
\subsection{What is rounding?}

The process of ``rounding", in its most basic form, produces a manifold \emph{with corners} from a \emph{smooth} analytic space with a normal crossings divisor. So the corners are not rounded but rather the opposite.  %Evidently then, ``rounding" is something of a misnomer. 
 On the other hand, these corners are rather round and shapely. Also, anybody who has seen the construction under any name and hears the name ``rounding" immediately knows what this is about. Evidently then,  even though ``rounding" might be  something of a misnomer, it is a very good name.
Its origin seems to be in work of Kajiwara, Nakayama and Ogus \cite{Kajiwara-Nakayama,Nakayama-Ogus}.

In various moduli problems, the rounding of the moduli space often has a more natural topological interpretation than the moduli space itself.  A good  example is the Deligne-Mumford moduli space $\overline{\mathcal{M}}_{g,n}$ of marked nodal curves, whose boundary is a normal crossings divisor.  The ``interior" $\mathcal{M}_{g,n}$ can be described topologically as a quotient of Teichmuller space by the appropriate mapping class group.  There are generalizations of Teichmuller space and the mapping class group involving $2$-manifolds decorated with circles, due to Bers and others; the analogous quotient yields a topological description of the \emph{rounding} of $\overline{\mathcal{M}}_{g,n}$, rather than the moduli space $\overline{\mathcal{M}}_{g,n}$ itself.\footnote{A reference is needed!!}  

Another example is that of twisted curves, discussed in section \ref{Roots}. A twisted curve is an algebraic stack with a log structure, so it is a bit exotic. But its rounding is a good old topological space. A similar 
example occurs in current work of one of us (Gillam): the relative Hilbert stack of a marked Riemann surface can be defined algebraically using Jun Li style expansions, but it is not representable (except in some trivial cases). However its rounding is a topological space (a manifold with corners, even) which is relatively easy to describe.  A similar phenomenon occurs in many moduli problems involving expansions, discussed briefly below.  

The topological preeminence of the rounding of moduli spaces might ultimately be traced back to the preference in topology for operations involving real codimension one subspaces (e.g.\ connected sum of manifolds) as opposed to the algebro-geometric preference for complex codimension one operations (e.g.\ pushout of two smooth varieties along a common divisor).  From this point of view, one might think of, say, log geometry as an attempt to speak algebraically about various ``real codimension one" phenomena.

\subsection{The oriented real blowup}
The most general rounding operation is the \emph{Kato--Nakayama logarithmic space} associated to a log analytic space \cite{Kato-Nakayama}.  In the basic example of a smooth analytic space with log structure from a normal crossings divisor, the Kato--Nakayama space can be described in terms of \emph{oriented real blowup}, which is a relatively simple rounding operation that can be described as follows.  Suppose $X$ is a topological space, $\pi : L \to X$ is a complex line bundle, and $s : X \to L$ is a section of $\pi$.  Locally on $X$ we can choose a trivialization $(\pi, \phi) : L \to X \times \CC$ and consider the subspace 
\be \B_{L,s,\phi} X & := & \bigg\{\, l \in L\  :\ \  |\phi(l)| \cdot(\phi s \pi)(l)\, = \ \phi(l) \cdot|(\phi s \pi)(l)|\, \bigg\} \ee
 of $L$.  A continuous function $u : X \to \CC^*$ yields a new trivialization $(\pi, u \cdot \phi)$, where $(u \cdot \phi)(l) := (u \pi)(l) \phi(l)$ .  The key observation is that $\B_{L,s,\phi} X= \B_{L,s,u \cdot \phi} X$, so the subspace $\B_{L,s, \phi} X$ is independent of the choice of $\phi$, hence one can define a subspace $\B_{L,s} X \subseteq L$ by defining it locally on $X$ using a trivialization, then gluing the locally defined subspaces.  From the local picture using a trivialization, it is clear that the subspace $\B_{L,s} X$ contains the zero section and $L|_{Z(s)}$ (where $Z(s) \subseteq X$ is the zero locus of $s$) and is invariant under the $\RR_{>0}$ action on $L$ inherited from the full $\CC^*$ scaling action.  We let $\B_{L,s}^* X$ be the complement of the zero section in $\B_{L,s}$ and we call \be \Blo_{L,s} X & := & (\B_{L,s}^* X ) / \RR_{>0} \ee the \emph{oriented real blowup} of $X$ along $(L,s)$.  

The space $\Blo_{L,s} X$ is a closed subspace of the oriented circle bundle $S^1 L := L^* / \RR_{>0}$ associated to $L$ and is, in particular, proper over $X$.  The projection $\tau : \Blo_{L,s} X \to X$ is an isomorphism away from $Z(s)$ and $\tau^{-1}(Z(s))$ is oriented circle bundle $S^1 L|_{Z(s)}$.  The spaces $\B_{L,s} X$ and $\Blo_{L,s} X$ are natural under pulling back line bundles and sections.

If $X$ is an analytic space and $D \subseteq X$ is a Cartier divisor, then $D$ determines a line bundle $\O_X(D)$ together with a section $s$ whose zero locus is $D$.  In this situation, we will write $\B_D X$, $\Blo_D X$, etc.\ and speak of the \emph{oriented real blowup of} $X$ \emph{along} $D$.  The space $\Blo_D X$ inherits a differentiable structure from its inclusion in $S^1 \O_X(D)$.

The basic example to keep in mind is the oriented real blowup $\Blo_0 \CC$ of the complex plane $\CC$ at the origin.  The origin is the zero locus of the identity map $\Id : \CC \to \CC$, hence \be \Blo_0 \CC & = & \big\{ (z,Z) \in \CC \times \CC^*\ :\ |z|Z = z |Z| \big\}\  /\  \RR_{>0} \\ & = & \big\{ (z,Z) \in \CC \times S^1\ :\ |z| Z = z \big\} \\ & \cong & \RR_{\geq 0} \times S^1 , \ee where the last isomorphism from $\RR_{\geq 0} \times S^1$ is given by $(\lambda, Z) \mapsto (\lambda Z , Z)$.  Evidently $\Blo_0 \CC$ is a half-infinite annulus whose boundary $S^1$ is the exceptional locus of $\tau : \Blo_0 \CC \to \CC$ (the fiber over the origin).


\subsection{The Kato--Nakayama space} 
Let $(X,\mathcal{M}_X)$ be a fine and saturated  logarithmic analytic space. 
\begin{definition}[\protect{\cite[1.2]{Kato-Nakayama}}]
we define its {\em canonical rounding}, or {\em Kato-Nakayama sapce}, denoted $X^{\log}$, as the space whose points are pairs $(x,F)$ where $x \in X$ and $F :  \mathcal{M}_{X,x} \to S^1$ is a monoid homomorphism satisfying  $F(u) = u(x) / |u(x)|$ for every $u \in \O_{X,x}^* \subseteq \mathcal{M}_{X,x}$. 
\end{definition}
This space can be easily given a topology. Let us describe the topology in the special case where $\cM_X$ is the canonical log structure associated to a Cartier divisor $D \subseteq X$, see Example \ref{NClog}. 
% Let \be \mathcal{M}_X & := & \bigg\{ f \in \O_X\, :\ f|_{X \setminus D} \in \O_{X \setminus D}^*\, \bigg\} \ee be the associated log structure.  
Locally on $X$ we can find $f_1,\dots,f_n \in \mathcal{M}_X(X)$ which, together with the units, generate $\mathcal{M}_X$.  The map \be X^{\log} & \to & X \times (S^1)^n \\ (x,F) & \mapsto & (x, F(f_{1,x}), \dots, F(f_{n,x})) \ee is then easily seen to be a monomorphism onto a closed subset of $X \times (S^1)^n$, so we give $X^{\log}$ the subspace topology so that this is a closed embedding.  Since one can check easily that this topology doesn't depend on the choice of generators $f_1,\dots,f_n$, the locally defined topologies glue to a topology on $X^{\log}$ making the projection $\tau : X^{\log} \to X$ given by $\tau(x,F) := x$ a proper map.

 

\subsection{Relating Kato--Nakayama spaces to oriented blowups}
There is a morphism \be \phi : X^{\log} & \to & \Blo_D X \\ (x,F) & \mapsto & (x, f \mapsto F(\overline{f})) \ee of topological spaces over $X$ which requires a little explanation.  Here $f \in S^1 \O_X(-D)|_x$ is in the circle bundle associated to the fiber $\O_X(-D)|_x \cong \CC$ and $\overline{f} \in \O_{X,x}(-D)$ is a lifting of $f$ to the stalk (one shows that any such $\overline{f}$ is actually in $\mathcal{M}_{X,x}$ and that $F(\overline{f})$ doesn't depend on this choice of lifting $\overline{f}$).  If we use the identification $$S^1 \O_X(D) = \Hom_{S^1}( S^1 \O_X(-D), S^1), $$ then we can think of $\phi$ as a map from $X^{\log}$ to $S^1 \O_X(D)$; one then shows that this $\phi$ is continuous and that $\phi$ factors through $\Blo_D X \subseteq S^1 \O_X(D)$.

When $X$ is a smooth analytic space and $D$ is a smooth divisor, the map $\phi$ is easily seen to be an isomorphism since one can reduce to the case $(X,D) = (\CC,0)$ on formal grounds.  Slightly more generally, if $X$ is smooth, but $D$ is only a normal crossings divisor, then locally we can write $D$ as a union of smooth divisors $D_1,\dots,D_i$ which look like the first $i$ coordinate hyperplanes in $\CC^n$ ($n = \dim X$), and we can define a variant of the oriented real blowup \be \Blo_D' X & := & (\Blo_{D_1} X ) \times_X \cdots \times_X (\Blo_{D_i} X)  \ee and a map $\phi : X^{\log}  \to  \Blo_D' X$.  In this local picture, the log structure $\mathcal{M}_X$ is the direct sum (in the category of log structures) of the log structures $\mathcal{M}_X^j$ from $D_1,\dots,D_i$, the associated Kato--Nakayama space $X^{\log}$ is the fibered product over $X$ of the $(X,\mathcal{M}_X^j)^{\log}$, and $\phi$ is just the fibered product over $X$ of the previously constructed isomorphisms $\phi_j : (X,\mathcal{M}_X^j)^{\log} \to \Blo_{D_j} X$.  The locally defined variants can be glued to define a global variant $\Blo_D' X$ of the oriented real blowup and an isomorphism $\phi : X^{\log} \cong \Blo_D' X$ of topological spaces over $X$.  

\subsection{Topology, cohomology, and the Kato--Nakayama space}
Locally, if $X = \CC^n$ and $D$ is the union of the first $i$ coordinate hyperplanes, then $D$ is the zero locus of $(z_1,\dots,z_n) \mapsto z_1 \cdots z_i \in \CC$ and we have \be \Blo_D X & = & \{ (z_1,\dots,z_n,Z) \in \CC^n \times S^1 : | z_1 \cdots z_i | Z = z_1 \cdots z_i \} \\ \Blo_D' X & = & \{ (\overline{z}, \overline{Z}) \in \CC^n \times (S^1)^i: |z_j| Z_j = z_j \; {\rm for} \; j=1,\dots, i \}. \ee In the general normal crossings divisor situation, the fiber of $\tau : X^{\log} \to X$ over a point $x \in X$ is naturally identified with $$S^1 N_{D_1/X}|_x \times \cdots \times S^1 N_{D_i / X}|_x,$$ where $D_1,\dots,D_i$ are the branches of $D$ containing $x$.  When $\dim X = n$, a point $y \in \tau^{-1}(x)$ has a neighborhood diffeomorphic to a neighborhood of the origin in $\RR_{\geq 0}^i \times \RR^{2n-i}$.  (Note $i \leq n$, so the depth of the corners in a Kato--Nakayama space is somewhat constrained.)  Recall that the \emph{topology} near the origin only depends on whether $i > 0$, but the differentiable structure depends on the actual value of $i$.  In particular, the topological boundary of the manifold $X^{\log}$ is given by $\tau^{-1}(D)$, and this manifold with boundary is homotopy equivalent to its interior, so $\H^*(X^{\log}) = \H^*(X \setminus D).$

The topology of morphisms of Kato--Nakayama spaces is also very nice, as shown by the following beautiful general result, see \cite[Theorem 0.3]{Nakayama-Ogus}:

\begin{theorem}
Let $f: X \to Y$ be a smooth and integral morphism of fine log analytic spaces. Then the associated map $X^{\log} \to Y^{\log}$ is a topological submersion.
\end{theorem}

In fact, Nakayama and Ogus prove a more general result, replacing integrality by K. Kato's notion of {\em exact morphisms}, see \cite[Definition 4.6]{KKato}. 

The fact that the topology is nice suggests that one expects nice cohomological implications. This is indeed the original motivation leading Kato and Nakayama to define $X^{\log}$, see \cite[Theorem 0.2 (1)]{Kato-Nakayama}:

\begin{theorem}
Let $(X,M)$ be a fine and saturated log scheme with $X$ of finite type over $\CC$. 
Let $F$ be a constructible sheaf on the log-\'etale site $X_{\text{log-\'et}}$, and let $F^{\log}$ be its pullback to the topological space $X^{\log}$. Then for all $q\in \ZZ$ we have 
$$ H^q(X_{\text{log-\'et}}, F) = H^q(X^{\log}, F^{\log}).$$
\end{theorem}

Very strong results hold true for de Rham cohomology. In fact the Kato--Nakayama space is a model for the log de Rham cohomology of $X$ in the sense that \be \H^*(X^{\log}, \CC) & = & \HH^*(X, \land^{\bullet} \Omega_X^\dagger) \ee under mild assumptions on $X$. We discuss this in  Section \ref{DeRham} below.

\subsection{Kato--Nakayama spaces of expanded pairs}
Given a pair $(X,D)$ consisting of a smooth variety $X$ over $\CC$ with a smooth divisor $D \subseteq X$, the notion of an \emph{expanded pair} $t : \mathcal{X} \to B$ over a base $B$ arises in various relative curve counting theories.  The fiber of $t$ over a point $b \in B$ always looks like \be X[n]_0 & = & X \coprod_D \Delta_1 \coprod_D \cdots \coprod_D \Delta_n \ee (for an appropriate $n$), where $\Delta_i =  \PP( N_{D/X} \oplus \O_D)$ is a $\PP^1$ bundle over $X$.  Both $\mathcal{X}$ and $B$ have natural log structures making $t$ a log smooth map of log schemes.  The fiber of $t^{\log} : \mathcal{X}^{\log} \to B^{\log}$ over a point $c \in \tau_B^{-1}(b)$ looks like $$ X^{\log} \coprod_{c_1 : S^1 N_{D/X} \cong S^1 N_{D/ \Delta_1}} \Delta_1^{\log} \cdots \coprod_{ c_n : S^1 N_{D/\Delta_{n-1}} \cong S^1 N_{D / \Delta_n}} \Delta_n^{\log} , $$ where the choice of $c \in \tau^{-1}(b) \cong (S^1)^n$ determines the choice of orientation reversing $S^1$ bundle isomorphisms $c_1,\dots,c_n$.  Here each $\Delta_j$ has the log structure from the two copies of $D$, and $\Delta_j^{\log}$ is a cylinder bundle over $D$ (better: an $I$-bundle over $D^{\log}$).

The action of $(\CC^*)^n$ on $X[n]_0$ given by scaling the fibers of the $\PP^1$ bundles $\Delta_i$ is an action by isomorphisms of log schemes, so it lifts to an action on Kato--Nakayama spaces.  This lifted action is nontrivial on $B^{\log}$ as the $(S^1)^n$ factor of $(\CC^*)^n$ acts simply transitively on $\tau_B^{-1}(b) \cong (S^1)^n$.  In the usual moduli problems involving expansions, the isotropy group of a point $b$ involves elements of $(\CC^*)^n$ such that the induced action on $X[n]_0$ respects a map from a curve to $X[n]_0$, a subscheme of $X[n]_0,$ etc., and this isotropy is usually required to be finite to have a good moduli problem.  Since $G \cap \RR_{>0} = \{ \Id \}$ for any finite subgroup $G$ of $\CC^*$, the Kato--Nakayama space of the moduli problem is often representable even if the moduli problem itself is not.  This is always the case for moduli problems involving, say, quotients of sheaves on $X[n]_0$ pulled back from $X$, since these quotients themselves have no automorphisms and the only isotropy comes from the subgroup of $(\CC^*)^n$ preserving the quotient.
