ1)(α+x)(2α+x)2 .
The euler class of the obstruction bundle is given by
e
 E∗
3⊗TP4|P1
23

=
 λ3−2xλ2
 λ3−(α+x)λ2+(α+x)2λ1−(α+x)3
×
 λ3−(2α+x)λ2+(2α+x)2λ1−(2α+x)32.
Since λ2
1 =2λ2, λ2
2 =2λ1λ3, and λ2
3 =0 on M3,1, the contribution of the fourth diagram in Figure 11
to (4.32) is thus
Z
M3,1×P1
α6(16α2λ4
1−16αλ5
1+9λ6
1−64α3λ3)
(α+ x−ψ1)(α+x)(2α+x)2
+
Z
M3,1
α2(128α4λ2
1−256α3λ3
1+424α2λ4
1−308αλ5
1+141λ6
1−768α3λ3)
8(α−ψ1)
= 1
8⟨x, P1⟩

69λ6
1ψ1−148λ5
1ψ2
1+232λ4
1ψ3
1−256λ3
1ψ4
1+128λ3ψ4
1+128λ2
1ψ5
1, M3,1

= −
1
2880 .
(4.36)
Combining the numbers in (4.33)-(4.36) and multiplying the result by 2 (to account for i=±), we
obtain the ﬁrst equality in (1.14). This conclusion agrees with A. Gathmann’s growi program.
We next apply [12, Theorem 3.6] with the action of T≡(C∗)2 on P4 given by
(c1, c2) · [Z1, Z2, Z3, Z4, Z5] = [Z1, c1Z2, c−1
1 Z3, c2Z4, c−1
2 Z5]
and its linear lift to OP4(1) deﬁned in the same way. The ﬁxed locus of this action consists of the
ﬁve points
p1 ≡[1, 0, 0, 0, 0],
. . . ,
p5 ≡[0, 0, 0, 0, 1].
Let
α1 = c1
 OP4(1)

p2 ∈H∗
T ,
α2 = c1
 OP4(1)

p4 ∈H∗
T ,
V =

[0, Z2, Z3, Z4, Z5]∈P4	
.
We denote by
M
V
3,1;(1)(P4, 1)p1 ⊂M
V
3,1;(1)(P4, 1)
the preimage of p1 under the evaluation morphism ev1. We will compute
Z
[M
V
3,1;(1)(P4,1)p1]vir1 .
(4.37)
The C∗-ﬁxed loci of this moduli space consist of maps sending the positive-genus components to p1
and at most one of the ﬁxed points pi with i=2, 3, 4, 5; the image of the non-contracted degree 1
38

rational tail runs between p1 and one of the ﬁxed points pi with i = 2, 3, 4, 5. The four types of
graphs possibly contributing to the integrals in (4.37) are shown in the right half of Figure 11. In
the computations below, we ﬁrst assume that i=2.
The locus represented by the ﬁrst diagram in the right half of Figure 11 is isomorphic to M3,2.
Its deformations consist of smoothing the node and turning the line around it away from p2; the
equivariant euler class of the space of deformations is thus
(−α1−ψ2)
 α1−(−α1)

(α1−α2)
 α1−(−α2)

= −2α1(α2
1−α2
2)(α1+ψ2) .
The obstruction bundle is as for the ﬁrst diagram in Figure 11, but its euler class is now given by
Y
j=1,2
 (α3
j −α2
jλ1+αjλ2−λ3)(α3
j +α2
jλ1+αjλ2+λ3)

= α6
1α6
2 ;
the equality holds by [30, (5.3)]. The contribution of the ﬁfth diagram in Figure 11 to (4.37) is thus
−
Z
M3,2
α6
1α6
2
2α1(α2
1−α2
2)(α1+ψ2) = −1
2 ·
α6
2
α4
1(α2
1−α2
2)

ψ8
2, M3,2

= −1
2 ·
α6
2
α4
1(α2
1−α2
2)

ψ7
1, M3,1

= −1
2 ·
1
82944 ·
α6
2
α4
1(α2
1−α2
2) .
(4.38)
The second equality above applies the dilaton equation [13, Exercise 25.2.7]; the last follows from
the ﬁrst column in Table 3.
The locus represented by the second diagram in the right half of Figure 11 is isomorphic to
F2 ≡M2,2 × M
0,∞
1,0;(1),(1)(P1, 1)∼.
The equivariant euler class of the space of deformations becomes
−2α1(α2
1−α2
2)(α1+ψt)(α1−ψb),
where ψt ∈H∗(M2,2) and ψb=ψ∞is on the rubber moduli space; see [12, Section 3.3]. The euler
class of the obstruction bundle is now given by
e
 E∗
2⊗Tp1P4
e
 E∗
1⊗Tp2V

=
Y
j=1,2
 (α2
j −αjλ1+λ2)(α2
j +αjλ1+λ2)

· (2α1−λb)(α1−α2−λb)(α1+α2−λb)
∼= −α4
1α4
2(5α2
1−α2
2)λb
mod H∗
T ⊂HT∗(F2) ,
where λb ∈H∗(F2) is the pull-back of λ∈H∗(M1,1) by either forgetful morphism f from the second
factor. Since
M
0,∞
1,0;(1),(1)(P1, 1)∼≈M1,1 × M
0,∞
0,1;(1),(1)(P1, 1)∼
as spaces and the last factor above is a point, ψb vanishes on the virtual class of the second factor
in F2. The second proofs of (4.18) and (4.31) readily show that
f∗

M
0,∞
1,0;(1),(1)(P1, 1)∼
vir =

M1,1

.
39

Thus, the contribution of the sixth diagram in Figure 11 to (4.37) is
Z
[F2]vir
α4
1α4
2(5α2
1−α2
2)λb
2α1(α2
1−α2
2)(α1+ψt)α1
= −1
2 · α4
2(5α2
1−α2
2)
α4
1(α2
1−α2
2) ·

ψ5
2, M2,2

⟨λ, M1,1⟩
= −1
2 · α4
2(5α2
1−α2
2)
α4
1(α2
1−α2
2) · 1
24

ψ4
2, M2,1

= −1
2 ·
1
27648 · α4
2(5α2
1−α2
2)
α4
1(α2
1−α2
2) .
(4.39)
The second equality above applies the dilaton equation [13, Exercise 25.2.7]; the last follows from
the ﬁrst column in Table 2.
The locus represented by the third diagram in the right half of Figure 11 is isomorphic to
F3 ≡M1,2 × M
0,∞
2,0;(1),(1)(P1, 1)∼≡F3;1 × F3;2 .
The equivariant euler class of its deformation space is as in the previous paragraph. The euler class
of the obstruction bundle is now given by
e
 E∗
1⊗Tp1P4
e
 E∗
2⊗Tp2V

=
Y
j=1,2
 (αj−λt)(αj +λt)

· (4α2
1−2α1λ1+λ2)
×
 (α1−α2)2−(α1−α2)λ1+λ2
 (α1+α2)2−(α1+α2)λ1+λ2

∼= −α3
1α2
2(9α2
1−α2
2)λ3
1 + 1
2α2
1α2
2(25α4
1−10α2
1α2
2+α4
2)λ2
1
mod H∗
T⊗H2(F3;2) ⊂HT∗(F3) ,
where λt ∈H∗(M1,2) and λ1, λ2 ∈H∗(F3;2) are the pull-backs of the Hodge classes λ1, λ2 ∈H∗(M2)
by the forgetful morphism f from the second factor. Since
M
0,∞
2,0;(1),(1)(P1, 1)∼≈M2,1 ×M0,∞
0,1;(1),(1)(P1, 1)∼∪
 M1,1×M1,1×M
0,∞
0,2;(1),(1)(P1, 1)∼

Z2
as spaces and the last factors in the two spaces on the RHS above are zero- and one-dimensional,
ψ2
b vanishes on the virtual class of F3;2.
Since λ3
1 vanishes on the divisor in M2 consisting of
two-component curves and

ψk−1
∞, M
0,∞
0,k;(1),(1)(P1, 1)∼

= 1
∀k ≥3,
(4.40)
the second proofs of (4.18) and (4.31) readily show that

λ3
1, [M
0,∞
2,0;(1),(1)(P1, 1)∼]vir
=

e(C2/TΣ2), Σ2

λ3
1, [M2]

= 2

λ3
1, [M2]

;

ψ∞λ2
1, [M
0,∞
2,0;(1),(1)(P1, 1)∼]vir
=

λ2
1, M1,1
2 .
Thus, the contribution of the seventh diagram in Figure 11 to (4.37) is
Z
[F3]vir
α3
1α2
2(9α2
1−α2
2)λ3
1
2α1(α2
1−α2
2)(α1+ψt)α1
−1
2
Z
[F3]vir
α2
1α2
2(25α4
1−10α2
1α2
2+α4
2)λ2
1ψ∞
2α1(α2
1−α2
2)(α1+ψt)α2
1
=
⟨ψ2
t , M1,2⟩
2 α4
1(α2
1−α2
2)

α2
1α2
2(9α2
1−α2
2) · 2

λ3
1, [M2]

−α2
2(25α4
1−10α2
1α2
2+α4
2) · ⟨λ, M1,1⟩2
2

= −1
2 ·
1
138240 · α2
2(89α4
1−46α2
1α2
2+5α4
2)
α4
1(α2
1−α2
2)
;
(4.41)
40

the last equality follows from Table 1.
The locus represented by the last diagram in Figure 11 is isomorphic to
F4 ≡M
0,∞
3,0;(1),(1)(P1, 1)∼.
The equivariant euler class of its deformation space is reduced to 2α1(α2
1−α2
2)(α1−ψb). The euler
class of the obstruction bundle becomes
e
 E∗
3⊗Tp2V

= (8α3
1−4α2
1λ1+2α1λ2−λ3)
 (α1−α2)3−(α1−α2)2λ1+(α1−α2)λ2−λ3

 (α1+α2)3−(α1+α2)2λ1+(α1+α2)λ2−λ3

∼= −1
2α2
1(9α2
1−α2
2)λ5
1 + 1
4α1(45α4
1−14α2
1α2
2+α4
2)λ4
1
−(18α6
1−20α4
1α2
2+2α2
1α4
2)λ3
1 −(17α6
1+45α4
1α2
2+3α2
1α4
2−α6
2)λ3
mod H∗
T⊗H≤4(F4),
where λi ∈H∗(F4) is the pull-back of the Hodge class λi ∈H∗(M3) by the forgetful morphism f.
Since
M
0,∞
3,0;(1),(1)(P1, 1)∼≈M3,1 ×M0,∞
0,1;(1),(1)(P1, 1)∼∪M2,1×M1,1×M
0,∞
0,2;(1),(1)(P1, 1)∼
∪
 (M1,1)3×M
0,∞
0,3;(1),(1)(P1, 1)∼

S3
as spaces and the last factors in the three spaces on the RHS above are zero-, one-, and two-
dimensional, respectively, ψ3
b vanishes on the virtual class of F4. Since λ5
1 vanishes on the divisor
in M3 consisting of two-component curves and and λ4
1 vanishes on the subvariety consisting of four-
component curves (three elliptic curves attached to a P1), (4.40) and the second proofs of (4.18)
and (4.31) give

λ5
1, [M
0,∞
3,0;(1),(1)(P1, 1)∼]vir
=

λ5
1e(E∗/TΣ3), [M3,1]

=

λ5
1ψ2
1−λ6
1ψ1, [M3,1]

;

ψ∞λ4
1, [M
0,∞
3,0;(1),(1)(P1, 1)∼]vir
= 8

λ, M1,1

λ3
1, M2

;

ψ2
∞λ3
1, [M
0,∞
3,0;(1),(1)(P1, 1)∼]vir
=

λ, M1,1
3 ,

ψ2
∞λ3, [M
0,∞
3,0;(1),(1)(P1, 1)∼]vir
= 1
6

λ, M1,1
3 .
Thus, the contribution of the last diagram in Figure 11 to (4.37) is
−1
2
Z
[F4]vir
α2
1(9α2
1−α2
2)λ5
1
2α1(α2
1−α2
2)α1
+ 1
4
Z
[F4]vir
α1(45α4
1−14α2
1α2
2+α4
2)λ4
1ψ∞
2α1(α2
1−α2
2)α2
1
−1
6
Z
[F4]vir
(125α6
1−75α4
1α2
2+15α2
1α4
2−α6
2)λ3
1ψ2
∞
2α1(α2
1−α2
2)α3
1
.
The preceding set of equations reduces this to
1
2 α4
1(α2
1−α2
2)
 
−1
2α4
1
 9α2
1−α2
2
 
λ5
1ψ2
1, [M3,1]

−4

λ6
1, [M3]

+ α2
1
 45α4
1−14α2
1α2
2+α4
2

· 2

λ, M1,1

λ3
1, M2

−
 125α6
1−75α4
1α2
2+15α2
1α4
2−α6
2

λ, M1,1
3
6
!
= −1
2 ·
1
2903040 · 1747α6
1−1577α4
1α2
2+441α2
1α4
2−35α6
2
α4
1(α2
1−α2
2)
.
(4.42)
41

The sum of (4.38), (4.39), (4.41), and (4.42) multiplied by 2 (to account for i=3) is
−
1
2903040 · 1747α6
1+292α4
1α2
2
α4
1(α2
1−α2
2)
.
Adding in the same expression with α1 and α2 interchanged (to account for i=4, 5), we ﬁnd that
GWP4,V1
3,1;(1)(pt) = −
97
193536 .
Along with the ﬁrst equality in (1.14), this conﬁrms the δ =1 case of the second equality in (1.14).
5
The Cieliebak-Mohnke approach to GW-invariants
Theorem 1 and Examples 1-3 answer a key question arising in recent attempts to adapt the idea
of [2] to constructing positive-genus GW-invariants geometrically. In this section, we review this
approach and discuss its connections with Theorem 1 and Examples 1-3.
Suppose (X, ω) is a compact symplectic manifold such that ω represents an integral cohomology
class. By [3, Theorem 1], the Poincare dual of every suﬃciently large integer multiple δω of ω can
be represented by a symplectic hypersurface V in (X, ω). If A∈H2(X; Z)−{0} can be represented
by a J-holomorphic map u: Σ−→X for some ω-tame almost complex structure, then
A · V = δ ω(A) > 0.
The idea of [2] is to deﬁne the primary genus 0 GW-invariants by counting J-holomorphic maps
P1 −→X that pass through generic representatives of constraints of the appropriate total dimension
and send A·V points of P1 to V and dividing the resulting number by (A·V )!. In order to ensure
that the sets of maps being counted are ﬁnite, the almost complex structure J on X is allowed to
vary with the domain of the map in a coherent way. For δ suﬃciently large and a generic coher-
ent family of J’s, every non-constant J-holomorphic map u : P1 −→X of ω-energy at most ω(A)
intersects X−V and sends at least three distinct points of the domain to V ; see [2, Proposition 8.13].
The almost complex structures J used in [2] are required to be compatible with V , in the sense
that J(TV ) ⊂TV .
A coherent family of such complex structures can be viewed as a special
case of a single pair (J, ν), with ν as (2.2) satisfying the ﬁrst condition in (2.14). By a standard
cobordism argument, the resulting count of (J, ν)-maps is independent of a generic pair (J, ν)
compatible with (ω, V ). As in [2, Section 10], the independence of the counts on V can be shown
by deﬁning such counts with respect to two transverse Donaldson’s hypersurfaces, V and V ′, that
are compatible with the same ω-tame almost complex structure J on X: the dimension-counting
argument at the beginning of Section 3.1 implies that the number of maps does not change if an
additional J-holomorphic hypersurface V ′ is added.
Remark 5.1. The counts deﬁned in [2] have not been directly shown to be invariant under de-
formations of ω, which is a central property of GW-invariants in symplectic topology. This could
be established by showing that two Donaldson’s divisors with respect to deformation equivalent
symplectic forms and of the same degree are deformation equivalent through Donaldson’s divisors.
While it remains unknown whether this is the case, the counts of [2] are directly shown in [19] to
be invariant under small deformations of ω.
42

Remark 5.2. Pairs (J, ν) as in Section 2 have been standard on the symplectic side of GW-theory
at least since [33, 34]. Using such pairs in [2] would have avoided the need for an elaborate coherency
condition on families of almost complex structure and would have simpliﬁed the transversality
issues, likely cutting down the paper by half to two thirds.
It would have also extended the
construction to genus 0 invariants with constraints pulled back from the Deligne-Mumford space.
By [20, Section 3.2], any topological component of the preimage of a J-holomorphic hypersurface V
in X under the limit u:Σ−→X of a sequence of J-holomorphic maps uk : Σk −→X from smooth
domains meeting X −V comes with a holomorphic section of the pull-back of the normal bundle
to V . By [16, Section 6], this conclusion also applies to (J, ν)-holomorphic maps, if J(TV ) = TV
and ν satisﬁes the ﬁrst condition in (2.14). If J and ν also satisfy (2.11) and the second condition
in (2.14), spaces of maps from stable domains that satisfy this limiting condition are of dimension
at least two less than the space of maps from smooth domains which meet X −V . If all relevant
domains are stable, invariants of (X, ω, V ) can then be deﬁned by counting such maps; see Section 2.
The attempts in [9, 18] to extend the approach of [2] to positive-genus GW-invariants utilize the
ideas outlined in the previous paragraph. A crucial claim of [9, 18] is that the resulting counts of
relative genus g degree d (J, ν)-maps are independent of the choice of (g, A)-hollow hypersurface V
(at least, if it is a Donaldson’s hypersurface). As illustrated by Theorem 1 and Examples 1-3, this
claim is often, but not always, true. As illustrated by the direct proof of Theorem 1 in Section 3.1,
it is true precisely when the ideas outlined in the previous paragraph are not needed to deﬁne the
relative counts. In these cases, the argument in Section 3.1 implies that the counts do not change
when a second J-holomorphic hypersurface V2 is added.
Remark 5.3. The Nijenhuis condition (2.11) on J and the second restriction on ν in (2.14) are
central to the setups in [9] and [18]. However, neither [9] nor [18] shows that these conditions
can be satisﬁed with respect to two Donaldson’s hypersurfaces simultaneously (and even more
complicated combinations of hypersurfaces are used in [18]); for transversality reasons, the second
condition in (2.14) needs to be achieved for any speciﬁed restriction of ν to the intersection of the
two hypersurfaces. These properties are used to show that the deﬁned counts are independent of
the choice of Donaldson’s divisor V . The attempt in [10] to address the condition (2.11) on J fails
at the ﬁrst and basic step, for which the reader was referred to the proof of [28, Theorem 6.17]; the
author of [10] has agreed with this and withdrawn his claim. However, it appears plausible that
some version of the intended argument in [10] could be carried out.
Remark 5.4. There have been extensive discussions of [9] and [18] at and since the SCGP work-
shop which took place during the week of March 17-21, 2014. Examples 2 and 3 directly contradict
the main premise behind the approaches in [9] and [18], formulated as Axiom A4 in [18, Deﬁni-
tion 1.5] and [18, Theorem 11.1]; these are relevant to [9] after evaluating on appropriate insertions
(Example 1 contradicts [18], but is excluded in [9]). This leaves little of substance in either [9]
or [18], as far as an alternative construction of counts of J-holomorphic maps is concerned. While
the crucial issues raised in Remark 5.3 are not even mentioned and the independence issue is
addressed incorrectly in [9] because of the author’s apparent misunderstanding of [33, 34], this
100-page manuscript consists mostly of setting up the approach based on the analytic construction
of the Deligne-Mumford moduli space in [32]. In contrast, the setup in [18] is carried out in a
standard way, similarly to our Section 2. On the other hand, [18] contains two additional cen-
tral premises that are intended to take the approach of [2] to the level of virtual classes (instead
43

of just numbers as in [9]): [18, Lemma 7.4], claiming that certain (never deﬁned) moduli spaces
are manifolds, and [18, (11.4)], a symplectic sum formula for virtual classes (instead of numbers);
the reader is referred to [16, 17] for the proofs of these statements. However, the two statements
require gluing maps with rubber components; such gluing constructions are skipped and claimed
to be unnecessary precisely because of a pre-print like [18] is in preparation (ﬁrst cited in 2001).
These issues are discussed in more detail in [5, Section 2]; unfortunately, the video of the discussion
of [18] itself at the SCGP workshop has not been made publicly available due to T. Parker’s veto.
In principle, the approach of [2] could be adapted to constructing positive-genus GW-invariants
by subtracting lower-genus contributions with appropriate coeﬃcients if the real dimension of the
target X is 8 or less. These coeﬃcients are determined by the chern classes of the divisor V , the
top intersections of λ-classes on Mg, and the relative GW-theory of P1. While all of these are
computable in some sense, it does not appear that the resulting coeﬃcients would have reasonably
simple expressions.
References
[1] K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), no. 1,
45-88.
[2] K. Cieliebak and K. Mohnke, Symplectic hypersurfaces and transversality in Gromov-Witten
theory, J. Symplectic Geom. 5 (2007), no. 3, 281–356.
[3] S. Donaldson, Symplectic submanifolds and almost-complex geometry, J. Diﬀerential Geom. 44
(1996), no. 4, 666-705.
[4] K. Faber and R. Pandharipande, Relative maps and tautological classes, J. EMS 7 (2005),
no. 1, 13–49.
[5] M. Farajzadeh Tehrani and A. Zinger, On Symplectic Sum Formulas in
Gromov-Witten Theory, arXiv/1404.1898.
[6] C. Faber, Algorithms for computing intersection numbers on moduli spaces of curves, with an
application to the class of the locus of Jacobians, New Trends in Algebraic Geometry (Warwick,
1996), 93-109, London Math. Soc. Lecture Note, Ser. 264, Cambridge Univ. Press, 1999.
[7] C. Faber and Pandharipande, Relative maps and tautological classes, J. Eur. Math. Soc. 7
(2005), no. 1, 13-49.
[8] K. Fukaya and K. Ono, Arnold conjecture and Gromov-Witten invariant, Topology 38 (1999),
no. 5, 933-1048.
[9] A.
Gerstenberger,
Geometric
transversality
in
higher
genus
Gromov-Witten
theory,
arXiv/1309.1426.
[10] A. Gerstenberger, Addendum to “Geometric transversality in higher genus Gromov-Witten
theory”, pre-print 2014.
[11] T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999),
487–518.
44

[12] T. Graber and R. Vakil, Relative virtual localization and vanishing of tautological classes on
moduli spaces of curves, Duke Math. J. 130 (2005), no. 1, 1-37.
[13] K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, and E. Zaslow,
Mirror Symmetry, Clay Math. Inst., AMS, 2003.
[14] J. Hu, T.-J. Li, and Y. Ruan, Birational cobordism invariance of uniruled symplectic manifolds,
Invent. Math. 172 (2008), no. 2, 231-275.
[15] E. Ionel, GW-invariants relative normal crossings divisors, arXiv/1103.3977.
[16] E. Ionel and T. Parker, Relative Gromov-Witten invariants, Ann. of Math. 157 (2003), no. 1,
45–96.
[17] E. Ionel and T. Parker, The symplectic sum formula for Gromov-Witten invariants, Ann. of
Math. 159 (2004), no. 3, 935–1025.
[18] E. Ionel and T. Parker, A natural Gromov-Witten virtual fundamental class, arXiv/1302.3472.
[19] A. Krestiachine, Donaldson hypersurfaces and Gromov-Witten invariants, Ph.D. Thesis HU
Berlin, in preparation.
[20] A.-M. Li and Y. Ruan, Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-
folds, Invent. Math. 145 (2001), no. 1, 151–218.
[21] J. Li, Stable morphisms to singular schemes and relative stable morphisms, J. Diﬀerential
Geom. 57 (2001), no. 3, 509–578.
[22] J. Li, A degeneration formula for GW-invariants, J. Diﬀerential Geom. 60 (2002), no. 1,
199–293.
[23] J. Li, Lecture notes on relative GW-invariants, Intersection Theory and Moduli, ICTP
Lect. Notes, XIX, 41-96, ICTP, 2004.
[24] J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of general symplectic
manifolds, in Topics in Symplectic 4-Manifolds, 47–83, Internat. Press 1998.
[25] E. Looijenga, Smooth Deligne-Mumford compactiﬁcations by means of Prym level structures,
J. Algebraic Geom. 3 (1994), 283-293.
[26] D. Maulik and R. Pandharipande, A topological view of Gromov-Witten theory, Topology 45
(2006), no. 5, 887–918.
[27] D. McDuﬀand D. Salamon, J-holomorphic curves and quantum cohomology, University Lec-
ture Series 6, AMS 1994.
[28] D. McDuﬀand D. Salamon, Symplectic Topology, 2nd Ed., Oxford University Press, 1998.
[29] D. McDuﬀand D. Salamon, J-Holomorphic Curves and Symplectic Topology, 2nd Ed., AMS
Colloquium Publications 52, AMS 2012.
[30] D. Mumford, Towards an enumerative geometry of the moduli space of curves, Arithmetic and
Geometry, Vol. II, 271-328, Progr. Math. 36, Birkh¨auser, 1983.
45

[31] A. Okounkov and R. Pandharipande, Virasoro constraints for target curves, Invent. Math. 163
(2006), no. 1, 47-108.
[32] J.
Robbin
and
D.
Salamon,
A
construction
of
the
Deligne-Mumford
orbifold,
J. Eur. Math. Soc. 8 (2006), no. 4, 611-699.
[33] Y. Ruan and G. Tian, A mathematical theory of quantum cohomology, J. Diﬀerential Geom. 42
(1995), no. 2, 259-367.
[34] Y. Ruan and G. Tian, Higher genus symplectic invariants and sigma models coupled with
gravity, Invent. Math. 130 (1997), no. 3, 455–516.
[35] A. Zinger, Enumerative vs. symplectic invariants and obstruction bundles, J. Symplectic
Geom. 2 (2004), no. 4, 445-543.
[36] A. Zinger, Enumeration of genus-two curves with a ﬁxed complex structure in P2 and P3,
J. Diﬀerential Geom. 65 (2003), no. 3, 341-467.
[37] A. Zinger, Basic Riemannian geometry and Sobolev estimates used in symplectic topology,
arXiv/1012.3980.
46
