2.1. Blow-Ups

In this section we define real oriented blow-ups in sections of line bundles and prove some results relating to these. These will be used to define the Kato–Nakayama analytification of a log scheme.

First, let $X$ be a topological space, let $\eta\colon E \to X$ be a rank n vector bundle and let $s\colon X\to E$ be a section. Furthermore, let $E_0$ denote the image of the zero section and let $E' := E\setminus E_0$. Note that the action by $\mathbb R_{ \gt 0}$ (and in fact the action by $\mathbb R^*$) on $E$ restricts to a free action on $\textstyle E$'. We define the real oriented blow-up of $X$ in s, denoted $\mathrm{Bl}^{\mathbb{R}}_{s}\ X$ as the space

\begin{equation*}\{p\in E'\ \vert\ \exists\ \alpha\in \mathbb R_{\geq 0} \colon\ p =\alpha \cdot (s\circ \eta)(p)\}/\mathbb R_{ \gt 0}\end{equation*}

and we define the blow-down map $\rho\colon~\mathrm{Bl}^{\mathbb{R}}_{s}\ X \to X$ as the map induced by passing $p\colon E\to X$ to the quotient. Note that ρ has an inverse away from the zero locus of s and for any x in the zero locus of s we have that $\rho^{-1}(x) \cong S^{n-1}$.

The “usual” definition of the real oriented blow-up of a smooth manifold $X$ in a smooth closed sub manifold $Y$, denoted $\mathrm{Bl}^{\mathbb{R}}_{Y}\ X$ is that it is the complement of a tubular neighbourhood of $Y$ in $X$ and the blow-down map is then taken to be a retraction of this neighbourhood onto $Y$. It is also possible to give a functorial definition as in [Reference Arone and Kankaanrinta1]. The following result, whose proof I will omit, relates this construction to the usual notion of a blow-up in a smooth closed submanifold.

Proposition 2.3. Let $M$ be a (smooth) manifold, let $E$ be a vector bundle, let $s\colon M\to E$ be a smooth section whose image intersects $E_0$ transversally, and let $Y = s^{-1}(E_0)$ be the zero locus of s. Then there is a unique isomorphism of spaces over $M$,

\begin{equation*}\mathrm{Bl}^{\mathbb{R}}_{s}\ \mathrm{M} \cong~\mathrm{Bl}^{\mathbb{R}}_{Y}\ \mathrm{M}.\end{equation*}

Despite these properties one should be aware that $\mathrm{Bl}^{\mathbb{R}}_{s}\ X$ does not always resemble the “usual” definition of a blow-up. For example if we take the blow-up in the zero section of $E$ the result is (up to canonical isomorphism) the sphere bundle of $E$ which, in the case where $X$ is a manifold, has higher dimension than $X$. Even stranger situations are also possible and the blow-up of a manifold is not even necessarily a manifold with corners. For example let $s\colon S^2\to \mathbb R^2$, considered as a section of a trivial bundle on $S^2$, be defined by $s(x,y,z) = (x-1,0)$. The real oriented blow-up $\mathrm{Bl}^{\mathbb{R}}_{s}\ S^2$ is the wedge sum $S^1\vee S^2$.

Corollary 2.6. Let $L_1,L_2,\dots, L_n$ be complex line bundles on a manifold $X$ and let $L=\bigotimes_{i=1}^nL_i^{\otimes e_i}$ where $e_i$ are integers and $\otimes$ is the complex tensor product. Let $\sigma_1,\dots, \sigma_n$ be sections $\sigma_i\colon X \to L_i$. Then, there is an isomorphism

\begin{equation*}\mathrm{Bl}^{\mathbb{R}}_{\tilde \sigma_0}~\mathrm{Bl}^{\mathbb{R}}_{\tilde \sigma_n}\ \dots~\mathrm{Bl}^{\mathbb{R}}_{\sigma_1}\ X \xrightarrow{\cong} \left (\mathrm{Bl}^{\mathbb{R}}_{\tilde \sigma_n}\ \dots~\mathrm{Bl}^{\mathbb{R}}_{\sigma_1}\ X \right )\times S^1\end{equation*}

where $\tilde \sigma_i$ denotes the pullback of σ_i through all previous morphisms and $\sigma_0\colon X\to L$ is the zero section.

2.2. Kato–Nakayama analytification

For a DF log scheme $\mathsf X =(X, (s_i\colon \mathcal O_X \to \mathcal L_i) _{1\leq i \leq n})$ such that $X$ is of finite type over $\mathbb C$ one can construct an associated topological space $\mathsf X^\text{KN}$ called its Kato–Nakayama analytification. This space is defined as the sequence of real oriented blow-ups

\begin{equation*}\mathrm{Bl}^{\mathbb{R}}_{\tilde s_n}~\mathrm{Bl}^{\mathbb{R}}_{\tilde s_{n-1}}\ \dots~\mathrm{Bl}^{\mathbb{R}}_{s_1}\ X^{\text{an}}\end{equation*}

where $\tilde s_i$ is the section s_i of the vector bundle $\mathcal L_i$ pulled back via all previous blow-ups. This is order independent up to canonical isomorphism since, at each step, we are blowing up in the pullback (i.e. total, not strict, transform) of the corresponding line bundle with section. Regardless of order we let $\rho_{\mathsf X}\colon \mathsf X^\text{KN} \to X^\text{an}$ denote the corresponding blow-down map. This construction can be made functorial in the following way.

First, if $f\colon (X,(s_i\colon \mathcal O_X \to \mathcal L_i) _{1\leq i \leq n}) \to (Y,(t_i\colon \mathcal O_X \to \mathcal M_i) _{1\leq i \leq n})$ is a strict morphism of log-schemes, i.e a morphism such that the log structure morphism is given by isomorphisms $f^*\mathcal M_i \to \mathcal L_i^{\otimes 1}$ for each i, then we can form the following commutative and cartesian (with respect to the downward vertical arrows) diagram

where we have $E_X^1 = \mathcal L_1^\text{an}$ and $E_Y^1 = \mathcal M_1^\text{an}$ and where $s,t$ are the corresponding sections. The blow-ups $\mathrm{Bl}^{\mathbb{R}}_{s_1}\ X(\mathbb C)$ and $\mathrm{Bl}^{\mathbb{R}}_{t_1}\ Y(\mathbb C)$ are subspaces of the quotients by an equivalence relation ∼ on $E_X^1$ and $E_Y^1$ respectively where two vectors in a fibre are identified if they are equal up to positive scalar multiplication. By commutativity of the diagram, the subspace corresponding to the blow-up of $X(\mathbb C)$ is mapped to the blow-up of $Y(\mathbb C)$ as subspaces of $E_X^1/\sim$ and $E_X^1/\sim$ by the induced quotient space map. Thus, we get an induced subspace map $\mathrm{Bl}^{\mathbb{R}}_{s_1}\ X(\mathbb C) \to~\mathrm{Bl}^{\mathbb{R}}_{t_1}\ Y(\mathbb C)$. Iterating this gives the desired map of the entire sequence of blow-ups.

Next, let $\phi \colon \mathfrak M =(t_j\colon \mathcal O_X \to \mathcal M_j) _{1\leq j \leq m} \to \mathfrak L = (s_i\colon \mathcal O_X \to \mathcal L_i) _{1\leq i \leq n}$ be a morphism of DF log structures on X given by integers $\{e_{ij}\}$ and isomorphisms $\phi_i \colon \mathcal M_i \to \bigotimes_{j}\mathcal L^{\otimes e_{ij}}$. For simplicity assume that all line bundles are trivial. The general case can be constructed from this by gluing. Furthermore, let $\pi\colon (X, \mathfrak L)^\text{KN} \to X(\mathbb C)$ and $\rho \colon (X,\mathfrak M)^\text{KN} \to X(\mathbb C)$ denote the corresponding blow-down maps. Then, by composing with the trivializing isomorphisms, we obtain a unique invertible algebraic function $\lambda_i\colon X \to \mathbb C$ such that we can identify the analytification of $\phi_i$ with a map $\tilde{\phi_i}\colon X(\mathbb C)\times\mathbb C \to X(\mathbb C)\times\bigotimes_{1\leq j\leq n} \mathbb C^{\otimes e_{ij}}$ which sends

\begin{equation*}(x,z) \mapsto (x, \lambda_i(x) z\bigotimes_{1\leq j\leq n} 1^{\otimes e_{ij}}).\end{equation*}

This gives a map of spaces over $X(\mathbb C)$, $X(\mathbb C)\times (\mathbb C^*)^n \to X(\mathbb C)\times (\mathbb C^*)^m$ given by

\begin{equation*}(x,(z_1,\dots, z_n)) \mapsto (x,(\lambda_1(x)\prod_{j} z_j^{e_{1j}}, \dots, \lambda_n(x)\prod_{j} z_j^{e_{nj}})).\end{equation*}

By definition of λ_i this map sends

\begin{equation*}(x,s_1(x),\dots, s_n(x))\mapsto (x,t_1(x),\dots, t_m(x))\end{equation*}

and hence induces a morphism of blow-ups as quotients of subspaces of these bundles.

Since any morphism of log schemes is the composition of a strict morphism and a morphism given by the identity on underlying spaces these two cases are enough to define the analytification of an arbitrary map of log schemes.

One can verify that this definition of the analytification functor for DF log scheme agrees with the analytification of its associated log scheme originally introduced by Kato and Nakayama in [Reference Kato and Nakayama13]. This can be checked through direct computation but for a less tedious argument see [Reference Bergström, Diaconu, Petersen and Westerland4]. From section 3 onward we will drop the “DF” prefix and let “log scheme” mean “DF log scheme”. All log schemes referenced in this article will indeed be DF log schemes but because of the above the distinction is not important for our purposes anyway.

We will not need to use this explicit definition of the analytification of a morphism at any point in this article. Instead, all we will need is that the analytification of a map exists, is functorial, and satisfies the following properties:

• • For a log scheme $\mathsf X$ with log structure $\mathfrak L = (s_i\colon \mathcal O_{\underline{\mathsf X}} \to \mathcal L_i) _{1\leq i \leq n}$ the analytification of the map $\mathsf X \to \underline{\mathsf X}$ given by the identity on underlying schemes is the blow-down map

  \begin{equation*}\mathsf X^\text{KN} =~\mathrm{Bl}^{\mathbb{R}}_{\tilde s_n}~\mathrm{Bl}^{\mathbb{R}}_{\tilde s_{n-1}}\ \dots~\mathrm{Bl}^{\mathbb{R}}_{s_1}\ \underline{\mathsf X}(\mathbb C) \to \underline{\mathsf X}(\mathbb C).\end{equation*}

• • For a strict morphism of log schemes $f\colon \mathsf X \to \mathsf Y$ the following is a Cartesian diagram:

  

Both of these properties are standard and easy to verify so we omit a proof and will use them freely without reference.

2.3. Virtual logarithmic geometry

In this section we will give a short introduction to virtual morphisms of log schemes which were introduced by Howell [Reference Howell12] and further studied by Dupont, Panzer, and Pym [Reference Dupont, Panzer and Pym8]. Such virtual morphisms will be irrelevant for most sections of this article but in section 5.4 we include a discussion on how one can generalize some of our results using virtual morphisms in ways that are not possible otherwise. In the language of normal, i.e. non DF, logarithmic geometry a log structure on a scheme $X$ is a sheaf of monoids $\mathcal M$ with a morphism of sheaves of monoids $\alpha\colon \mathcal M \to \mathcal O_X$ which restricts to an isomorphism $\alpha^{-1}(\mathcal O_X^\times) \to \mathcal O_X^\times$. A map of log structures is then a map of sheaves of monoids over $\mathcal O_X$, $\mathcal M_1 \to \mathcal M_2$. Note that such a map commutes with the maps from $\mathcal O^\times$ by definition.

A virtual morphism of log structures is a morphism of the groupifications $\mathcal M_1^{\text{gp}} \to \mathcal M_2^{\text{gp}}$ which makes the diagram

commute. Note that the groupifications $\mathcal M_1^{\text{gp}}, \mathcal M_2^{\text{gp}}$ do not have maps to $\mathcal O_X$ which is why we use this weaker condition. Also note that the category of virtual log structures has more morphisms but we define it to have the same objects as the category of log structures. For a short motivation regarding why it makes sense to introduce such virtual morphisms it is worth mentioning that the log Betti and log de Rham cohomology functors are well defined for log schemes with virtual morphisms and the same is also true for the Kato–Nakayama analytification functor.

Although Dupont, Panzer, and Pym do define virtual morphisms of DF log structures their definition of a DF log structure is different from ours. Thus, we give a somewhat different definition here.

Definition 2.7. A virtual morphism of DF log structures on a scheme $X$

\begin{equation*}(s_i\colon \mathcal O_X \to \mathcal L_i) _{1\leq i \leq n} \to (t_j\colon \mathcal O_X \to \mathcal M_j) _{1\leq j \leq m}\end{equation*}

is a collection $\{e_{ij}\}$ of integers together with isomorphisms of line bundles

\begin{equation*}\mathcal L_i \xrightarrow{\cong} \bigotimes_{1\leq j \leq m}\mathcal M_j^{\otimes e_{ij}}\end{equation*}

for each $1\leq i \leq n$ satisfying the following if $s_i \neq 0$

• • $e_{ij}\geq 0$ for every j.

• • the sections s_i are mapped to the corresponding sections $\bigotimes_{1\leq j \leq m}t_j^{\otimes e_{ij}}$

The category of DF log schemes with virtual morphisms, ${}^{\text{\textbf{V}}}~\text{\textbf{DF-Log}}$ fits into a commutative diagram of categories:

We will omit the construction of the functor ${}^{\text{\textbf{V}}}~\text{\textbf{DF-Log}}_X \to ^{\text{\textbf{V}}}~\text{\textbf{Log}}_X$ but the interested reader is encouraged to construct it themselves and/or verify that our definition of the Kato–Nakayama analytification functor is still well defined for virtual morphisms. Doing this hopefully sheds some light on why this definition is a reasonable one.

Example 2.9. Let $X$ be a smooth scheme and let $D$ be a smooth effective Cartier divisor. By definition

\begin{equation*}(X, s_D\colon \mathcal O_X \to \mathcal O_X(D))^\text{KN} =~\mathrm{Bl}^{\mathbb{R}}_{D(\mathbb C)}\ X(\mathbb C).\end{equation*}

By our definition of the real oriented blow-up there is an embedding of $\mathrm{Bl}^{\mathbb{R}}_{D(\mathbb C)}\ X(\mathbb C)$, into the unit circle bundle of (the analytification of) $\mathcal O_X(D)$ with some arbitrarily chosen metric. This unit circle bundle is, again by definition, the KN analytification of $(X, 0\colon \mathcal O_X \to \mathcal O_X(D))$. The inclusion

\begin{equation*}(X, s_D\colon \mathcal O_X \to \mathcal O_X(D))^\text{KN} \hookrightarrow (X, 0\colon \mathcal O_X \to \mathcal O_X(D))^\text{KN}\end{equation*}

is not the analytification of any map of log schemes since the zero section cannot be pulled back to a non zero section. It is however the analytification of a virtual map given by the identity on both underlying spaces and $\mathcal O_X(D) \to \mathcal O_X(D)^{\otimes 1}$.

3.1. The Fulton–MacPherson compactification of a manifold

Let $X$ be a smooth manifold, let n be a positive integer and let $P_2({n})$, denote the set of subsets of $[n]$ with at least 2 elements. For a set $I\in P_2({n})$ let $\Delta_I$ denote the corresponding diagonal in $X^n$, i.e.

\begin{equation*}\Delta_I = \{(x_1,\dots, x_n)\in X^n \vert\ x_i = x_j \ \forall\ i,j \in I\}.\end{equation*}

By abuse of notation we will also use $\Delta_I$ to denote the (small) diagonal in $X^I$.

Definition 3.1. The Fulton–MacPherson compactification, $\text{FM}^\text{top}_n(X)$ is the closure of the image of the map

\begin{equation*}\text{Conf} _n(X) \hookrightarrow \prod_{I\in P_2({n})}~\mathrm{Bl}^{\mathbb{R}}_{\Delta_I}\ X^I.\end{equation*}

We let $f_I\colon~\text{FM}^\text{top}_n(X) \to~\mathrm{Bl}^{\mathbb{R}}_{\Delta_I}\ (X)$ denote the restrictionof the corresponding projection to $\text{FM}^\text{top}_n(X)$ and we let $\rho\colon~\text{FM}^\text{top}_n(X) \to X^n$ denote the surjective extension of the embedding $\text{Conf} _n(X) \hookrightarrow X^n$ to $\text{FM}^\text{top}_n(X)$.

The fibre over the origin of $\mathrm{Bl}^{\mathbb{R}}_{\Delta_I}\ (\mathbb R^D)^I$ is $S^{D(\vert I \vert-1)-1}$. Thus, $f_I\colon~\text{FM}^\text{top}_n(\mathbb R^D) \to~\mathrm{Bl}^{\mathbb{R}}_{\Delta_I}\ (\mathbb R^D)^I$ restricts to a map $\pi_I\colon~\text{K}_{D,n} \to S^{D(\vert I \vert-1)-1}$. These maps give a closed embedding

\begin{equation*}\text{K}_{D,n} \hookrightarrow \prod_{I\in P_2({n})}S^{D(\vert I \vert-1)-1},\end{equation*}

i.e. an element $x\in~\text{K}_{D,n}$ is uniquely determined by its components $x_I = \pi_I(x)$.

Note that while we have chosen to use $[n] = \{1,\dots,n\}$ as coordinate indices in the definitions above we could have used any (finite) index set N instead. We can thus define $\text{FM}^\text{top}_N(X)$ and $\text{K}_{d,N}$ with maps $\rho\colon~\text{FM}^\text{top}_N(X)\to X^N$, $f_{I}\colon~\text{FM}^\text{top}_N(X) \to~\mathrm{Bl}^{\mathbb{R}}_{\Delta_I}\ (X)^I$, and $\pi_{I}\colon~\text{K}_{D,N} \to S^{D(\vert I \vert-1)-1}$ in the analogous way. Here $I$ is a subset of $N$.

3.2. The Fulton–MacPherson operad

For a fixed dimension D the collection $ \{\text{K}_{D,n}\} _{n\in \mathbb N}$ can be given the structure of a topological operad as follows.

The permutation action of $\Sigma_n$ on $\text{Conf} _n(\mathbb R^D)$ extends to a free action on $\text{FM}^\text{top}_n(\mathbb R^D)$ which gives an induced action on the fibre over the origin, $\text{K}_{D,n}$. This is the symmetry action of the operad.

Next, any surjection $\varphi \colon I\twoheadrightarrow J$ of sets induces a monomorphism $(\mathbb R^D)^J \hookrightarrow (\mathbb R^D)^I$ such that the inverse image of $\Delta_I$ is $\Delta_J$. This gives a map $\mathrm{Bl}^{\mathbb{R}}_{\Delta_J}\ (\mathbb R^D)^J\to~\mathrm{Bl}^{\mathbb{R}}_{\Delta_I}\ (\mathbb R^D)^I$ which restricts to a map of fibres over the origin

\begin{equation*}g_\varphi\colon S^{D(\vert J \vert-1)-1} \to S^{D(\vert I \vert-1)-1}.\end{equation*}

Let $q\colon M\twoheadrightarrow [n]$ be a surjection of finite sets. By abuse of notation we will also let q denote the restriction of this function $I\twoheadrightarrow q(I)$ for any $I \subseteq M$. The above construction allows us to define a map

\begin{equation*}\gamma\colon~\text{K}_{D,[n]} \times \prod_{i\in [n]}~\text{K}_{D,q^{-1}(i)} \to~\text{K}_{d,M}\end{equation*}

by

\begin{equation*}\gamma(x,y^1,\dots,y^n)_I = \begin{cases} y^i_I & I \subseteq q^{-1}(i)\\ g_q(x_{q(I)}) &~\text{else} \end{cases}.\end{equation*}

This defines the composition maps of the operad.

Finally, $\text{K}_{d,1}$ is the one point space so there is only one choice for the identity map, $\eta\colon * \to~\text{K}_{d,1}$. The operad axioms can be verified manually for this collection of maps.

By a result of Salvatore, the Fulton–MacPherson operad is weakly equivalent to the operad of little d-dimensional disks [Reference Salvatore23, proposition 3.9].

The action of $\text{SO}(D)$ on $\mathbb R^D$ induces an action on $\text{Conf} _n(\mathbb R^D)$ which extends to $\text{FM}^\text{top}_n(\mathbb R^D)$. Since all rotations map the zero vector to itself this restricts to an action on the fibre $\text{K}_{D,n}$. One can check that this gives an action by $\text{SO}(D)$ on the $\text{\textbf{FM}}_D$ operad in the sense of [Reference Salvatore and Wahl24]. Furthermore the weak equivalence $\text{\textbf{FM}}_D \sim~\text{\textbf{LD}}_D$ can be shown to commute with this action which gives a weak equivalence between $\text{\textbf{FM}}_D\rtimes~\text{SO}(D)$ and $\text{\textbf{LD}}_D \rtimes~\text{SO}(D)$, the framed little disks operad.

With the notable exception of the case D = 2, the framed little disks operad is, however, not the operad studied in this article. Instead we are interested in the following construction. When the dimension is an even number $D = 2d$ there is an embedding of topological groups

\begin{equation*}S^1\cong U(1)\stackrel{diag.}{\hookrightarrow}U(d)\hookrightarrow \text{SO}(2d)\end{equation*}

This induces an action of S ¹ on the $\text{\textbf{FM}}_{2d}$ operad and we can form the semidirect product $\text{\textbf{FM}}_{2d} \rtimes S^1$.

The main goal of this article is to construct a (non-unital) operad of log schemes whose Kato–Nakayama analytification is $\text{\textbf{FM}}_{2d}\rtimes S^1$ which is weakly equivalent to the S ¹-framed little disks operad.

4.1. Fulton–MacPherson compactifications

Let us recall some important aspects of the Fulton–MacPherson compactification [Reference Fulton and MacPherson9]. Let $X$ be a smooth $\mathbb{k}$-variety. For an integer n and a subset $I \subseteq [n]$ with $\vert I \vert\geq 2$ let $\mathcal I_I$ denote the sheaf of ideals on $X^n$ corresponding to the $I$-diagonal, $\Delta_I \subseteq X^n$. Recall that we denote the set of subsets of $[n]$ with at least 2 elements by $P_2({n})$.

Let $\rho\colon~\text{FM}^{\text{alg}}_n(X) \to X^n$ denote the Fulton–MacPherson compactification of $\text{Conf} _n(X)$. We call this a “compactification” of $\text{Conf} _n(X)$ since ρ restricted to $\rho^{-1}(\text{Conf} _n(X))$ is an isomorphism and $\text{FM}^{\text{alg}}_n(X)$ is compact provided that $X$ is. The remainder of this section will be dedicated to recalling some important results relating to the Fulton–MacPherson compactification. First, we just state the following facts, all of which can be found with proofs in [Reference Fulton and MacPherson9].

• • The scheme $\text{FM}^{\text{alg}}_n(X)$ is a smooth variety.

• • There is an ordering of the elements in $P_2({n})$, such that $\rho \colon~\text{FM}^{\text{alg}}_n(X) \to X^n$ is given by a sequence of blow-ups where we start with $X^n$ and at step t of the sequence we blow-up in the dominant transform of the tth diagonal through all previous blow-ups.

• • The closed subset $D(I) \subseteq~\text{FM}^{\text{alg}}_n(X)$ given by the dominant transform of $\Delta(X)$ through all these blow-ups is a smooth, effective Cartier divisor.

• • The divisors $D(I)$ form a strict normal crossings divisor, i.e. any set of the divisors meets transversely.

• • The intersection $D(I_1,\dots, I_k):= D(I_1) \cap \dots \cap D(I_k)$ is non-empty if and only if the sets $I_1, \dots, I_k$ are nested i.e. for each $i,j$ we have $I_i \subseteq I_j$, $I_j \subseteq I_i$ or $I_i\cap I_j = \emptyset$.

• • We have $\text{FM}^{\text{alg}}_n(X) \setminus~\text{Conf} _n(X) = \bigcup_{I \in P_2({n})} D(I)$.

• • We have that $\rho^{-1}(\Delta(I)) = \bigcup_{I \subseteq I'} D(I')$, i.e. the pullback of the ideal inclusion $\mathcal I_I \hookrightarrow \mathcal O_{X^n}$ factors via a quotient

  \begin{equation*}\rho^* \mathcal I_I \twoheadrightarrow \prod_{I \subseteq I'} \mathcal I_{D(I')} \cong \bigotimes_{I \subseteq I'} \mathcal O_{\text{FM}^{\text{alg}}_n(X)}(-D(I')).\end{equation*}

• • The composition $\text{FM}^{\text{alg}}_n(X) \to X^n \to X^I$ factors as $\text{FM}^{\text{alg}}_n(X) \xrightarrow{f_I}~\mathrm{Bl}_{\Delta_I}\ X^I \xrightarrow{\rho_I} X^I,$ where ρ_I is the blow-down map.

Note that by the last two points we have that if $E_{\Delta_I} \subseteq~\mathrm{Bl}_{\Delta_I}\ X^I$ denotes the exceptional divisor then there is an isomorphism

\begin{equation*}f_I^* \mathcal O(E_{\Delta_I}) \cong \bigotimes_{I \subseteq I'} \mathcal O_{\text{FM}^{\text{alg}}_n(X)}(D(I'))\end{equation*}

which sends corresponding sections to each other.

In addition to this geometric description of $\text{FM}^{\text{alg}}_n(X)$ there is also a functorial description which we will make great use of in this article.

Definition 4.1. A screen on a scheme $H$ is a map $f\colon H \to X^n$ together with the following data

• • an invertible quotient $f^* \mathcal I_{I} \twoheadrightarrow \mathcal L_I$ for each $I\in P_2({n})$

• • a map $\mathcal L_I \to \mathcal L_J$ for each $I \subseteq J$

such that the following diagram commutes for every $I \subseteq J$

where $f^*\mathcal I_I \to f^* \mathcal I_J$ is the pullback of the corresponding inclusion.

We say a screen satisfies the $I$-vanishing property for some $I\in P_2({n})$ if $\text{pr}_i\circ f\colon H \to X =~\text{pr}_j\circ f\colon H \to X$ for each $i,j\in I$ and, for every $J\in P_2({n})$, if $J\not \subseteq I$ and $\vert I\cap J \vert \geq 2$ then $\mathcal L_{I\cap J}\to \mathcal L_J$ is the zero morphism.

We can define a screen on $\text{FM}^{\text{alg}}_n(X)$ by taking $f=\rho$, $\mathcal L_I = \bigotimes_{I \subseteq I'}\mathcal O_{\text{FM}^{\text{alg}}_n(X)}(-D(I'))$ and letting the invertible quotients be the maps

\begin{equation*}\rho^*\mathcal I_I \twoheadrightarrow \bigotimes_{I \subseteq I'}\mathcal O_{\text{FM}^{\text{alg}}_n(X)}(-D(I')).\end{equation*}

The maps $\mathcal L_{I} \to \mathcal L_J$ here are induced by the ideal inclusions $\mathcal O_{\text{FM}^{\text{alg}}_n(X)}(-D(I))\hookrightarrow \mathcal O_{\text{FM}^{\text{alg}}_n(X)}$. We call this screen the universal screen on $\text{FM}^{\text{alg}}_n(X)$.

4.2. Pointed rooted trees of projective spaces

Chen, Gibney, and Krashen [Reference Chen, Gibney and Krashen5] introduced stable pointed rooted trees of projective spaces as a higher-dimensional analogue of stable pointed rational curves, and showed that $T_{d,n}$ (defined in section 4.3) is the moduli space of stable rooted n-pointed trees of d-dimensional projective spaces. Here we recall the definition and describe the morphisms in the operad $\text{\textbf{CGK}}_d$ (defined in section 4.3), with $\text{\textbf{CGK}}_d(n) =T_{d,n}$ in terms of natural operations on such trees. The discussion is intended to convey geometric intuition for section 4.3, and we therefore suppress most proofs and technical details.

Definition 4.4. A rooted tree of d-dimensional projective spaces is a connected, reduced, projective scheme $T$ with a closed embedding $r_0\colon \mathbb P^{d-1} \hookrightarrow T$, called the root hyperplane, obtained by the following finite inductive process:

1. 1. Start with $T \cong \mathbb{P}^d$ and choose a hyperplane embedding

   \begin{equation*}r_0 \colon \mathbb{P}^{d-1} \hookrightarrow T.\end{equation*}

   This embedding is the root hyperplane.

2. 2. Given a tree of d-dimensional projective spaces T produced in a previous step, choose a smooth point $x \in T$ disjoint from the singular locus and the root hyperplane $r_0\colon \mathbb P^{d-1} \hookrightarrow T$. The next iteration $T$ʹ is defined by forming the blow-up $\mathrm{Bl}_{x}\ T$ and gluing along its exceptional divisor $\mathbb{P}^{d-1}$ a new copy of $\mathbb{P}^d$ along some hyperplane $\mathbb{P}^{d-1} \subseteq \mathbb{P}^d.$ Since x was chosen to be disjoint from the root hyperplane r ₀ induces an embedding $r_0'\colon \mathbb P^{d-1} \hookrightarrow T'$, which we take to be the new root hyperplane. Repeat finitely many times.

An n-pointed rooted tree is such a rooted tree of projective spaces $T$ together with n distinct points

\begin{equation*}p_1, \dots, p_n :~\text{Spec } k \hookrightarrow T\end{equation*}

lying outside the singular locus $\text{Sing}(T)$ and outside the root hyperplane.

Decompose $T=\bigcup_i T_i$ into irreducible components and let $T_0$ denote the root component, i.e. the component containing the root hyperplane. Each $T_i$ is isomorphic to $\mathbb{P}^d$ blown up at a closed subscheme Q_i consisting of finitely many disjoint $\mathbb{k}$-points, hence admits a canonical blow-down map

\begin{equation*}b_i: T_i \longrightarrow \mathbb{P}^d.\end{equation*}

Furthermore, there is an embedding $r_i\colon \mathbb P^{d-1}\hookrightarrow T_i$ for each T_i such that $b_i\circ r_i\colon \mathbb P^{d-1} \to \mathbb P^d$ is the inclusion of a hyperplane. For the root component r ₀ is the root hyperplane and for other components $r_i$ is the embedding of the hyperplane along which the component was attached. Note that the disjoint union of $r_i$ and the exceptional divisor $E_i$ of the blow-up $T_i \to \mathbb P^d$ is precisely the intersection $T_i\cap~\text{Sing}(T)$.

By theorem 3.4.4. in [Reference Chen, Gibney and Krashen5] the smooth varieties $T_{d,n}$ are moduli spaces of stable n-pointed rooted trees of d-dimensional projective spaces. There are no stable 1-pointed trees but in this case we define $T_{d,1}=~\text{Spec } \mathbb{k}$. As mentioned before, a hyperplane in $\mathbb P^1$ is just a point and thus we have $T_{1,n} \cong \overline{M}_{0,n+1}$, the moduli space of stable n + 1-pointed rational curves where the additional point is the root hyperplane.

Each variety $T_{d,n}$ comes with a canonical action by the symmetric group $\Sigma_n$ corresponding to permutation of the marked points. Furthermore, for each $1\leq i \leq n$ there is a morphism

\begin{equation*}T_{d,n}\times T_{d,m}\xrightarrow{\circ_i} T_{d,n+m-1}\end{equation*}

which takes an n-pointed tree $T$ and an m-pointed tree $T$ʹ and maps them to the $n+m-1$ pointed tree formed by blowing-up $T$ in the ith marked point and gluing the resulting exceptional divisor to the root hyperplane in $T$ʹ. It is clear from definitions that these maps give rise to an operad of varieties, which we denote $\text{\textbf{CGK}}_d$, with spaces $\text{\textbf{CGK}}_d(n) = T_{d,n}$. In the case d = 1 this construction is classical, see for example [Reference Getzler and Kapranov11].

4.3. Definition and properties of $T_{d,n}$

In this section we recall the definition and some important properties of the $T_{d,n}$ spaces from [Reference Chen, Gibney and Krashen5]. We also introduce a functor of points for these spaces and use this to define the operad $\text{\textbf{CGK}}_d$ with spaces $\text{\textbf{CGK}}_d(n) = T_{d,n}$ which was informally described in section 4.2. We will not explicitly prove that $\text{\textbf{CGK}}_d$ is isomorphic to the operad described in section 4.2 but it is not hard to show using theorem 3.4.4. in [Reference Chen, Gibney and Krashen5].

If $\text{pr}_i\colon X^n \to X$ denotes the projection to the ith component it is clear from definitions that the composition

\begin{equation*}D([n])\hookrightarrow~\text{FM}^{\text{alg}}_n(X) \xrightarrow{\rho} X^n \xrightarrow{\text{pr}_i} X\end{equation*}

is independent of the integer i. We denote this morphism by $\pi \colon D([n]) \to X$. In [Reference Chen, Gibney and Krashen5], Chen, Gibney, and Krashen show that for any $\mathbb{k}$-valued point $x\in X$ the fibre $\pi^{-1}(x)$, depends only on n and $\dim X$ up to (non-canonical) isomorphism provided that X is smooth. We define $T_{d,n}$ as $\pi^{-1}(0)$ for the map $\pi\colon D([n]) \to \mathbb A^d$ but by the above we could equally well have taken any other smooth variety instead of $X= \mathbb A^d$.

The variety $T_{d,n}$ is smooth and comes equipped with a smooth closed subscheme $T_{d,n}(I)$ for every non empty $I \subseteq [n]$. For $I \in P_2({n})$, i.e. $I\neq [n]$ and $\vert I \vert\geq 2$, we define $T_{d,n}(I)$ as the pullback of $D(I)\subseteq~\text{FM}^{\text{alg}}_n(X)$ to $T_{d,n}$. For other $I$ (i.e. $I$ is $[n]$ or a one point set) we take $T_{d,n}(I) := T_{d,n}$. Similarly, for a collection of (distinct) subsets $I_1,\dots, I_k \subseteq [n]$ we define $T_{d,n}(I_1,\dots, I_k) := \bigcap_{i}T_{d,n}(I_i)$. For $I\neq [n]$ and $\vert I \vert\geq 2$ the subscheme $T_{d,n}(I)$ is a smooth divisor of $T_{d,n}$ and the union of all these divisors is a normal crossings divisor. Additionally, the intersection $T_{d,n}(I_1,\dots, I_k)$ is non empty if and only if $(I_1,\dots, I_k)$ is nested. All this essentially follows from the analogous results for $\text{FM}^{\text{alg}}_n(X)$ recalled in section 4.1. Details can be found in [Reference Chen, Gibney and Krashen5].

The first goal of this section will be to give a functor of points description of the schemes $T_{d,n}(I_1,\dots, I_k)$. For this we introduce the following.

Definition 4.6. For a scheme $H$, an integer d, and a finite, non-empty, set $I$ with at least two element we let $\mathcal F^{H,d}_I$ denote the quasi-coherent sheaf of modules defined by

\begin{equation*}\mathcal F^{H,d}_I := \left\langle \{t_{ij}^k\}_{i,j \in I}^{1\leq k\leq d}\right \rangle \Bigm / \sim.\end{equation*}

Here $\left\langle \{t_{ij}^k\}_{i,j \in I}^{1\leq k\leq d}\right \rangle$ denotes the free module with generators $t_{ij}^k$ and ∼ is generated by sections of the form $t_{ij}^k +t_{jl}^k-t_{il}^k$. We will also denote this by $\mathcal F_I^d$ or $\mathcal F_I$ if H or $H,d$ are clear from context.

For every function of sets $\varphi\colon I\to J$ we let $F_{\varphi}\colon \mathcal F_{I}^{H,d} \hookrightarrow \mathcal F_J^{H,d}$ denote the sheaf morphism defined by sending $t_{ij}^k \mapsto t_{\varphi(i)\varphi(j)}^k.$

Definition 4.7. Let $\mathcal T_{d,n}\colon Sch^{op}\to Set$ be the functor which sends $H$ to the set of collections of invertible quotients,

\begin{equation*} \{\phi_I\colon \mathcal F^{H,d}_I \twoheadrightarrow \mathcal L_I\} _{I \in P_2({n})},\end{equation*}

such that for every $I \subseteq J$ there is a morphism $\mathcal L_I \to \mathcal L_J$ making the following diagram commute.

Such a collection will sometimes be referred to as a simple screen. Similarly let $\mathcal T_{d,n}(I_1,\dots, I_k)$ denote the subfunctor of $\mathcal T_{d,n}$ with the added condition that for every $I_r$, the simple screens satisfy the $I_r$-vanishing property, i.e. if $J\nsubseteq I_r$ and $\vert J\cap I_r\vert \geq 2$ then $\mathcal L_{J\cap I_r} \to \mathcal L_J$ is the zero morphism.

There is a natural simple screen on $T_{d,n}$ which can be constructed as follows. Let $i\colon T_{d,n}\hookrightarrow~\text{FM}^{\text{alg}}_n(\mathbb A^d)$ denote the inclusion of the fibre over the origin in $(\mathbb A^d)^n$. The ideal sheaves $\mathcal I_I$ pull back to $i^* \mathcal I_I \cong \mathcal F_I^d$ and for $I\subseteq J$ the inclusion $\mathcal I_I \hookrightarrow \mathcal I_J$ pulls back to the map $F_{I\subseteq J}\colon \mathcal F_I^d\to \mathcal F_J^d$. Thus, the universal screen on $\text{FM}^{\text{alg}}_n(\mathbb A^d)$ pulls back to a simple screen on $T_{d,n}$ via i. We will refer to this as the universal simple screen on $T_{d,n}$. Let $\mathcal F_I\twoheadrightarrow \mathcal M_I^{d,n}$ denote the invertible quotient corresponding to $I \in P_2({n})$ of the universal simple screen on $T_{d,n}$. We will also denote this by $\mathcal M_I$ if $d,n$ are clear from context.

Lemma 4.8. Let $i\colon T_{d,n} \to~\text{FM}^{\text{alg}}_n(\mathbb A^d)$ denote the inclusion of the fibre over the origin. There are unique isomorphisms

\begin{equation*}\bigotimes_{I \subseteq I'} i^* \mathcal O(-D(I')) \xrightarrow{\cong} \mathcal M_I\end{equation*}

such that for $I \subseteq R$ the following diagram commutes

where $t_{R'}\colon O(-D(R')) \hookrightarrow \mathcal O$ denotes the ideal inclusion.

Proof. First note that if $i_0\colon~\text{Spec } \mathbb{k} \to (\mathbb A^d)^n$ denotes the inclusion of the origin then we have seen that $i_0^*\mathcal I_I = \mathcal F_I$ for every $I \in P_2({n})$, where $\mathcal I_I$ is the ideal sheaf corresponding to the I-diagonal in $(\mathbb A^d)^n$. Since

\begin{equation*}T_{d,n} = \pi^{-1}(0) = D([n])\times_{\mathbb A^d}~\text{Spec } \mathbb{k} \cong D([n])\times_{(\mathbb A^d)^n}~\text{Spec } \mathbb{k}\end{equation*}

the result follows immediately from proposition 4.3. The same argument also gives the $T_{d,n}(I_1,\dots, I_k)$ case.

This functorial description simplifies the proof of many of the following results greatly. Let n be a positive integer, and let $q\colon [n]\twoheadrightarrow M$ be a surjection of sets. We also let q to denote the restriction of this map $J \to q(J)$ for any $J\subseteq M$.

Proposition 4.10. The composition map described in section 4.2 is well defined and gives an isomorphism

\begin{equation*}T_{d,n}\times \prod_{r=1}^n T_{d,q^{-1}(r)} \cong T_{d,M}(q^{-1}(1),\dots, q^{-1}(n)).\end{equation*}

Proof. We will construct an explicit natural isomorphism between the functors represented by the left and the right side. Identifying the induced isomorphism of schemes with the map described in section 4.2 is left as an unimportant exercise for the reader.

We define this natural transformation

\begin{equation*}\mathcal T_{d,n}\times \prod_{r=1}^n \mathcal T_{d,q^{-1}(r)}\to \mathcal T_{d,M}(q^{-1}(1),\dots,q^{-1}(n))\end{equation*}

by sending simple screens (on some $\mathbb{k}$-scheme $H$)

\begin{equation*} \{\phi_I\colon \mathcal F_{I}\!\to\! \mathcal L^0_I\} _{I\in P_2({n})} \times \prod_{r=1}^{n} \{\psi^r_{J}\colon \mathcal F_{J}\!\to\! \mathcal L^r_{J}\} _{J\in P_2({q^{-1}(r)})} \!\to\! \{\rho_K\colon \mathcal F_K\to \mathcal L_K\} _{K\in P_2({M})},\end{equation*}

where $ \{\rho_K\colon \mathcal F_K\to \mathcal L_K\} _{K\in P_2({M})}$ is defined as follows. We define the line bundles $\mathcal L_K$ as

\begin{equation*}\mathcal L_K = \begin{cases} \mathcal L^r_{K} & K \subseteq q^{-1}(r)~\text{some } r\\ \mathcal L^0_{q(K)}&~\text{else} \end{cases}\end{equation*}

and the corresponding quotients as

\begin{equation*}\rho_K = \begin{cases} \psi^r_{K} & K \subseteq q^{-1}(r),~\text{some } r\\ \phi_{q(K)} \circ F_{q} &~\text{else} \end{cases}.\end{equation*}

For the definition of $F_q\colon \mathcal F_{K}\to \mathcal F_{q(K)}$, see definition 4.6. We define the maps $\mathcal L_{K_1} \to \mathcal L_{K_2}$, for $K_1 \subseteq K_2$, to be

\begin{equation*}\begin{cases} \mathcal L^r_{K_1} \to \mathcal L^r_{K_2} & K_2 \subseteq q^{-1}(r)~\text{some } r\\ \mathcal L^0_{q(K_1)} \to \mathcal L^0_{q (K_2)} & K_1 \not\subseteq q^{-1}(r)~\text{any } r\\ 0 & K_1 \subseteq q^{-1}(r)~\text{and } K_2 \not \subseteq q^{-1}(r) \end{cases}.\end{equation*}

It is easy to check that $ \{\rho_K\} _{K\in P_2({M})}$ with the maps $\mathcal L_{K_1} \to \mathcal L_{K_2}$ above is a simple screen with the $q^{-1}(r)$-vanishing property for every $r$ and so this transformation is well defined on the level of sets. Furthermore, the map is clearly natural and so we have defined a natural transformation as desired.

To show that this gives a natural isomorphism first note that that this map is clearly injective. Surjectivity follows from the fact that if $ \{\rho_K\colon \mathcal F_K\to \mathcal L_K\} _{K\in P_2({M})}$ is a simple screen satisfying the $q^{-1}(r)$-vanishing property for every r then, for every K not contained in any $q^{-1}(r)$ the map $\rho_K\colon \mathcal F_K\to \mathcal L_K$ factorizes uniquely as

by the universal property of the quotient.

Corollary 4.11. For any $I \in P_2({n})$, the isomorphism of the proposition restricts to an isomorphism of closed subvarieties

\begin{equation*}T_{d,n}(I)\times \prod_{r=1}^nT_{d,q^{-1}(r)}\cong T_{d,M}(q^{-1}(1), \dots, q^{-1}(n), q^{-1}(I)).\end{equation*}

Similarly, for any $I_r\in P_2({q^{-1}(r)})$, the isomorphism of the proposition restricts to an isomorphism of closed subschemes

\begin{equation*}T_{d,n}\times T_{d,q^{-1}(1)}\times\dots\times T_{d,q^{-1}(r)}(I_r)\times \dots\times T_{d,q^{-1}(n)} \cong T_{d,M}(q^{-1}(1), \dots, q^{-1}(n), I_r).\end{equation*}

Corollary 4.12. Let $\gamma\colon T_{d,n} \times \prod_{i=1}^n T_{d,q^{-1}(i)} \to T_{d,M}$ be the composition of the isomorphism above with the inclusion into $T_{d,M}$ and let $\pi_0\colon T_{d,n} \times \prod_{i=1}^n T_{d,q^{-1}(i)} \to T_{d,n}$ and $\pi_r \colon T_{d,n} \times \prod_{i=1}^n T_{d,q^{-1}(i)} \to T_{d,q^{-1}(r)}$ denote the corresponding projections. There are unique isomorphisms relating the elements of the corresponding universal screens

\begin{equation*}\gamma^*\mathcal M_I^{d,M} \xrightarrow{\cong} \begin{cases} \pi_r^* \mathcal M^{d,q^{-1}(r)}_{I} & I \subseteq q^{-1}(r)~\text{some } r\\ \pi^*_0\mathcal M^{d,n}_{q(I)}&~\text{else} \end{cases}.\end{equation*}

making (one of) the following diagrams commute:

The next relevant property of the $T_{d,n}$ spaces is that the $\Sigma_n$ action informally described in section 4.2 is well defined. Indeed, this action is precisely the restriction to $T_{d,n}$ of the $\Sigma_n$ action on the Fulton–MacPherson compactification $\text{FM}^{\text{alg}}_n(\mathbb A^d)$. For each $\sigma \in \Sigma_n$, it is easy to verify that this action corresponds to the action on $\mathcal T_{d,n}$ given by

\begin{equation*} \{\phi_I\colon \mathcal F_I \!\twoheadrightarrow\! \mathcal L_I\} _{I \in P_2({n})} \!\in\! \mathcal T_{d,n}(H) \mapsto \{\phi_{\sigma^{-1}(I)} \circ\sigma^{-1}_I \!\colon\! \mathcal F_I \!\twoheadrightarrow\! \mathcal L_{\sigma^{-1}(I)}\} _{I \in P_2({n})} \!\in\! \mathcal T_{d,n}(H),\end{equation*}

where

\begin{equation*}\sigma_I \colon \mathcal F_I \to \mathcal F_{\sigma(I)}\end{equation*}

is defined by sending $t_{ij}^k \mapsto t^k_{\sigma(i)\sigma(j)}$.

With the maps we have constructed so far we can define the operad of schemes $\text{\textbf{CGK}}_d$ which was informally described in section 4.2. Its objects are $\text{\textbf{CGK}}_d(n) = T_{d,n}$, its composition maps are the morphisms of proposition 4.10 (composed with the inclusion $T_{d,M}(q^{-1}(1),\dots, q^{-1}(n)) \hookrightarrow T_{d,M}$), and its symmetry action is the one described above. There is only one candidate for the unit morphism of the operad since $T_{d,1}\cong~\text{Spec } \mathbb{k}$. It is easy to, at least intuitively, see why these maps satisfy the operad axioms using their n-pointed rooted tree descriptions from section 4.2. Verifying that the axioms hold using the functorial descriptions of all relevant maps here is a more tedious, but easy, exercise.

The case d = 1 here is of special interest. The operad $\text{\textbf{CGK}}_1$ is isomorphic to the “Deligne–Mumford”-operad whose objects are the moduli spaces of stable n + 1-pointed curves of genus 0, $\overline{\mathcal M}_{0,n+1}$. Indeed, as mentioned earlier, Chen, Gibney, and Krashen show that there are isomorphisms $T_{1,n} \cong \overline{\mathcal M}_{0,n+1}$ and one can verify that these isomorphisms can be chosen such that they commute with the respective operad morphisms.

5.1. The log schemes $\text{FM}^{\text{log}}_n(X)$, $\mathsf K_{d,n}$ and $\mathsf T_{d,n}$

First, let us define the log geometric Fulton–MacPherson compactification which has the nice property that its analytification is the topological Fulton–MacPherson compactification of $\text{Conf} _n(\mathbb C^d)$, $\text{FM}^\text{top}_n(\mathbb C^d)$.

Note that all of the line bundles with sections $\mathcal O_{\text{FM}^{\text{alg}}_n(X)}(D(I))$ pull back to the trivial line bundle with a nowhere-vanishing section on $\text{Conf} _n(X)$. This induces a morphism of log schemes $\text{Conf} _n(X) \to~\text{FM}^{\text{log}}_n(X)$. Furthermore, the following diagram commutes:

Similarly, we can define a log geometric analog of $T_{d,n}$, denoted $\mathsf K_{d,n}$, by taking the underlying scheme to be $T_{d,n}$ and the log structure given by the pullback of this log structure on $\text{FM}^{\text{alg}}_n(\mathbb A^d)$ via the inclusion $T_{d,n} \hookrightarrow~\text{FM}^{\text{alg}}_n(\mathbb A^d)$. Note that for each $I \in P_2({n})$ with $I \neq [n]$ the line bundle with section corresponding to the divisor $D(I)$ pulls back to the line bundle with section corresponding to $T_{d,n}(I)$ and for $I = [n]$ the pullback of the corresponding line bundle has the zero section. We will let $s_I\colon \mathcal O_{T_{d,n}}\to \mathcal O_{T_{d,n}}(I)$ denote the pullback of $s_I\colon \mathcal O_{\text{FM}^{\text{alg}}_n(\mathbb A^d)} \to \mathcal O_{\text{FM}^{\text{alg}}_n(\mathbb A^d)}(D(I))$ for each $I \in P_2({n})$.

The operad composition morphisms of the $\text{\textbf{CGK}}_d$ operad do not extend to maps of log-schemes for the objects $\mathsf K_{d,n}$. Instead, in order to extend $\text{\textbf{CGK}}_d$ to a (pseudo)-operad of log-schemes we must let its objects be the log-schemes $\mathsf T_{d,n}$, whose log structures are given by one line bundle with section $s_I\colon \mathcal O_{T_{d,n}}\to \mathcal O_{T_{d,n}}(I)$ for each non-empty $I \subseteq [n]$ defined as above for $\vert I \vert\geq 2$ and defined by

\begin{equation*}\mathcal O_{T_{d,n}}(\{i\}) := \bigotimes_{I \ni i} \mathcal O_{T_{d,n}}(I)^\vee,\end{equation*}

with $s_{\{i\}} = 0$ for one element sets. For n = 1 we define $\mathcal O_{T_{d,1}}(\{1\}) := \mathcal O_{T_{d,1}}$ with the zero section.

5.2. The log geometric CGK operad

As mentioned, the symmetry and composition morphisms of the $\text{\textbf{CGK}}_d$ operad can be extended to morphisms of log schemes $\mathsf T_{d,n}$. To see this for the symmetry morphisms note that, for $\sigma \in \Sigma_n$ we have that the corresponding automorphism on $\text{FM}^{\text{alg}}_n(X)$, sends each divisor $D(I)$ isomorphically to $D(\sigma(I))$. This induces isomorphisms of line bundles (with sections) $\sigma^*\mathcal O_{\text{FM}^{\text{alg}}_n(X)}(D(I)) \to \mathcal O_{\text{FM}^{\text{alg}}_n(X)}(D(\sigma^{-1}(I)))$. These maps extend the symmetry action on $\text{FM}^{\text{alg}}_n(X)$ to a symmetry action on $\text{FM}^{\text{log}}_n(X)$. This in turn gives symmetry actions on $\mathsf K_{d,n}$ and $\mathsf T_{d,n}$ in the same way.

Extending the composition morphisms is slightly trickier. Again, let n be a positive integer and let $q\colon M\twoheadrightarrow [n]$ be a surjection of sets. As before, let $\pi_0\colon T_{d,n}\times \prod_{i=1}^n T_{d,q^{-1}(i)} \to T_{d,n}$ and $\pi_r \colon T_{d,n}\times \prod_{i=1}^n T_{d,q^{-1}(i)} \to T_{d,q^{-1}(r)}$ denote the corresponding projections, and let

\begin{equation*}\gamma \colon T_{d,n}\times \prod_{i=1}^n T_{d,q^{-1}(i)} \to T_{d,M}\end{equation*}

be the composition morphism of $\text{\textbf{CGK}}_d$. We can immediately show the following.

This alone is almost enough to extend the operad composition morphisms to maps of corresponding log-schemes but we must also express $\gamma^* \mathcal O_{T_{d,M}}(I)$ as a tensor product of the line bundles on $T_{d,n}\times \prod_{i=1}^n T_{d,q^{-1}(i)}$ for $I=M$, $I= q^{-1}(r)$, and $\vert I \vert = 1$. Note that in all of these cases the pullback of the sections of these corresponding line bundles are all 0. To this end, recall that that lemma 4.8 gives isomorphisms of line bundles on $T_{d,n}$

\begin{equation*}\mathcal M_I \xrightarrow{\cong} \bigotimes_{I \subseteq I'} \mathcal O_{T_{d,n}}(I')^\vee,\end{equation*}

where $\mathcal M_I$ are the line bundles of the universal simple screen on $T_{d,n}$ (section 4.3), such that the following diagram commutes for every $I\subseteq J$,

where the bottom arrow is the map

\begin{equation*}\bigotimes_{I\subseteq I',\ J\not \subseteq I'} s_{I'}^\vee.\end{equation*}

In particular, this means that we have isomorphisms

\begin{equation*}\mathcal O_{T_{d,n}}(I) \cong \mathcal M_I^\vee \bigotimes_{I \subsetneq I'} \mathcal O_{T_{d,n}}(I)^\vee.\end{equation*}

Proof. First note that by the above we have $\mathcal O_{T_{d,M}}(M) \cong \mathcal (M_{M}^{d,M})^\vee$. By Corollary 4.12 we have $\gamma^*\mathcal M_{M}^{d,M} \cong \pi_0^*\mathcal M_{[n]}^{d,n}$. From this part (a) follows. For part (b) note that lemma 5.3 gives an isomorphism

\begin{equation*}\bigotimes_{q^{-1}(r)\subsetneq S} \gamma^*\mathcal O_{T_{d,M}}(I) \cong \gamma^*\mathcal O_{T_{d,M}}(M)\bigotimes_{r\in J \subsetneq [n]} \pi_0^*\mathcal O_{T_{d,n}}(J) \cong \bigotimes_{r\in J \subseteq [n]} \pi_0^*\mathcal O_{T_{d,n}}(J)\end{equation*}

where the last isomorphism follows from part (a). By definition, this is just $\pi_0^*\mathcal O_{T_{d,n}}(\{r\})^\vee$ and so we get an isomorphism

\begin{align*}\gamma^* \mathcal O_{T_{d,M}}(q^{-1}(r)) &\xrightarrow{\cong} \gamma^*(\mathcal M_{q^{-1}(r)}^{d,M})^\vee \bigotimes_{q^{-1}(r)\subsetneq I'} \mathcal \gamma^*O_{T_{d,M}}(I')^\vee\\ &\xrightarrow{\cong}\pi_r^*\mathcal O_{T_{d,q^{-1}(r)}}(q^{-1}(r))\otimes \pi_0^*\mathcal O_{T_{d,q^{-1}(r)}}(\{r\}).\end{align*}

This gives part (b). Part (c) follows from a similar argument.

With this we have defined isomorphisms of line bundles extending the operad composition maps of $\text{\textbf{CGK}}_d$ to morphisms of log-schemes

\begin{equation*}\gamma^{\log} \colon \mathsf T_{d,n}\times \prod_{i=1}^n \mathsf T_{d,q^{-1}(i)} \to \mathsf T_{d,M}.\end{equation*}

Unfortunately, there are no maps of log schemes

\begin{equation*}\text{Spec } \mathbb{k} \to (\text{Spec } \mathbb{k} ,\mathcal O_{\mathbb{k}} \xrightarrow{0} \mathcal O_{\mathbb{k}})\end{equation*}

and therefore we cannot extend our unit map to a map of log schemes. However, it is easy to verify that despite this the $\circ_i$-operations, $T_{d,n}\times T_{d,m} \xrightarrow{\circ_i} T_{d,m+n-1}$, do extend to maps of log schemes $\mathsf T_{d,n}\times \mathsf T_{d,m} \xrightarrow{\circ_i} \mathsf T_{d,m+n-1}$. One can verify that these maps, together with the symmetry morphisms satisfy the pseudo-operad axioms.

5.3. Kato–Nakayama analytifications

In this section we show that the Kato–Nakayama Analytification of $\text{\textbf{CGK}}^{\log}_d$ is homeomorphic to the $S^1$-framed Fulton–MacPherson operad in dimension 2d, $\text{\textbf{FM}}_{2d}\rtimes S^1$, which in turn is a model for the $S^1$-framed little 2d-disks operad $\text{\textbf{LD}}_{2d} \rtimes S^1$ as mentioned in section 3.

We begin by showing that there is a homeomorphism of spaces over $X^n(\mathbb C)$,

\begin{equation*}\text{FM}^{\text{log}}_n(X)^\text{KN} \cong~\text{FM}^\text{top}_n(X(\mathbb C)),\end{equation*}

which commutes with the corresponding inclusions of $\text{Conf} _n(X(\mathbb C))$. The rough idea behind our proof of this statement is that the left hand side is the Kato–Nakayama space of a sequence of logarithmic blow-ups of $X^n$ and the right hand side is a sequence of real blow-ups of $X(\mathbb C)^n$ and under sufficiently nice circumstances these two agree.

Proposition 5.7. Let $X$ be a smooth $\mathbb C$-variety and let $Y$ be a smooth subvariety of codimension k. If Y is the zero locus of a section of a rank k vector bundle on X then there is an isomorphism of spaces over $X(\mathbb C)$

\begin{equation*}(\mathrm{Bl}^{\log}_{Y}X)^{\text{KN}}\cong~\mathrm{Bl}^{\mathbb{R}}_{Y(\mathbb C)}\ X(\mathbb C).\end{equation*}

Corollary 5.8. Let $\mathsf X = (X,\mathfrak L)$ be a log-scheme whose log structure is given by line bundles with sections cutting out smooth divisors $D_1,\dots, D_n$ such that $D_1\cup \dots \cup D_n$ is a strict normal crossings divisor and let $Y$ be as in the proposition with the additional constraint that $Y$ intersects all intersections $D_{i_1}\cap \dots D_{i_k}$ transversally. Then there is an isomorphism of spaces over $\mathsf X^{\text{KN}}$

\begin{equation*}(\mathrm{Bl}^{\log}_{Y}X)^{\text{KN}}\cong~\mathrm{Bl}^{\mathbb{R}}_{\tilde Y}\ \mathsf X^{\text{KN}}\end{equation*}

where $\tilde Y$ denotes the strict transform of $Y(\mathbb C)$ under the sequence of blow-down maps

\begin{equation*}\mathsf X^{\text{KN}} \to X(\mathbb C).\end{equation*}

Proof. Let $t\colon \mathcal O_X \to \mathcal V$ denote the vector bundle section cutting out $Y$. It is clear from the proposition that there is an isomorphism

\begin{equation*}(\mathrm{Bl}^{\log}_{Y}X)^{\text{KN}}\cong~\mathrm{Bl}^{\mathbb{R}}_{\tilde t}\ \mathsf X^{\text{KN}}\end{equation*}

where $\tilde t$ denotes the pullback (i.e. total transform) to $\mathsf X^\text{KN}$ of the vector bundle section t seen as a section of a topological vector bundle $t \in \Gamma(X(\mathbb C),{\mathcal V})$. The transversallity conditions for intersections with $Y$ imply that $\tilde Y$ is the zero locus of the section $\tilde t$. This completes the proof.

Proposition 5.9. There is a unique homeomorphism of spaces over $X^n(\mathbb C)$,

\begin{equation*}\text{FM}^{\text{log}}_n(X)^\text{KN} \cong~\text{FM}^\text{top}_n(X(\mathbb C)),\end{equation*}

which commutes with the corresponding inclusions of $\text{Conf} _n(X(\mathbb C))$.

Proof. First, recall from [Reference Li18] that the Fulton–MacPherson space $\text{FM}^{\text{alg}}_n(X)$ is isomorphic to the sequence of blow-ups

\begin{equation*}\mathrm{Bl}_{\tilde \Delta_{I_N}}\ \dots~\mathrm{Bl}_{\tilde \Delta_{I_1}}\ X^n\end{equation*}

where $I_1, \dots, I_N$ are the sets in $P_2({[n]})$ in any order satisfying the conditions of theorem 1.3 in [Reference Li18] and $\tilde \Delta_{I_i}$ denotes the dominant transform of the diagonal $\Delta_{I_i} \subseteq X^n$ under all previous blow-ups. Furthermore, recall that the divisors $D(I_i)$ are the dominant transforms of the exceptional divisor $\tilde D(I_i)$ from the blow-up

\begin{equation*}\mathrm{Bl}_{\tilde \Delta_{I_i}}~\mathrm{Bl}_{\tilde \Delta_{I_{i-1}}}\ \dots~\mathrm{Bl}_{\tilde \Delta_{I_1}}\ X^n \to~\mathrm{Bl}_{\tilde \Delta_{I_{i-1}}}\ \dots~\mathrm{Bl}_{\tilde \Delta_{I_1}}\ X^n\end{equation*}

taken under all subsequent blow-ups $\mathrm{Bl}_{\tilde \Delta_{I_N}}\ \dots\mathrm{Bl}_{\tilde \Delta_{I_{i+1}}}\ $. Additionally, the topological Fulton–MacPherson space $\text{FM}^\text{top}_n(X(\mathbb C))$ can also be written as a sequence of real oriented blow-ups of $X^n(\mathbb C)$ in its diagonals in a similar manner.

The proposition would now follow from corollary 5.8 by induction if there was an ordering $I_1, \dots, I_N$ of the sets in $P_2({[n]})$ satisfying the aforementioned conditions such that for every $1\leq i \leq n$

\begin{equation*}\tilde \Delta_{I_{i+1}} \subseteq~\mathrm{Bl}_{\tilde \Delta_{I_i}}~\mathrm{Bl}_{\tilde \Delta_{I_{i-1}}}\ \dots~\mathrm{Bl}_{\tilde \Delta_{I_1}}\ X^n\end{equation*}

meets any intersection of the divisors $\tilde D(I_1) \dots \tilde D(I_i)$ transversally. It is not hard to show that any ordering where the sets $I\in P_2({[n]})$ appear in decreasing order of size, i.e. $\vert I_i \vert\geq \vert I_{i+1} \vert$, satisfies all criteria and thus we are done.

Proof. We have already constructed the isomorphism $\text{FM}^{\text{log}}_n(X)^\text{KN} \cong~\text{FM}^\text{top}_n(X(\mathbb C))$. Since the morphism $\mathsf K_{d,n} \hookrightarrow~\text{FM}^{\text{log}}_n(\mathbb A^d)$ is strict, the top square in the following diagram is Cartesian.

We already know that the bottom square is Cartesian and thus the entire diagram is Cartesian, i.e. $\mathsf K_{d,n}^\text{KN} \to (\text{FM}^{\text{log}}_n(\mathbb A^d))^{\text{KN}}$ is the inclusion of the fibre over the origin in $(\mathbb C^d)^n$ and thus $(\text{FM}^{\text{log}}_n(\mathbb A^d))^\text{KN} \cong~\text{FM}^\text{top}_n(\mathbb C^d)$ restricts to an isomorphism $\mathsf K_{d,n}^\text{KN} \cong~\text{K}_{2d,n}.$ Notice that we get 2d on the right hand side since $\mathbb C\cong \mathbb R^2$. Furthermore, we know that

\begin{equation*}\mathsf T_{d,n}^\text{KN} =~\mathrm{Bl}^{\mathbb{R}}_{0\in \mathcal O(\{1\})}\ \dots\mathrm{Bl}^{\mathbb{R}}_{0\in \mathcal O(\{n\})}\ \mathsf K_{d,n}^\text{KN}.\end{equation*}

By corollary 2.6 this is homeomorphic to $(S^1)^n\times~\text{K}_{2d,n},$ i.e. the nth space in the $\text{\textbf{FM}}_{2d}\rtimes S^1$ operad. It is easy to show that these choices of isomorphisms make all relevant maps commute as claimed.

Next, we will show that the maps $f_I\colon~\text{FM}^\text{top}_n(X(\mathbb C)) \to~\mathrm{Bl}^{\mathbb{R}}_{\Delta_I}\ X^I$, $\pi_I\colon~\text{K}_{2d,n} \to S^{2d(\vert I \vert-1)-1}$ (which were defined in section 3) and the projections to the $S^1$ components $\pi_i\colon (S^1)^n\times~\text{K}_{2d,n} \to S^1$ are all identified with analytifications of explicit maps of corresponding log schemes under the isomorphisms described in the proof of corollary 5.10.

Let $\mathsf S_{m}$ denote the log scheme $(\mathbb P^m, 0\colon \mathcal O \to \mathcal O(-1))$. Its Kato–Nakayama space $\mathsf S_m^{\text{KN}}$ homeomorphic to $S^{2m+1}$ and the map $S^{2m+1}\cong S_m^{\text{KN}} \to \mathbb P^m(\mathbb C)$ is the Hopf fibration (proving this using proposition 5.7 may provide some insight into the idea behind the arguments to come). Recall from section 4.1 that the composition

\begin{equation*}\text{FM}^{\text{alg}}_n(X) \to X^n \to X^I\end{equation*}

factors via a map $f_I \colon~\text{FM}^{\text{alg}}_n(X) \to~\mathrm{Bl}_{\Delta_I}\ X^I$ for any $I \in P_2({n})$, and furthermore that there is an isomorphism of line bundles with sections

\begin{equation*}f_I^*\mathcal O(E_{\Delta_I}) \cong \bigotimes_{I \subseteq I'}\mathcal O(D(I')).\end{equation*}

This isomorphism defines a map of log schemes

\begin{equation*}f_I^{\log}\colon~\text{FM}^{\text{log}}_n(X) \to~\mathrm{Bl}^{\log}_{\Delta_I}X^I.\end{equation*}

Also recall that the fibre over the origin for the blow-down map $\mathrm{Bl}_{\Delta_I}\ X^I \to X^I$ is $\mathbb P^{d(\vert I \vert-1)-1}$ and that, if i denotes the inclusion of this fibre, we have that $i^* \mathcal O(E_{\Delta_I}) \cong \mathcal O(-1)$. This means that $\mathsf S_{d(\vert I \vert-1)-1} = (\mathbb P^{d(\vert I \vert-1)-1}, 0\colon \mathcal O \to \mathcal O(-1))$ is the fibre over the origin of the log blow-down map $\mathrm{Bl}^{\log}_{\Delta_I} X^I \to X^I$ and $f_I^{\log}$ restricts to a map $\pi_I^{\log}\colon \mathsf K_{d,n} \to \mathsf S_{d(\vert I \vert-1)-1}$.

The maps we have described fit together in the following diagram:

The analytification of this diagram is:

Since $\text{Conf} _n(\mathbb R^{2d})$ is dense in $\text{FM}^\text{top}_n(\mathbb R^{2d})$ and since the blow-down map $\rho_I\colon~\mathrm{Bl}^{\mathbb{R}}_{\Delta_{I}}\ (\mathbb R^{2d})^{I} \to (\mathbb R^{2d})^I$ has an inverse on $\text{Conf} _I(\mathbb R^{2d})$ there is only one possible set of maps $(f_I^{\log})^\text{KN}$ and $(\pi_I^{\log})^\text{KN}$ which can make this diagram commute. By definition, this diagram also commutes if we insert the maps $f_I, \pi_I$ defined in section 3 and thus we must have $(f_I^{\log})^\text{KN} = f_I$ and $(\pi_I^{\log})^\text{KN} = \pi_I$.

Finally, let $\pi_i^{\log}$ be the map

\begin{equation*}\pi_i^{\log}\colon \mathsf T_{d,n} \to \mathsf S_0 = (\text{Spec } \mathbb C, 0\colon \mathcal O_{\mathbb{C}}\to \mathcal O_{\mathbb{C}})\end{equation*}

given by the only possible map on the level of schemes and the isomorphism of log structures

\begin{equation*}(\pi_i^{\log})^* \mathcal O_{\mathbb C} \xrightarrow{\cong} \bigotimes_{i\in S} \mathcal O_{T_{d,n}}(I).\end{equation*}

It is clear that the analytification of this map is the projection map $\pi_i\colon (S^1)^n\times~\text{K}_{2d,n} \to S^1$ as desired.

We are now ready to show that the analytifications of the composition and symmetry maps in $\text{\textbf{CGK}}^{\log}_d$ are the composition and symmetry maps of $\text{\textbf{FM}}_{2d}\rtimes S^1$. For the symmetry action this is trivial as we can again use that $\text{Conf} _n(\mathbb C^d) \subseteq~\text{FM}^\text{top}_n(\mathbb C^d)$ is dense to conclude this using an argument similar to the one above.

A somewhat more complicated argument is needed to prove that the analytifications of the $\text{\textbf{CGK}}^{\log}_n$ composition morphisms and $\circ_i$-operations are the corresponding maps of the $\text{\textbf{FM}}_{2d}\rtimes S^1$ operad. We merely give an outline of this argument. First note that for any surjection of finite sets $\varphi\colon I\twoheadrightarrow J$ the induced map $(\mathbb A^d)^J \to (\mathbb A^d)^I$ extends uniquely to a map $\mathrm{Bl}^{\log}_{\Delta_J}(\mathbb A^d)^J \to~\mathrm{Bl}^{\log}_{\Delta_I}(\mathbb A^d)^I$ which restricts to a map $g_\varphi^{\log}\colon \mathsf S_{d(\vert J \vert-1)-1}\to \mathsf S_{d(\vert I \vert-1)-1}$. By the same argument as above the analytification of this map must be the map $g_\varphi\colon S^{2d(\vert J \vert-1)-1} \to S^{2d(\vert I \vert-1)-1}$ defined in section 3. Using this one can verify that the analytification of $\gamma^{\log}$ is the composition morphism of $\text{\textbf{FM}}_{2d}\rtimes S^1$ componentwise. That is, if $\gamma$ denotes the composition map in $\text{\textbf{FM}}_{2d}\rtimes S^1$ we can verify that

\begin{equation*}\pi_I \circ \gamma = \pi_I\circ (\gamma^{\log})^\text{KN} = (\pi_I^{\log}\circ\gamma^{\log})^\text{KN}\end{equation*}

for every $I \subseteq M$ and from this $\gamma = (\gamma^{\log})^\text{KN}$ immediately follows. The $\circ_i$ case is analogous. We include the proof for one important step which hopefully also provides some insight into the methods used to fill in the remaining steps of the argument.

Let $f\colon \mathsf S_0\times \mathsf S_n\to \mathsf S_n$ be the map given by the canonical isomorphism $f\colon~\text{Spec } \mathbb C\times \mathbb P^n\to \mathbb P^n$ on underlying schemes and by the obvious isomorphism $f^*\mathcal O_{\mathbb P^n}(-1) \xrightarrow{\cong} \pi_1^*\mathcal O_{\text{Spec } \mathbb C}\otimes \pi_2^*\mathcal O_{\mathbb P^n}(-1)$ for the log structures. Here $\pi_1,\pi_2$ denote the corresponding projection maps from the product $\text{Spec } \mathbb C \times \mathbb P^n$.

Proof. First, let $F\colon \mathbb A^1 \times \mathbb A^n \to \mathbb A^n$ be the map $(z,(x_1,\dots, x_n))\mapsto (zx_1,\dots, zx_n)$. By the universal property of the blow-up this induces a map $\tilde F \colon \mathbb A^1\times~\mathrm{Bl}_{p}\ \mathbb A^n \to~\mathrm{Bl}_{p}\ \mathbb A^n$, where $p$ denotes the origin in $\mathbb A^n$. Furthermore, $\tilde F^{-1}(E_p)$, where $E_p$ is the exceptional divisor in $\mathrm{Bl}_{p}\ \mathbb A^n$, is the scheme theoretic union $o\times\mathrm{Bl}_{p}\ \mathbb A^n \cup \mathbb A^1 \times E_p$. By lemma 5.1 this gives a commutative diagram of log varieties:

where $\tilde F^{\log}$ denotes the map $\tilde F$ extended to a map log varieties in the obvious way and where $i,j$ are the strict morphisms induced by the inclusions of fibres over the origin for the underlying schemes. The analytification of this diagram is of the following form:

where $\tilde F' = \tilde F^\text{KN}$ and $f' = f^\text{KN}$. Since ρ is a homeomorphism on a dense subset there is only one pair $f',\tilde F'$ which can make the diagram commute by the same argument as above. It is easy to construct a function $\tilde F'$ such that the diagram commutes with $f$' equal to the group action morphism and from this the lemma follows.

The results of this section taken together give our main theorem.

5.4. Virtual log geometric generalizations

We end this article by discussing some apparent generalizations of these results in the category of virtual log schemes. The most immediate improvement we get with virtual morphisms is that, by example 2.8, we there are virtual morphisms

\begin{equation*}\text{Spec } \mathbb C \to \mathsf S_0.\end{equation*}

It is not hard to show that we can choose such a morphism such that it satisfies the unit axiom for our operad. Hence, in the category of log schemes with virtual morphisms we can define $\text{\textbf{CGK}}^{\log}_d$ to be an operad, not just a pseudo-operad. It is clear that theorem 5.12 still holds for this operad.

Virtual morphisms also allow us to construct the (unframed) Fulton–MacPherson operad $\text{\textbf{FM}}_{2d}$ as the analytification of an operad of log schemes with virtual morphisms. Indeed, the operad composition map

\begin{equation*}\gamma \colon T_{d,n}\times \prod_{i=1}^n T_{d,q^{-1}(i)} \to T_{d,M}\end{equation*}

extends to a virtual morphism of log schemes

\begin{equation*}\mathsf K_{d,n}\times \prod_{i=1}^n \mathsf K_{d,q^{-1}(i)} \to \mathsf K_{d,M}.\end{equation*}

This map is defined using lemmas 5.3 and 5.4 but this time we replace the isomorphism in lemma 5.4, (b) with the isomorphism

\begin{equation*}\gamma^*\mathcal O_{T_{d,M}}(q^{-1}(r)) \xrightarrow{\cong} \left (\bigotimes_{r\in I}\pi_0^* \mathcal O_{T_{d,n}}(I)^{\otimes(-1)}\right ) \otimes \pi_r^* \mathcal O_{d, q^{-1}(r)}(q^{-1}(r)).\end{equation*}

This gives a well defined virtual morphism of log structures by our definitions. These maps define an operad $\text{\textbf{CGK}}^{\text{V-log}}_d$ in log schemes with virtual morphisms. Using the same methods as before one can show that the analytification of the resulting operad is $\text{\textbf{FM}}_{2d}$.