DERIVED LOG STACKS AFTER OLSSON

J.P.PRIDHAM

Abstract. In this note, we give a formulation of log structures for derived stacks using Olsson’s log stack. The derived cotangent complex is then Olsson’s logarithmic cotangent complex, which (unlike Gabber’s) is just given by log differential forms in the log smooth case. The derived moduli stack of log stable maps then produces the desired virtual tangent space and obstruction theory on the underlying underived stack.

Contents

1. Olsson’s log stacks 1 2. Derived log stacks 3 3. Logarithmic derived moduli stacks 5 3.1. Moduli of log curves 5 3.2. Moduli of log maps 7 References 9

1. Olsson’s log stacks

Definition 1.1. Recall from [Ols2, §2] that the algebraic stack 𝔏^{0} classifies fine log structures. In other words, for any scheme U, 𝔏^{0}(U) is the groupoid of fine log structures on U.

The algebraic stack 𝔏^{1} classifies morphisms of fine log structures, so 𝔏^{1}(U) consists of pairs (ℳ_{0}, ℳ_{1}) of log structures on U with a morphism ℳ_{0} → ℳ_{1}. The algebraic stacks 𝔏^{n} classify n-strings of morphisms of fine log structures, so 𝔏^{n}(U) consists of strings

ℳ_{0} → ℳ_{1} → ℳ_{2} → … → ℳ_{n}.

More generally, for a finite category Γ, the algebraic stack 𝔏^{Γ} classifies Γ-diagrams of log structures.

Beware that these stacks are neither quasi-compact nor quasi-separated.

Definition 1.2. For 0 < j < n, we define δ_{j}^{∗}: 𝔏^{n} → 𝔏^{n−1} to be given by composition at the jth place, sending

ℳ_{0} → ℳ_{1} → ℳ_{2} → … → ℳ_{n}

to

ℳ_{0} → … → ℳ_{j−1} → ℳ_{j+1} → … → ℳ_{n}.

The map δ_{0}^{∗} is defined by forgetting ℳ_{0} →, and the map δ_{n}^{∗} by forgetting → ℳ_{n}.

When no map is specified, we adopt the convention that 𝔏^{n+1} → 𝔏^{n} is understood to be the map δ_{n}^{∗}.

Definition 1.3. Given a flat log scheme S, recall from [Ols1] that the algebraic stack 𝔏𝔬𝔤_{S} classifies fine log schemes. In other words, for any affine scheme U, 𝔏𝔬𝔤_{S}(U) is the groupoid of fine log schemes over S with underlying scheme U.

This work was supported by the Engineering and Physical Sciences Research Council [grant numbers EP/F043570/1 and EP/I004130/1].

Writing S̊ for the scheme underlying a log scheme S, the log structure can be interpreted as a morphism S̊ → 𝔏^{0}, and then [Ols2, §2] observes that

𝔏𝔬𝔤_{S} = 𝔏^{1} ×_{𝔏^{0}} S̊,

with a map U → 𝔏𝔬𝔤_{S} corresponding to a morphism f^{∗}ℳ_{S} → ℳ_{U} of log structures on U, for f: U → S̊.

Lemma 1.4. A morphism X → S of fine log schemes is log smooth if and only if X̊ → 𝔏𝔬𝔤_{S} is smooth.

Proof. This part of [Ols1, Theorem 4.6]. □

As we will only be interested in locally free log structures, we make the following definition:

Definition 1.5. Define 𝔏_{f}^{n} ⊂ 𝔏^{n} to be the open substack parametrising diagrams of locally free log structures.

Definition 1.6. Given a monoid P, we follow [Ols1] in writing P^{Gp} for the group completion of P.

Proposition 1.7. The algebraic stacks 𝔏_{f}^{n} are smooth for all n.

Proof. By [Ols1, Corollary 5.25], the stack S_{P} := [S[P]/S[P^{Gp}]] has the property that if S has a local chart Q → ℳ_{S}|_{U} on U, then

∐_{Q→P} S_{P} ×_{S_{Q}} U → 𝔏𝔬𝔤_{S}

is étale and surjective whenever U → S̊ is so.

Writing Z_{P} := [Spec ℤ[P]/Spec ℤ[P^{Gp}]], it follows that Z_{Q} → 𝔏^{0} ≃ 𝔏𝔬𝔤_{(Spec ℤ,𝒪^{∗})} is étale, and (since 𝔏^{1} ≃ 𝔏𝔬𝔤_{𝔏^{0}}), it also follows that

∐_{Q→P} Z_{P} → 𝔏^{1}

is étale and surjective.

In general, we have 𝔏^{n} ≃ 𝔏𝔬𝔤_{𝔏^{n−1}}, giving us an étale surjection

∐_{P_{0}→...→P_{n}} Z[P_{n}] → 𝔏^{n}.

If P is locally free, then Spec ℤ[P] is smooth and an atlas for Z[P] (the latter following because Spec ℤ[P^{Gp}] is also locally free). Thus

∐_{P_{0}→...→P_{n}} Spec ℤ[P_{n}] → 𝔏_{f}^{n}

is a smooth atlas, where the coproduct is now taken over locally free monoids P_{0}, …, P_{n}. □

Remark 1.8. In fact, an easy generalisation of [Ols1, Corollary 5.25] shows that if a finite category Γ has final object γ, then there is an étale surjection

∐_{\underline{P}} Z_{\underline{P} (γ)} → 𝔏^{Γ},

where \underline{P} runs over all functors from Γ to fine saturated monoids.

Definition 1.9. Given a log flat scheme S, define 𝔏𝔬𝔤_{S,f}^{n} := 𝔏_{f}^{n+1} ×_{𝔏_{f}^{0}} S.

Lemma 1.10. For j < n, the map δ_{j}^{∗}: 𝔏^{n} → 𝔏^{n−1} is relatively Deligne–Mumford and étale.

Proof. This is given in [Ols2, Prop 2.11]. □

In fact, Remark 1.8 ensures that 𝔏^{Δ} → 𝔏^{Γ} is relatively Deligne–Mumford and étale and étale whenever Γ and Δ both have final objects, and the functor Γ → Δ preserves the final object.

Beware that Lemma 1.10 does not apply to δ_{n}^{∗}, which is the morphism we use most frequently.

2. Derived log stacks

Denote the category of simplicial sets by 𝕊.

There is a natural embedding of the homotopy category of stacks into the homotopy category of derived stacks, and we essentially now just define a derived log stack 𝔛 to be a derived stack 𝔛̊ equipped with a morphism to 𝔏^{0}. A morphism 𝔛 → 𝔜 of derived log stacks is then a commutative diagram

\begin{matrix}X̊ & ⟶ & 𝔏^{1} \\ ↓ & & ↓ \\ Y̊ & ⟶ & 𝔏^{0}, \end{matrix}

and so on.

However, for a singular stack, the associated derived stack tends to be fairly unnatural, so we restrict to the smooth stacks 𝔏_{f}^{n} and consider only locally free log structures on derived stacks. We now recall from [Pri] an explicit description of the derived stack associated to a smooth stack.

Definition 2.1. For a commutative ring R, define sN_{R}^{♭} to be the category of simplicial R-algebras A for which

(1) the map A → π_{0}A has nilpotent kernel;

(2) the Dold–Kan normalisation NA is bounded (i.e. N_{i}A = 0 for all i ≫ 0).

Definition 2.2. Given a functor F : Alg_{R} → 𝕊 on R-algebras, we define a functor \underline{F} : sN_{R}^{♭} → s𝕊 to the category of bisimplicial sets by

\underline{F} (A)_{n} := F (A_{n}).

By [Pri, Theorem 2.17], a derived geometric stack over R is determined by its restriction to sN_{R}^{♭}.

Proposition 2.3. Given a smooth algebraic stack 𝔛 over R, the associated derived stack is given on sN_{R}^{♭} by the diagonal diag B\underline{𝔛} of the nerve of \underline{fX}. Explicitly,

diag B\underline{𝔛}(A)_{n} = (B_{n}𝔛)(A_{n}).

Proof. This is a straightforward consequence of [Pri, Theorem 3.16]. □

Note that diag B\underline{𝔛}(A) is just a model for the homotopy colimit ho\underrightarrow{lim}_{n} B𝔛(A_{n}), so we could use other model (such as W̄ B\underline{𝔛}(A) for the codiagonal W̄ of [CR]) for the homotopy colimit.

In the setting of [TV], we can form the simplicial presheaf B𝔛 on sN_{R}^{♭} by B𝔛(A) := B𝔛(A_{0}), and then Proposition 2.3 says that diag B\underline{𝔛} is the ∞-stackification (B𝔛)^{♯} of B𝔛.

Definition 2.4. For a commutative ℚ-algebra R, define dg_{+}N_{R}^{♭} to be the category of differential graded-commutative R-algebras in non-negative chain degrees for which

(1) the map A_{0} → H_{0}A has nilpotent kernel;

(2) A is bounded (i.e. A_{i} = 0 for all i ≫ 0).

Definition 2.5. Set Ω_{n} = Ω(Δ^{n}) to be the chain algebra

ℚ[t_{0}, t_{1}, …, t_{n}, dt_{0}, dt_{1}, …, dt_{n}]/(∑ t_{i} − 1, ∑ dt_{i})

of rational differential forms on the n-simplex Δ^{n}, with t_{i} in degree 0. These fit together to form a simplicial diagram Ω_{•} of chain algebras, and for a chain algebra A, we define A^{Δ^{n}} as the good truncation A^{Δ^{n}} := τ_{≥0}(A ⊗ Ω_{n}).

Definition 2.6. Given a functor F : Alg_{R} → 𝕊, we define a functor \underline{F} : dg_{+}N^{♭} → s𝕊 to the category of bisimplicial sets by

\underline{F} (A)_{n} := F ((A^{Δ^{n}})_{0}).

By [Pri, Theorem 2.17], a derived geometric stack over a ℚ-algebra R is determined by its restriction to dg_{+}N_{R}^{♭}.

Proposition 2.7. Given a smooth algebraic stack 𝔛 over a ℚ-algebra R, the associated derived stack is given on dg_{+}N_{R}^{♭} by diag B\underline{𝔛} (or equivalently W̄ B\underline{𝔛}).

Proof. Again, this is a straightforward consequence of [Pri, Theorem 3.16]. □

Definition 2.8. Given a derived ∞-stack F: dg_{+}N_{R}^{♭} → 𝕊 or F: sN_{R}^{♭} → 𝕊, define the underlying underived ∞-stack π^{0}F: Alg_{R} → 𝕊 by π^{0}F (A) := F (A).

Definition 2.9. Define a locally free log structure on a derived n-geometric stack 𝔛 to be a morphism 𝔛 → W̄ B\underline{𝔏_{f}^{0}} of derived log stacks. A morphism of free log structure on a

derived n-geometric stack 𝔛 is then defined to be a morphism 𝔛 → W̄ B\underline{𝔏_{f}^{1}}.

Remark 2.10. A homotopy class of locally free log structures on a derived n-geometric stack 𝔛 will be a homotopy class of morphisms 𝔛 → W̄ B\underline{𝔏_{f}^{0}}, but we should really regard the simplicial set

\underline{Hom}(𝔛, W̄ B\underline{𝔏_{f}^{0}}) ≃ R\underline{Hom}(𝔛, B𝔏_{f}^{0})

as the ∞-groupoid of locally free log structures on 𝔛, so π_{0} gives the homotopy classes, and π_{1} the homotopy classes of automorphisms, and so on.

Of course, we are interested in all morphisms of log structures, not just automorphisms, so we should look at the bisimplicial set

n ↦ \underline{Hom}(𝔛, W̄ B\underline{𝔏_{f}^{n}}),

which is a Segal space (in the sense of [Rez]) and hence a model for an (∞, 1)-category. This gives us an object we can regard as the ∞-category of locally free log structures on 𝔛.

In order to understand the difference between this ∞-category and the ∞-category of log structures considered in [SSV] (which was based on Gabber’s log cotangent complex), we can look at connected components of the ∞-groupoid of log structures on a derived affine scheme. For A ∈ sN_{ℤ}^{♭}, the set π_{0}\underline{𝔏_{f}^{0}}(A) is given by log structures on A_{0} modulo isomorphisms and homotopies given by log structures on A_{1}. In the setting of [SSV], a homotopy class of log structures on RSpec A is given by a simplicial log structure on A, modulo isomorphisms and simplicial log structures on A^{Δ^{1}}.

Definition 2.11. Write R𝔏_{f}^{n} for the derived stackification W̄ B\underline{𝔏_{f}^{n}} of the smooth stack 𝔏_{f}^{n}.

Given a derived log stack V, define R𝔏𝔬𝔤_{V,f} to be the homotopy fibre product ℝ𝔏𝔬𝔤_{V,f} := V̊ ×_{R𝔏_{f}^{0}}^{h} R𝔏_{f}^{1}. Note that when V is a log-flat underived stack, this agrees with Definition 1.3, ℝ𝔏𝔬𝔤_{V,f} is just the derived stackification of 𝔏𝔬𝔤_{V,f}. Likewise, define R𝔏𝔬𝔤_{V,f}^{n} := V̊ ×_{R𝔏_{f}^{0}}^{h} R𝔏_{f}^{n+1}.

Thus working with derived stacks is one way to resolve the base change issues arising for Olsson’s log stack when the base is not log-flat.

3. Logarithmic derived moduli stacks

We now fix a log-flat log stack V with locally free log structure as the base, and consider moduli of log curves over V, and then moduli of log maps from log curves to a fixed target, as features in the study of stable maps. The universal case takes V̊ = 𝔏^{0}, with log structure given by the identity map 𝔏^{0} → 𝔏^{0}.

3.1. Moduli of log curves.

3.1.1. The underived moduli stack. We first consider moduli of log curves. For any affine log scheme U over V with locally free log structure, we wish to parametrise proper logsmooth log curves over U. Now, an affine log scheme U over V is the same as an affine scheme Ů equipped with a morphism

Ů → 𝔏𝔬𝔤_{V,f}.

Given such data, a log-smooth log curve over U with locally free log structure is smooth morphism

C̊ → 𝔏𝔬𝔤_{U,f}

of relative dimension 1.

If M′_{g} denotes the (non-algebraic) moduli stack of smooth, not necessarily proper, genus g curves, then the groupoid of such curves is equivalent to the groupoid of stack homomorphisms

𝔏𝔬𝔤_{U} → M′_{g}.

Since 𝔏𝔬𝔤_{U,f} = 𝔏𝔬𝔤_{V,f}^{1} ×_{𝔏𝔬𝔤_{V,f}} Ů, this is the same as a map from Ů to the Hom-stack

N̊′_{g,V} := Hom_{𝔏𝔬𝔤_{V,f}} (𝔏𝔬𝔤_{V,f}^{1}, M′_{g} × 𝔏𝔬𝔤_{V,f}).

Projection to 𝔏𝔬𝔤_{V,f} gives us a log stack N′_{g,V} with underlying stack N̊′_{g,V}.

We now define

N̊_{g,V} ⊂ N̊′_{g,V}

to consist of those C̊ → 𝔏𝔬𝔤_{U} for which C̊ → Ů is proper. Because properness is deformation invariant, this is an open substack. Note that this substack is not the same as

Hom_{𝔏𝔬𝔤_{V,f}} (𝔏𝔬𝔤_{V,f}^{1}, M_{g} × 𝔏𝔬𝔤_{V,f}),

because properness of C̊ → Ů does not imply properness of C̊ → 𝔏𝔬𝔤_{U}. Write N_{g,V} for the log stack over V given by the morphism N̊_{g,V} → 𝔏𝔬𝔤_{V,f}.

We simply write N_{g} := N_{g,𝔏^{0}} when working with the universal base 𝔏^{0}. Note that N̊_{g,V} = N̊_{g} ×_{𝔏^{0}} V.

Since stability is invariant under deformations, the log moduli stack of stable log curves over V forms an open substack of N_{g,V}. We make no claims about algebraicity of this stack, but note that formal smoothness of M′_{g} and 𝔏𝔬𝔤_{V,f} ensure that the morphism

N̊′_{g,V} → 𝔏𝔬𝔤_{V,f}

is formally smooth, hence N̊_{g,V} → 𝔏𝔬𝔤_{V,f} is also formally smooth.

Moreover, the stacks N̊′_{g} and N̊_{g} are homogeneous. This means that for any nilpotent extension A → B of rings, and any morphism C → B, the maps

N̊′_{g}(A ×_{B} C) → N̊′_{g}(A) ×_{N̊′_{g}(B)} N̊′_{g}(C), N̊_{g}(A ×_{B} C) → N̊_{g}(A) ×_{N̊_{g}(B)} N̊_{g}(C)

are equivalences of groupoids.

For any open substack of N̊_{g} (and in particular for the moduli stack of log stable curves), this means that satisfying Artin’s conditions for algebraicity is just a matter of verifying formal existence, local finite presentation and finiteness of tangent spaces.

3.1.2. The universal curve. A map Ů → N̊′_{g} necessarily gives rise to a log curve C over U, so there is a universal such curve over N̊′_{g}. If we write C′_{g} → M′_{g} for the universal curve over M′_{g}, then C̊ is formed as the pullback

C̊ ⟶ C′_{g}
↓  ↓
𝔏𝔬𝔤_{U} ⟶ M′_{g},

which we may rewrite as

C̊  ⟶ C′_{g}
↓  ↓
𝔏𝔬𝔤_{V}^{1} ×_{𝔏𝔬𝔤_{V}} Ů ⟶ M′_{g}.

The universal case is given by taking Ů = N̊′_{g,V}, giving a universal curve D̊′_{g,V} as the pullback

D̊′_{g,V}  ⟶ C′_{g}
↓  ↓
𝔏𝔬𝔤_{V}^{1} ×_{𝔏𝔬𝔤_{V}} N̊′_{g,V} ⟶ M′_{g},

with log stack D′_{g,V} over N′_{g,V} given by the map from D̊′_{g,V} to

𝔏𝔬𝔤N̊′_{g,V} = 𝔏𝔬𝔤_{V}^{1} ×_{𝔏𝔬𝔤_{V}} N̊′_{g,V} = 𝔏𝔬𝔤_{V}^{1} ×_{𝔏𝔬𝔤_{V}} Hom_{𝔏𝔬𝔤_{V,f}} (𝔏𝔬𝔤_{V,f}^{1}, M′_{g} × 𝔏𝔬𝔤_{V,f}).

We then construct the universal curve D_{g,V} over N_{g,V} as the pullback

D_{g,V} := D′_{g,V} ×_{N′_{g,V}} N_{g,V} .

3.1.3. Derived moduli and the tangent space. We may also construct a derived moduli stack by the same method. Since M′_{g} is a formally smooth stack, it gives rise to a derived stack RM′_{g} := W̄ B\underline{M′}_{g} defined as in Proposition 2.3 or 2.7.

We then set

RN̊′_{g,V} := R\underline{Hom}_{R𝔏𝔬𝔤_{V,f}} (R𝔏𝔬𝔤_{V,f}^{1}, RM′_{g} × R𝔏𝔬𝔤_{V,f}),

which is a derived Hom-stack with underlying stack π^{0}RN̊′_{g,V} = BN̊′_{g,V}. Then take RN̊_{g,V} ⊂ RN̊′_{g,V} to be the open derived substack on the underlying stack BN̊_{g,V}. In other words

RN̊_{g,V} (A) := RN̊′_{g,V} (A) ×_{BN̊′_{g,V} (H_{0}A)}^{h} BN̊_{g,V} (H_{0}A).

For open substacks 𝔛 of N̊_{g,V}, such as derived moduli of log maps, this also allows us to define a derived stack R𝔛 by the same formula.

For a derived affine scheme U over R𝔏𝔬𝔤_{V,f}, we can reinterpret RN̊′_{g,V} (U) as the ∞- groupoid of smooth genus g curves C̊ → R𝔏𝔬𝔤_{U,f}. This follows essentially because RM′_{g} is the derived moduli stack of smooth genus g curves, since the cotangent complex of a smooth morphism has no higher terms and a curve has no cohomology above degree 1 (see [Lur, §8] for details).

Proposition 3.1. The derived stacks RN̊′_{g,V} and W̄ B\underline{N̊}′_{g,V} are equivalent, as are RN̊_{g,V} and W̄ \underline{N̊}_{g,V}.

Proof. Since N̊′_{g,V} and N̊_{g,V} are formally smooth and homogeneous, the conclusion of

Propositions 2.3 and 2.7 still applies, meaning that W̄ B\underline{N̊}′_{g,V} and W̄ \underline{N̊}_{g,V} are the derived ∞-stackifications of BN̊′_{g,V} and BN̊_{g,V} respectively.

It therefore suffices to show that the map

BN̊′_{g,V} → RN̊′_{g,V}

is another choice of derived ∞-stackification. Since the map is an isomorphism on π^{0}, it suffices by [Pri, Proposition 1.17 and Lemma 3.15] to show that this induces an isomorphism on derived tangent complexes.

At a point [C] ∈ N̊′_{g,V} (U) corresponding to a log curve C → U, formal smoothness, adjunction and properties of M′_{g} show that the tangent complex of N̊′_{g,V} (A) over 𝔏𝔬𝔤_{V} is given by

T_{[C]}(N̊′_{g,V} (A)/𝔏𝔬𝔤_{V} , M) = R\underline{Hom}_{𝒪C̊} (Ω(C̊/𝔏𝔬𝔤_{U}), 𝒪C̊ ⊗_{A} M)[1],

where U = Spec A and M is an A-module.

Calculation of the derived tangent complex of RN̊′_{g,V} over R𝔏𝔬𝔤_{V} gives the same answer, essentially because M′_{g} is formally smooth. □

Note that the tangent complex can be rewritten as

T_{[C]}(RN̊′_{g,V} (A)/R𝔏𝔬𝔤_{V} , M) ≃ R\underline{Hom}_{𝒪C̊} (Ω_{C/U}^{log}, 𝒪C̊ ⊗_{A} M)[1] = RΓ(C̊, 𝒯_{C/U}^{log} ⊗_{A} M)[1].

Beware that this would not be true for a derived moduli functor based on the approach of [SSV], since in that setting the tangent complex would be

R\underline{Hom}_{𝒪C̊} (𝕃_{C/U}^{G}, 𝒪C̊ ⊗_{A} M)[1],

for Gabber’s cotangent complex 𝕃^{G} as in [Ols2, §8], and in general 𝕃_{C/U}^{G} → Ω_{C/U}^{log} is not a quasi-isomorphism.

3.2. Moduli of log maps.

3.2.1. The underived moduli stack. As in [AMW, Definition 3.5.1], we now fix a log-flat log stack X over a base log stack V, and consider commutative diagrams

\begin{matrix}C & ⟶ & X \\ ↓ & & ↓ \\ U & ⟶ & V. \end{matrix}

of log stacks as a functor in U, where C → U ranges over proper log-smooth log curves of genus g. We will take all the log structures to be locally free. In particular, note that this holds for both the types of morphisms X → V considered in [AMW, §2.1]. For smooth pairs, the log structure on X is free of rank 1, and that on V is trivial. For acceptable degenerations, the log structure on X is free of rank 2, with that on V free of rank 1.

The datum C → U can be regarded as a map U → N_{g} of log stacks, for N_{g} as above, and then we seek a map

C → X

of log stacks over V. The log structures on C and X over V can be interpreted as morphisms C̊, X̊ → 𝔏𝔬𝔤_{V,f}, and then we need a compatible map C̊ → 𝔏𝔬𝔤_{X,f} = 𝔏𝔬𝔤_{V,f}^{1} ×_{𝔏𝔬𝔤_{V,f}} X̊. Explicitly, this means that the diagram

\xymatrix{C̊ \ar[r] \ar[dr] & 𝔏𝔬𝔤_{V,f}^{1} ×_{𝔏𝔬𝔤_{V,f}} X̊ \ar[d]^{δ_{0}^{∗}∘pr_{1}} \\  & _{𝔏𝔬𝔤_{V,f}}}

must commute.

Now,

C = U ×_{N_{g}} D_{g},

so we seek a map

Ů ×N̊_{g} D̊_{g} → 𝔏𝔬𝔤_{X,f}

over 𝔏𝔬𝔤_{V,f}.

Thus the stack describing this problem is the Hom-stack

N̊_{g}(X) := HomN̊_{g}(D̊_{g}, 𝔏𝔬𝔤_{X,f} ×_{𝔏𝔬𝔤_{V,f}} N̊_{g}),

which has a log structure given by pulling back the log structure on N̊_{g}, in other words by the morphism

N̊_{g}(X) → N̊_{g} → 𝔏𝔬𝔤_{V} ,

and we denote the resulting log stack N_{g}(X)

Since stability is a deformation-invariant condition, the log stack of stable maps is an open substack of N_{g}(X).

3.2.2. Derived moduli and its tangent space. We also wish to understand the derived moduli stack associated to this moduli problem. This should parametrise the ∞-groupoid of commutative diagrams

\begin{matrix}C & ⟶ & X \\ ↓ & & ↓ \\ U & ⟶ & V. \end{matrix}

of derived log stacks, where C → U ranges over proper log-smooth log curves of genus g. Reasoning exactly as above, we see that this problem is governed by the derived moduli stack

RN̊_{g}(X) := R\underline{Hom}_{RN̊_{g}}(RD̊_{g}, R𝔏𝔬𝔤_{X,f} ×_{R𝔏𝔬𝔤_{V,f}}^{h} RN̊_{g}),

where RD̊_{g} := W̄ \underline{D̊_{g}} is the derived stackification of D̊_{g}.

When X and V are underived log stacks, then by construction we have

π^{0}RN̊_{g}(X) ≃ HomN̊_{g}(D̊_{g}, π^{0}(R𝔏𝔬𝔤_{X,f} ×_{R𝔏𝔬𝔤_{V,f}}^{h} RN̊_{g}))
≃ HomN̊_{g}(D̊_{g}, 𝔏𝔬𝔤_{X,f} ×_{𝔏𝔬𝔤_{V,f}} N̊_{g})
= N̊_{g}(X),

the first equivalence coming because RD̊_{g} → N̊_{g} is fibred in underived schemes (curves, in fact). Thus RN̊_{g}(X) is a derived enhancement of N̊_{g}(X).

Now, properties of derived Hom-stacks give the tangent complex of RN̊_{g}(X) over RN̊_{g} at a map m: C̊ → 𝔏𝔬𝔤_{X} as

T_{[(C,m)]}(RN̊_{g}(X)/RN̊_{g}, M) ≃ R\underline{Hom}_{𝒪_{R𝔏𝔬𝔤_{X,f}}} (𝕃^{R𝔏𝔬𝔤_{X,f} /R𝔏𝔬𝔤_{V,f}} , Rm_{∗}𝒪C̊ ⊗ M).

By Lemma 1.10, the map δ_{0}^{∗}: 𝔏𝔬𝔤_{V,f}^{1} → 𝔏𝔬𝔤_{V,f} is étale, so

𝕃^{R𝔏𝔬𝔤_{X,f} /R𝔏𝔬𝔤_{V,f}} ≃ 𝕃^{R𝔏𝔬𝔤_{X,f} /R𝔏𝔬𝔤_{V,f}^{1}} ,

which is the pullback along 𝔏𝔬𝔤_{V,f}^{1} → 𝔏𝔬𝔤_{V,f}^{0} of 𝕃^{X̊/R𝔏𝔬𝔤_{V,f}}. Thus by adjunction

T_{[(C,m)]}(RN̊_{g}(X)/RN̊_{g}, M) ≃ R\underline{Hom}_{𝒪X̊} (𝕃^{X̊/R𝔏𝔬𝔤_{V,f}} , Rm_{∗}𝒪C̊ ⊗ M),

where we also write m for the induced morphism C̊ → X̊.

Establishing representability of an open substack of this derived stack (such as that parametrising stable maps) is then just a matter of verifying the finiteness and formal effectivity conditions of [Pri, Theorem 2.17], since homotopy-homogeneity follows automatically from our construction, and a derived stack is necessarily a homotopy-preserving hypersheaf.

When X is an underived stack, 𝕃^{X̊/R𝔏𝔬𝔤_{V,f}} ≃ 𝕃^{X̊/𝔏𝔬𝔤_{V,f}} is just Olsson’s log cotangent complex 𝕃^{X/V}, and we have

T_{[(C,m)]}(RN̊_{g}(X)/RN̊_{g}, M) ≃ R\underline{Hom}_{𝒪X̊} (𝕃^{X/V} , Rm_{∗}𝒪C̊ ⊗ M).

When X is moreover log smooth over V, as happens in all the cases of interest in [AMW, §2.1], this reduces to

R\underline{Hom}_{𝒪X̊} (Ω_{X/V}^{log} , Rm_{∗}𝒪C̊ ⊗ M)

This recovers the relative obstruction theory of [AMW, §3.2.3] as the first cohomology group.

References

[AMW] Dan Abramovich, Steffen Marcus, and Jonathan Wise. Comparison theorems for Gromov–Witten invariants of smooth pairs and of degenerations. arXiv:1207.2085v1 [math.AG], 2012.

[CR] A. M. Cegarra and Josué Remedios. The relationship between the diagonal and the bar constructions on a bisimplicial set. Topology Appl., 153(1):21–51, 2005.

[Lur] J. Lurie. Derived Algebraic Geometry. PhD thesis, M.I.T., 2004. www.math.harvard.edu/∼lurie/papers/DAG.pdf or http://hdl.handle.net/1721.1/30144.

[Ols1] Martin C. Olsson. Logarithmic geometry and algebraic stacks. Ann. Sci. École Norm. Sup. (4), 36(5):747–791, 2003.

[Ols2] Martin C. Olsson. The logarithmic cotangent complex. Math. Ann., 333(4):859–931, 2005.

[Pri] J. P. Pridham. Representability of derived stacks. J. K-Theory, 10(2):413–453, 2012. arXiv:1011.2742v4 [math.AG].

[Rez] Charles Rezk. A model for the homotopy theory of homotopy theory. Trans. Amer. Math. Soc., 353(3):973–1007 (electronic), 2001.

[SSV] Stefan Sagave, Timo Schürg, and Gabriele Vezzosi. Logarithmic derived geometry I. arXiv:1307.4246v1 [math.AG], 2013.

[TV] Bertrand Toën and Gabriele Vezzosi. Homotopical algebraic geometry. II. Geometric stacks and applications. Mem. Amer. Math. Soc., 193(902):x+224, 2008. arXiv math.AG/0404373 v7.