Let X be a smooth projective variety and L a line bundle on X. Let Y = PX(L⊕ O), which has a natural projection
π : Y → X.
π has two sections Y0, Y∞. Denote by
i0: Y0→ Y and i∞: Y∞→ Y the inclusions.
Recall some terminologies used in [23]. Relative invariants coming from (Y, Y0) and (Y, Y∞) are called type I ; those from (Y, Y0, Y∞) are called type II.
A variant of type II relative invariants are the invariants of the non-rigid targets, called rubber targets, whose relative invariants are called rubber invariants. See [9, Section 2.4] and [23, Section 1.5] for precise definitions and references. The rubbers naturally occur in the expanded targets of the usual relative maps.
A cohomology class of the form i0∗(ω) or i∞∗(ω) is called distinguished.
Note that
[Y0]· π∗α = i0∗(α), [Y∞]· π∗α = i∞∗(α).
In this section, we will use ω to denote a non-distinguished class, i.e.
ω∈ π∗H∗(X)⊂ H∗(Y ).
3.2. Relative invariants with rigid targets 3.2.1. Log notations. We have
A1(Y ) = i0∗A1(X)⊕Z[P1],
where [P1] is the class for the fiber of π. For an effective curve class β of Y , it is determined by θ = π∗(β) and
βY∞ by β = i0∗(θ) +
β
Y∞
[P1].
We use (Y, Y0, Y∞) to denote the log scheme whose underlying scheme is Y equipped with the divisorial log structure determined by the divisor
Y0 Y∞. Locally it is the product of U (a Zariski open subset of Y ) with the trial log structure and the log scheme (P1,{0}, {∞}). Similarly we have a log scheme (Y, Y∞). They are log smooth and integral.
Let M0,n((Y, Y0, Y∞), β; μ, ν) be the log stack of stable log maps from genus zero, n-pointed log curves to (Y, Y0, Y∞). Here β is the curve class, μ a partition of d0 =
βY0 and ν a partition of d∞=
βY∞. This is equivalent to specifying the contact orders of the marked points with Y0and Y∞(see [2, Section 3.2] and [8]). As μ and ν encode the log structure we are considering on Y and ν determines d∞, we will use the notation M0,n(Y ; μ, ν) when θ = π∗(β) is clear from the context.
For relative invariants of (Y, Y∞) with class β = i0∗(θ) + d∞[P1] and a partition ν of d∞, we have the log stack
M0,n((Y, Y∞), β; ν) orM0,n(Y ; ν) for short.
3.2.2. A virtual dimension count. View X as a log scheme with the trivial log structure. The projections
(Y, Y0, Y∞)→ X and (Y, Y∞)→ X
are log maps. When θ is nonzero or n≥ 3, we have induced maps between log stacks:
pX :M0,n(Y ; μ, ν)→ M0,n(X, θ), qX :M0,n(Y ; ν)→ M0,n(X, θ).
(3.2.1)
The following lemma follows from virtual dimension count.
Lemma 3.2.1.
1. dim [Mg,n(Y ; μ, ν)]vir = dim [Mg,n(X, θ)]vir+ 1− g.
2. dim [Mg,n(Y ; ν)]vir = dim [Mg,n(X, θ)]vir+ 1− g +
βY0.
When g = 0, the lemma suggests we might prove strong virtual push-forward properties for pX and, when 1 +
βY0 ≥ 0, for qX.
3.2.3. Compatibility of obstruction theories. Let X and X be log smooth projective varieties. Consider a strict map
i : X → X;
assume the underlying map of i is either a closed immersion or induces an injective map on the Chow group A1 as in [21, Section 5].
The map i induces a map between log stacks
¯i :M0,n(X)→ M0,n(X),
where M0,n(X) (resp. M0,n(X)) is the log stack of stable log maps to X (resp. X) from genus zero, n-pointed log curves. (We do not specify curve class or contact orders for ease of notation.) By our assumption on i, there is a commutative diagram
M0,n(X) ¯i
ρ
M0,n(X)
ρ
M0,n
and the horizontal arrow is strict. This induces a commutative diagram between stacks
M0,n(X) ¯i M0,n(X) TorM0,n
.
Define
E¯i∨ :=Rπ∗f∗(L∨i),
where π, f are maps from the universal curve C over M0,n(X) in the follow-ing diagram
C f
π
X
Mg,n(X).
Then it is straightforward to check we have compatible obstruction theories
(3.2.2) ¯i∗E E E¯i
¯i∗Lρ Lρ L¯i , where
E →Lρ and E →Lρ
are the perfect obstruction theories for ρ and ρ respectively.
The bottom row of (3.2.2) can be identified with the transitivity triangle of Olsson’s log cotangent complexes, while the top row is related to the transitivity triangle on X
¯i∗ΩX → ΩX →Li.
3.2.4. Strong virtual pushforward property. The proof of the fol-lowing proposition is modeled on [21, Proposition 5.22 and Corollary 5.27], where the absolute invariants are treated.
Proposition 3.2.2. Let pX, qX be morphisms defined in (3.2.1). We have 1.
(pX)∗[M0,n(Y ; μ, ν)]vir= 0 in A∗(M0,n(X, θ)) and
(pX)∗([M0,n(Y ; μ, ν)]vir∩ ev1∗[Y0]) = N (μ, ν)[M0,n(X, θ)]vir, where N (μ, ν) is a rational number determined by μ and ν.
2. Assume
βY0 ≥ 0, then
(qX)∗[M0,n(Y ; ν)]vir= 0.
Proof. We will prove the strong virtual pushforward property [22, Defini-tion 4.1] for pX and qX, which consists of mainly checking certain compati-bility of perfect obstruction theories. Then the above equations follow from the virtual dimension counts in Lemma3.2.1.
For (1), we embed X into a homogeneous variety. Choose two line bun-dles M and L on X such that both M and L⊗ M are very ample. These line bundles induce an embedding
i : X →P|M|×P|L⊗M|
such that L is the pullback of O(−1, 1) onP|M|×P|L⊗M|. Then we have a cartesian diagram
Y j P(O(−1, 1) ⊕ O)
X i P|M|×P|L⊗M|, which induces a cartesian diagram between log stacks
M0,n(Y ; μ, ν) ¯j
pX
M0,n(P(O(−1, 1) ⊕ O); μ, ν)
p
M0,n(X, θ) ¯i M0,n(P|M|×P|L⊗M|, (
θM,
θL⊗ M)).
Here we use p for pP|M|×P|L⊗M| and ¯i, ¯j to denote the horizontal maps. As ¯i is strict, the underlying diagram between stacks is also cartesian.
Recall we have defined obstruction theory E¯i(resp. E¯j) for ¯i (resp. ¯j) in (3.2.2). They fit in the following diagram
p∗XE¯i ≈ E¯j
p∗XL¯i L¯j . (cf. [21, Propsition 5.4 (ii)].)
As P|M|×P|L⊗M| is homogeneous, E¯i is perfect in [−1, 0]. This implies there exists a virtual pullback ¯i! such that
¯i![M0,n(P|M|×P|L⊗M|, (
θM,
θL⊗ M))]vir= [M0,n(X, θ)]vir and
¯i![M0,n(P(O(−1, 1) ⊕ O); μ, ν)]vir = [M0,n(Y ; μ, ν)]vir.
Note that by Lemma 3.2.1, p satisfies the strong virtual pushforward prop-erty sinceP|M|×P|L⊗M| is homogeneous. Then we can transfer this property to pX using ¯i!.
To determine the number N , consider a point l :P1 →P|M|×P|L⊗M|
in is injective, we have a cartesian diagram
M0,n(P(O(d0− d∞)⊕ O); μ, ν)
The number N is determined by (pP1)∗
[M0,n(P; μ, ν)]vir∩ ev1∗[P0]
= N (μ, ν)[M0,n(P1, 1)]vir. This completes the proof of (1). The proof of (2) is entirely similar and is omitted.
Remark 3.2.3. Using localization, one can show similar vanishing results.
Consider the fiberwise C∗ action on Y and the trivial action on X. Under these actions, π : Y → X is C∗ equivariant, and the induced map pX and qX are C∗ equivariant. Assume β satisfies
βY0≥ 0,
βY∞≥ 0.
For type II invariants, assume 2g− 2 + n > 0, then the pushforward of the equivariant virtual class under pX lies in t−1A∗(Mg,n(X, γ))[[t−1]]
by dimension count. Since pushforward of equivariant class should be an equivariant class, i.e. in A∗(Mg,n(X, γ))[[t]], it must vanish.
For type I, when g = 0, n ≥ 3, the pushforward of the equivariant virtual class under qX lies in t−1A∗(M0,n(X, γ))[[t−1]] by dimension count and therefore vanishes.
3.2.5. Special two-pointed fiber integrals. When θ is zero, β is a fiber class for π and d0 = d∞> 0 for type II invariants. In particular,
n≥ l(μ) + l(ν) ≥ 1 + 1 = 2.
If n = 2, we see l(μ) = l(ν) = 1 and there are no non-relative marked points.
Let d := d0 = d∞.
Lemma 3.2.4. When the partitions μ = (d), ν = (d) are totally ramified and β a fiber class, M0,2(Y ; (d), (d)) is isomorphic to the root stack d
L/X ([4, Appendix B.1]). In particular, it is smooth with virtual class equal to its fundamental class. Consequently, for α1, α2 ∈ H∗(X),
[M0,2(Y ;(d),(d))]vir
ev∗1(α1)∪ ev∗2(α2) = 1 d
X
α1∪ α2.
Proof. We show that first that the source curve has no contracted com-ponent. Assume there are v contracted components and h non contracted components, then there are v + h− 1 nodes. Consider the number of special points (nodes or marked points) on each component. On a contracted com-ponent, there are at least 3 of them. There are at least 2 special points on a non contracted component, which are points mapped into Y0 and Y∞. As each node is counted twice, we have
2(v + h− 1) + 2 ≥ 3v + 2h.
Thus v = 0. This implies in fact the source curve must be smooth as this is a fiber integral with 2 totally ramified relative points.
It is then easy to see a stable log map C → Y is determined by its underlying map. The moduli space being unobstructed follows from
H1(C,OC) = 0.
Since β is a fiber class for π, the last integral can be evaluated by first integrating over the fiber of
e :M0,2(Y ; (d), (d)) = r
L/X → X which has degree 1/d. The last statement then follows.