Holomorphic curves in Exploded Torus Fibrations: Compactness

Functions on exploded fibrations will sometimes take values in the following semirings:

The tropical semiring is a semiring $\mathfrak{t}^{\mathbb{R}}$ which is equal to $\mathbb{R}$ with ‘multiplication’ being the operation of usual addition and ‘addition’ being the operation of taking a minimum. We will write elements of $\mathfrak{t}^{{\mathbb{R}}}$ as $\mathfrak{t}^{x}$ where $x\in\mathbb{R}$. Then we can write the operations as follows

$$\mathfrak{t}^{x}\mathfrak{t}^{y}:=\mathfrak{t}^{x+y}$$

$$\mathfrak{t}^{x}+\mathfrak{t}^{y}:=\mathfrak{t}^{\min\{x,y\}}$$

$\mathfrak{t}^{1}$ can be thought of as something which is infinitesimally small. With this in mind, we have the following order on $\mathfrak{t}^{\mathbb{R}}$:

$$\mathfrak{t}^{x}>\mathfrak{t}^{y}\text{ when }x<y$$

Given a ring $R$, the exploded semiring $R\mathfrak{t}^{{\mathbb{R}}}$ consists of elements $c\mathfrak{t}^{x}$ with $c\in R$ and $x\in\mathbb{R}$. Addition and multiplication are as follows:

$$c_{1}\mathfrak{t}^{x}c_{2}\mathfrak{t}^{y}=c_{1}c_{2}\mathfrak{t}^{x+y}$$

$$c_{1}\mathfrak{t}^{x}+c_{2}\mathfrak{t}^{y}=\begin{cases}&c_{1}\mathfrak{t}^{x}\text{ if }x<y\\ &(c_{1}+c_{2})\mathfrak{t}^{x}\text{ if }x=y\\ &c_{2}\mathfrak{t}^{y}\text{ if }x>y\end{cases}$$

It is easily checked that addition and multiplication are associative and obey the usual distributive rule. We will mainly be interested in $\mathbb{C}\mathfrak{t}^{\mathbb{R}}$ and $\mathbb{R}\mathfrak{t}^{\mathbb{R}}$.

There are semiring homomorphisms

$$R\xrightarrow{\iota}R\mathfrak{t}^{\mathbb{R}}\xrightarrow{\lfloor\cdot\rfloor}\mathfrak{t}^{\mathbb{R}}$$

defined by

$$\iota(c):=c\mathfrak{t}^{0}$$

$$\lfloor c\mathfrak{t}^{x}\rfloor:=\mathfrak{t}^{x}$$

The homomorphism $\lfloor\cdot\rfloor$ is especially important. We shall call $\lfloor c\mathfrak{t}^{x}\rfloor=\mathfrak{t}^{x}$ the tropical part of $c\mathfrak{t}^{x}$. There will be an analogous ‘tropical part’ of ‘basic’ exploded fibrations which can be thought of as the large scale.

Define the positive exploded semiring $R\mathfrak{t}^{\mathbb{R}^{+}}$ to be the sub semiring of $R\mathfrak{t}^{\mathbb{R}}$ consisting of elements of the form $c\mathfrak{t}^{x}$ where $x\geq 0$. There is a semiring homomorphism

$$\lceil\cdot\rceil:R\mathfrak{t}^{\mathbb{R}^{+}}\longrightarrow R$$

given by

$$\lceil c\mathfrak{t}^{x}\rceil:=\begin{cases}&c\text{ if }x=0\\ &0\text{ if }x>0\end{cases}$$

Note that the above homomorphism can be thought of as setting $\mathfrak{t}^{1}=0$, which is intuitive when $\mathfrak{t}^{1}$ is thought of as infinitesimally small.

We shall use the following order on $(0,\infty)\mathfrak{t}^{\mathbb{R}}$, which again is intuitive if $\mathfrak{t}^{1}$ is thought of as being infinitesimally small and positive:

$$x_{1}\mathfrak{t}^{y_{1}}<x_{2}\mathfrak{t}^{y_{2}}\text{ whenever }y_{1}>y_{2}\text{ or }y_{1}=y_{2}\text{ and }x_{1}<x_{2}$$

The following definition of abstract exploded fibrations is far too general, but it allows us to talk about local models for exploded torus fibrations as abstract exploded fibrations without giving too many definitions beforehand.

We shall use the terminology of an open subset $U$ of an abstract exploded fibration $\mathfrak{B}$ to mean an open subset $U\subset[\mathfrak{B}]$ along with the set of points in $[\cdot]^{-1}(U)$ and the sheaf $\mathcal{E}^{\times}(U)$. The strange notation for the set of points in $\mathfrak{B}$, $\{p\rightarrow\mathfrak{B}\}$ is used because it is equal to the set of morphisms of a point into $\mathfrak{B}$.

Note that the reader should not attempt to work directly with the above definition as it is too general. It simply enables us to talk about examples without giving a complete definition beforehand. The next sequence of examples will give us local models for exploded $\mathbb{T}$ fibrations.

Any smooth manifold $M$ is a smooth exploded $\mathbb{T}$ fibration. In this case, the underlying toplogical space $[M]$ is just $M$ with the usual topology, the set of points $\{p\rightarrow M\}$ the usual set of points in $M$, and $[\cdot]:\{p\rightarrow M\}\longrightarrow[M]$ the identity map. The sheaf $\mathcal{E}^{\times}(M)$ consists of all functions of the form $f\mathfrak{t}^{a}$ where $f\in C^{\infty}(M,\mathbb{C}^{*})$ and $a$ is a locally constant $\mathbb{R}$ valued function.

The reader should convince themselves that this is just a strange way of encoding the usual data of a smooth manifold, and that a exploded morphism between smooth manifolds is simply a smooth map. This makes the category of smooth manifolds a full subcategory of the category of exploded $\mathbb{T}$ fibrations.

The above example should be considered as a ‘completely smooth’ object. At the other extreme, we have the following ‘completely tropical’ object.

The exploded fibration $\mathfrak{T}^{n}$ is best described using coordinates $(\tilde{z}_{1},\dotsc,\tilde{z}_{n})$ where each $\tilde{z}_{i}\in\mathcal{E}^{\times}(\mathfrak{T}^{n})$. The set of points in $\mathfrak{T}^{n}$, $\{p\rightarrow\mathfrak{T}^{n}\}$ are then identified with $\left(\mathbb{C}^{*}\mathfrak{t}^{\mathbb{R}}\right)^{n}$ by prescribing the values of $(\tilde{z}_{1},\dotsc,\tilde{z}_{n})$. The underlying topological space $[\mathfrak{T}^{n}]$ is $\mathfrak{t}^{\mathbb{R}^{n}}$ (given the usual topology on $\mathbb{R}^{n}$), and

$$[(\tilde{z}_{1},\dotsc,\tilde{z}_{n})]=(\lfloor\tilde{z}_{1}\rfloor,\dotsc,\lfloor\tilde{z}_{n}\rfloor)\text{ or }[(c_{1}\mathfrak{t}^{a_{1}},\dotsc,c_{n}\mathfrak{t}^{a_{n}})]=(\mathfrak{t}^{a_{1}},\dotsc,\mathfrak{t}^{a_{n}})$$

Note that there is a $(\mathbb{C}^{*})^{n}$ worth of points $p\longrightarrow\mathfrak{T}^{n}$ over every topological point $[p]\in[\mathfrak{T}^{n}]$.

Exploded functions in $\mathcal{E}^{\times}\left(\mathfrak{T}^{n}\right)$ can be written in these coordinates as

$$c\mathfrak{t}^{y}\tilde{z}^{\alpha}:=c\mathfrak{t}^{y}\prod\tilde{z}_{i}^{\alpha_{i}}$$

where $c\in\mathbb{C}^{*}$, $y\in\mathbb{R}$ and $\alpha\in\mathbb{Z}^{n}$ are all locally constant functions.

The reader should now be able to verify the following observations:

1. 1.

   A morphism from $\mathfrak{T}^{n}$ to a smooth manifold is simply given by a map $[\mathfrak{T}^{n}]\longrightarrow M$ which has as its image a single point in $M$.

2. 2.

   A morphism from a connected smooth manifold $M$ to $\mathfrak{T}^{n}$ has the information of a map $[f]$ from $M$ to a point $[p]\in[\mathfrak{T}^{n}]$ and a smooth map $f$ from $M$ to the $(\mathbb{C}^{*})^{n}$ worth of points over $[p]$. The map $f$ is determined by specifying the $n$ exploded functions

   $$f^{*}(\tilde{z}_{i})\in\mathcal{E}^{\times}\left(M\right)$$

Note in particular that $\mathcal{E}^{\times}\left(M\right)$ is equal to the sheaf of smooth morphisms of $M$ to $\mathfrak{T}$. This will be true in general. A smooth morphism $f:\mathfrak{B}\longrightarrow\mathfrak{T}^{n}$ is equivalent to the choice of $n$ exploded functions in $\mathcal{E}^{\times}\left(\mathfrak{B}\right)$ corresponding to $f^{*}(\tilde{z}_{i})$.

Now, a hybrid object, part ‘smooth’, part ‘tropical’.

The exploded fibration $\mathfrak{T}^{1}_{[0,\infty)}:=\mathfrak{T}^{1}_{1}$ is more complicated. We shall describe this using the coordinate $\tilde{z}\in\mathcal{E}^{\times}(\mathfrak{T}^{1}_{1})$. The set points $\{p\rightarrow\mathfrak{T}^{1}_{1}\}$ is identified with $\mathbb{C}^{*}\mathfrak{t}^{\mathbb{R}^{+}}$ by specifying the value of $\tilde{z}(p)\in\mathbb{C}^{*}\mathfrak{t}^{\mathbb{R}^{+}}$.

The underlying topological space is given as follows:

$$\begin{split}[\mathfrak{T}^{1}_{1}]&:=\frac{\mathbb{C}\coprod\mathfrak{t}^{\mathbb{R}^{+}}}{\{0\in\mathbb{C}\}=\{\mathfrak{t}^{0}\in\mathfrak{t}^{\mathbb{R}^{+}}\}}\\ &:=\{(z,\mathfrak{t}^{a})\in\mathbb{C}\times\mathfrak{t}^{\mathbb{R}^{+}}\text{ so that }za=0\}\end{split}$$

\psfrag{ETF1math}{$\lfloor\mathfrak{T}^{1}_{1}\rfloor:=\mathfrak{t}^{\mathbb{R}^{+}}$}\psfrag{ETF1mathC}{$\lceil\mathfrak{T}^{1}_{1}\rceil:=\mathbb{C}$}\psfrag{ETF1mathtot}{A picture of $[\mathfrak{T}^{1}_{1}]$}\psfrag{ETF1Cs}{$\mathbb{C}^{*}$}\includegraphics{ETF1}

Our map from points to $[\mathfrak{T}^{1}_{1}]$ is given by

$$[\tilde{z}]:=\left(\lceil\tilde{z}\rceil,\lfloor\tilde{z}\rfloor\right)$$

The above expression uses the maps defined in section 2.1, $\lfloor c\mathfrak{t}^{a}\rfloor=\mathfrak{t}^{a}$, and $\lceil c\mathfrak{t}^{a}\rceil$ is equal to $c$ if $a=0$ and $0$ if $a>0$. Note that there is one point $p\rightarrow\mathfrak{T}^{1}_{1}$ over points of the form $(c,\mathfrak{t}^{0})\in[\mathfrak{T}^{1}_{1}]$, a $\mathbb{C}^{*}$ worth of points over $(0,\mathfrak{t}^{a})\in[\mathfrak{T}^{1}_{1}]$ for $a>0$ and no points over $(0,\mathfrak{t}^{0})\in[\mathfrak{T}^{1}_{1}]$.

This object should be thought of as a copy of $\mathbb{C}$ which we puncture at $0$, and consider this puncture to be an asymptoticaly cylindrical end. Each copy of $\mathbb{C}^{*}$ over $(0,\mathfrak{t}^{a})$ should be thought of as some ‘cylinder at $\infty$’. Note that even though there is a $(0,\infty)$ worth of two dimensional cylinders involved in this object, it should still be thought of being two dimensional. This strange feature is essential for the exploded category to have a good holomorphic curve theory. (Actually, we could just have easily had a $\mathbb{Q}^{+}$ worth of cylinders at infinity, and worked over the semiring $\mathbb{C}\mathfrak{t}^{\mathbb{Q}}$ instead of $\mathbb{C}\mathfrak{t}^{\mathbb{R}}$, but this author prefers the real numbers.)

We shall use the notation

$$\lceil\mathfrak{T}^{1}_{1}\rceil:=\frac{[\mathfrak{T}^{1}_{1}]}{\mathfrak{t}^{\mathbb{R}^{+}}=0}=\mathbb{C}$$

$$\lfloor\mathfrak{T}^{1}_{1}\rfloor:=\frac{[\mathfrak{T}^{1}_{1}]}{\mathbb{C}=0}=\mathfrak{t}^{\mathbb{R}^{+}}$$

We shall call $\lceil\mathfrak{T}^{1}_{1}\rceil$ the smooth or topological part of $\mathfrak{T}^{1}_{1}$ and $\lfloor\mathfrak{T}^{1}_{1}\rfloor$ the tropical part of $\mathfrak{T}^{1}_{1}$.

We can write any exploded function $h\in\mathcal{E}^{\times}\left(\mathfrak{T}^{1}_{1}\right)$ as

$$h(\tilde{z})=f(\lceil\tilde{z}\rceil)\mathfrak{t}^{y}\tilde{z}^{\alpha}\text{ for }f\in C^{\infty}(\mathbb{C},\mathbb{C}^{*})\text{, and }y\in\mathbb{R},\alpha\in\mathbb{Z}\text{ locally constant.}$$

We can now see that $\mathfrak{T}^{1}_{1}$ is a kind of hybrid of the last two examples. Restricting to an open set contained inside $\{\lfloor\tilde{z}\rfloor=\mathfrak{t}^{0}\}\subset[\mathfrak{T}^{1}_{1}]$ we get part of a smooth manifold. Restricting to an open set contained inside $\{\lceil\tilde{z}\rceil=0\}\subset[\mathfrak{T}^{1}_{1}]$, we get part of $\mathfrak{T}$.

We can define ${}^{+}\mathcal{E}^{\times}\left(\mathfrak{B}\right)$ to consist of all functions in $\mathcal{E}^{\times}(\mathfrak{B})$ which take values only in $\mathbb{C}^{*}\mathfrak{t}^{\mathbb{R}^{+}}$. Note that a smooth morphism $f:\mathfrak{B}\longrightarrow\mathfrak{T}^{1}_{1}$ from any exploded $\mathbb{T}$ fibration $\mathfrak{B}$ that we have described in examples so far is given by a choice of exploded function $f^{*}(\tilde{z})\in{}^{+}\mathcal{E}^{\times}\left(\mathfrak{B}\right)$. This will be true for exploded $\mathbb{T}$ fibrations in general.

We can build up ‘exploded curves’ using coordinate charts modeled on open subsets of $\mathfrak{T}^{1}_{1}$. For an example of this see the picture on page 2.18. For an example of a map of this to $\mathfrak{T}^{n}$ see example 2.39 on page 2.39. Those interested in tropical geometry may like the following example 2.40. For a local example of a family of such objects, see example 2.47 on page 2.47.

We shall use the special notation $\mathfrak{T}^{m}_{n}$ to denote $(\mathfrak{T}^{1}_{1})^{n}\times\mathfrak{T}^{(m-n)}$. (These spaces will be described rigorously below.) Before giving the most general local model for exploded $\mathbb{T}$ fibrations, we give the following definition which can be thought of as a local model for the tropical part of our exploded fibrations.

\psfrag{ETF1math}{$\lfloor\mathfrak{T}^{1}_{1}\rfloor:=\mathfrak{t}^{\mathbb{R}^{+}}$}\psfrag{ETF1mathC}{$\lceil\mathfrak{T}^{1}_{1}\rceil:=\mathbb{C}$}\psfrag{ETF1mathtot}{A picture of $[\mathfrak{T}^{1}_{1}]$}\psfrag{ETF1Cs}{$\mathbb{C}^{*}$}\includegraphics{ETF2}

The following is the most general local model for exploded $\mathbb{T}$ fibrations.

Given any integral cone $A\subset\mathfrak{t}^{\mathbb{R}^{m}}$ which generates $\mathfrak{t}^{\mathbb{R}^{m}}$ as a group, we will now describe the exploded fibration $\mathbb{R}^{n}\times\mathfrak{T}^{m}_{A}$. We shall use coordinates

$$(x_{1},\dotsc,x_{n},\tilde{z}_{1},\dotsc,\tilde{z}_{m})$$

The set of points $\{p\rightarrow\mathbb{R}^{n}\times\mathfrak{T}^{m}_{A}\}$ is then given by

$$\left\{(x_{1},\dotsc,x_{m},c_{1}\mathfrak{t}^{a_{1}},\dotsc,c_{m}\mathfrak{t}^{a_{m}})\in\mathbb{R}^{n}\times\left(\mathbb{C}^{*}\mathfrak{t}^{\mathbb{R}}\right)^{m};(\mathfrak{t}^{a_{1}},\dotsc,\mathfrak{t}^{a_{m}})\in A\right\}$$

We will describe the underlying topological space $[\mathbb{R}^{n}\times\mathfrak{T}^{m}_{A}]$ using some extra coordinate choices. Let $\{\alpha^{1},\dotsc,\alpha^{k}\}$ generate the dual cone $A^{*}\subset\mathbb{Z}^{m}$ as a semigroup with the relations

$$\sum_{i}r_{i,j}\alpha^{i}=0$$

Label coordinates for $\mathbb{C}^{k}$ as $(z^{\alpha^{1}},\dotsc,z^{\alpha^{k}})$, and use $\mathfrak{t}^{a}$ to denote points in $A$. We can describe the underlying topological space, $[\mathbb{R}^{n}\times\mathfrak{T}^{m}_{A}]$ as a subspace

$$[\mathbb{R}^{n}\times\mathfrak{T}^{m}_{A}]=\left\{(x,z^{\alpha^{i}},\mathfrak{t}^{a});\ \prod_{i}(z^{\alpha^{i}})^{r_{i,j}}=1,(\alpha^{i}\cdot a)z^{\alpha^{i}}=0\right\}\subset\mathbb{R}^{n}\times\mathbb{C}^{k}\times A$$

Again we shall use the notation

$$\lfloor\mathbb{R}^{n}\times\mathfrak{T}^{m}_{A}\rfloor:=A,\text{ the tropical part }$$

$$\lceil\mathbb{R}^{n}\times\mathfrak{T}^{m}_{A}\rceil:=\left\{\prod_{i}(z^{\alpha^{i}})^{r_{i,j}}=1\right\}\subset\mathbb{R}^{n}\times\mathbb{C}^{k},\text{ the smooth part}$$

(This smooth part is equal to $\mathbb{R}^{n}$ times the toric variety corresponding to the cone $A$.) Using the shorthand $\tilde{z}^{\alpha}:=\prod_{i}\tilde{z}_{i}^{\alpha_{i}}$, and recalling that $\lceil c\mathfrak{t}^{a}\rceil$ is equal to $c$ if $a=0$, and equal to $0$ if $a>0$, we can write the map from the set of points down to the smooth part as

$$\lceil(x,\tilde{z})\rceil:=(x,\lceil\tilde{z}^{\alpha^{1}}\rceil,\dotsc,\lceil\tilde{z}^{\alpha^{k}}\rceil)$$

The map from the set of points down to the tropical part is given by

$$\lfloor(x,\tilde{z})\rfloor:=(\lfloor\tilde{z}_{1}\rfloor,\dotsc,\lfloor\tilde{z}_{m}\rfloor)\text{ or }\lfloor(x,c_{1}\mathfrak{t}^{a_{1}},\dotsc,c_{m}\mathfrak{t}^{a_{m}})\rfloor=(\mathfrak{t}^{a_{1}},\dotsc,\mathfrak{t}^{a_{m}})$$

The map from the set of points down to the underlying topological space is then defined by

$$[(x,\tilde{z})]:=\left(\lceil(x,\tilde{z})\rceil,\lfloor(x,\tilde{z})\rfloor\right)$$

(Note that we can also describe $[\mathbb{R}^{n}\times\mathfrak{T}^{m}_{A}]$ as the closure of the image of the above map inside $\mathbb{R}^{n}\times\mathbb{C}^{k}\times A$.) Exploded functions $h\in\mathcal{E}^{\times}\left(\mathbb{R}^{n}\times\mathfrak{T}^{m}_{A}\right)$ are all functions that can be written as

$$h(x,\tilde{z}):=f\left(\lceil(x,\tilde{z})\rceil\right)\tilde{z}^{\alpha}\mathfrak{t}^{y}$$

where $f\in C^{\infty}(\mathbb{R}^{n}\times\mathbb{C}^{k},\mathbb{C}^{*})$, and $\alpha\in\mathbb{Z}^{m}$ and $y\in\mathbb{R}$ are locally constant. It is important to note that two exploded functions are equal if they have the same values on points (which means that we can represent the same exploded function using different smooth functions $f$.) Note also that this definition is independent of the choice of basis $\{\alpha^{i}\}$ for $A^{*}$.

The earlier examples $\mathfrak{T}^{1}_{1}=\mathfrak{T}^{1}_{\mathfrak{t}^{[0,\infty)}}$, and $\mathfrak{T}^{n}=\mathfrak{T}^{n}_{\mathfrak{t}^{\mathbb{R}^{n}}}$, and

$$\mathfrak{T}^{m}_{n}=(\mathfrak{T}^{1}_{1})^{n}\times\mathfrak{T}^{(m-n)}:=\mathfrak{T}^{m}_{\mathfrak{t}^{(\mathbb{R}^{+})^{n}\times\mathbb{R}^{(m-n)}}}$$

These examples are much easier to understand than the most general model, so if the above definition was confusing, it is worthwhile reading through it again with these examples in mind. In this case, we can choose the obvious basis for $A^{*}$ given by the first $n$ standard basis vectors, so $\tilde{z}^{\alpha^{i}}:=\tilde{z}_{i}$. The smooth part of $\mathfrak{T}^{m}_{n}$, is then $\lceil\mathfrak{T}^{m}_{n}\rceil:=\mathbb{C}^{n}$ with coordinates $(\lceil\tilde{z}_{1}\rceil,\dotsc,\lceil\tilde{z}_{n}\rceil)$. An important example of exploded fibrations locally modeled on open subsets of $\mathfrak{T}^{m}_{n}$ is given in example 2.37 on page 2.37. We need the more confusing models $\mathfrak{T}^{m}_{A}$ because they arise naturally in the intersection theory of exploded fibrations. They also occur in moduli spaces of holomorphic curves. We need to consider families of holomorphic curves with total spaces locally modeled on these more complicated $\mathfrak{T}^{m}_{A}$.

There are some extra sheaves of functions which are defined on any smooth exploded $\mathbb{T}$ fibration $\mathfrak{B}$ which will come in useful:

Recall that $\lfloor c\mathfrak{t}^{a}\rfloor:=\mathfrak{t}^{a}$. We shall also use the terminology that the order of $c\mathfrak{t}^{a}$ is $a$. The sheaf $\mathcal{A}^{\times}(\mathfrak{T}^{m}_{A})$ corresponds to functions on $A$ of the form $\mathfrak{t}^{a}\mapsto\mathfrak{t}^{y+\alpha\cdot a}$ where $\alpha\in\mathbb{Z}^{m}$ and $y\in\mathbb{R}$.

For example $\mathcal{A}(\mathfrak{T}^{n})$ consists of functions of the form $\mathfrak{t}^{f}$ where $f:\mathbb{R}^{n}\longrightarrow\mathbb{R}$ is some continuous, concave piecewise integral affine function. Functions of this form are usually called tropical functions. We shall usually not use $\mathcal{E}(\mathfrak{B})$ because the addition is strange, and $\mathcal{E}^{\times}(\mathfrak{B})$ is naturally defined as the sheaf of morphisms to $\mathfrak{T}$. The only place that $\mathcal{E}(\mathfrak{B})$ will be used in this paper will be in the definition of the tangent space of $\mathfrak{B}$. The operation of addition is needed here to state the usual property of being a derivation. The other reason that addition was mentioned in this paper was to emphasize the links with tropical geometry.

The inclusion $\iota:\mathbb{C}\longrightarrow\mathbb{C}\mathfrak{t}^{\mathbb{R}}$ defined by $\iota(c)=c\mathfrak{t}^{0}$ induces an inclusion of functions

$$\iota:C^{\infty}(\mathfrak{B})\longrightarrow\mathcal{E}(\mathfrak{B})$$

$$\iota(f)(p):=\iota(f(p))$$

We can consider an arbitrary Hausdorff topological space to be a $C^{0}$ exploded fibration. A continuous morphism of $\mathfrak{B}$ to a Hausdorff topological space $X$ is given by a map $f:\mathfrak{B}\longrightarrow[\mathfrak{B}]\longrightarrow X$ so that the pullback of continuous real valued functions on $X$ are $C^{0}$ functions on $\mathfrak{B}$. Note that in general, the map $[\cdot]:\mathfrak{B}\longrightarrow[\mathfrak{B}]$ is not a continuous morphism in this sense.

We can construct $\lceil\mathfrak{B}\rceil$ as the image of $\mathfrak{B}$ in $\mathbb{R}^{C^{0}(\mathfrak{B})}$. The universal property of $\lceil\cdot\rceil$ makes it clear that taking the topological part of $\mathfrak{B}$ is a functor from the exploded category to the category of Hausdorff topological spaces.

Note that limits in the above sense are of course not unique. A manifold is topologically compact when considered as an exploded fibration if and only if it is compact. Another example is any connected open subset of $\mathfrak{T}^{n}$. Sometimes the topological part of $\mathfrak{B}$ gives slightly misleading information about $\mathfrak{B}$, as the following example is intended to demonstrate. We shall give a more useful analogue of ‘compactness’ in definition 2.19 shortly.

Our local model $\mathbb{R}^{n}\times\mathfrak{T}^{m}_{A}$ has a smooth part $\lceil\mathbb{R}^{n}\times\mathfrak{T}^{m}_{A}\rceil$ which is equal to the product of $\mathbb{R}^{n}$ with the toric variety corresponding to the integral affine cone $A$. The following two examples have ‘misbehaved’ smooth parts: Consider the exploded fibration formed by quotienting $\mathbb{R}\times\mathfrak{T}$ by $(x,\tilde{z})=(x+1,\tilde{z})$ and $(x,\tilde{z})=(x+\pi,\mathfrak{t}^{1}\tilde{z})$. The smooth part of this is a single point, even though it is made up of pieces which have smooth parts equal to a circle. Another example is given by the subset of $\mathbb{R}\times\mathfrak{T}$ which is the union of the set where $\lfloor\tilde{z}\rfloor<\mathfrak{t}^{0}$ with the set where $\lfloor\tilde{z}\rfloor>\mathfrak{t}^{1}$ and $x>0$. (Note that these sets overlap because $\mathfrak{t}^{0}>\mathfrak{t}^{1}$.) The smooth part of this is equal to $\mathbb{R}$.

For many of our proofs, we shall restrict to a class of exploded fibrations in which the smooth part and the tropical part of the exploded fibration give better information than the above two examples. To explain this class of ‘basic’ exploded fibrations, we shall first construct some elementary examples.

Suppose that $P\subset\mathfrak{t}^{\mathbb{R}^{m}}$ is a convex integral affine polygon with nonempty interior defined by some set of inequalities:

$$P:=\{\mathfrak{t}^{a}\in\mathfrak{t}^{\mathbb{R}^{m}}\text{ so that }\mathfrak{t}^{c_{i}+a\cdot\alpha^{i}}\leq\mathfrak{t}^{0},\mathfrak{t}^{c_{i^{\prime}}+a\cdot\alpha^{i^{\prime}}}<\mathfrak{t}^{0}\}$$

where $\alpha^{i}\in\mathbb{Z}^{m}$ and $c_{i}\in\mathbb{R}$. This can simply be viewed as a convex polyhedron in $\mathbb{R}^{m}$ which has faces with rational slopes. View $P$ as a stratified space in the standard way, (so the strata of $P$ consist of the vertices of $P$, the edges of $P$ minus the vertices of $P$, the two dimensional faces of $P$ minus all vertices and edges, and so on, up to the interior of $P$). Note that as some of the defining inequalities may be strict, $P$ may be missing some faces or lower dimensional strata.

Then there is an exploded fibration $\mathfrak{T}^{m}_{P}$ associated with $P\subset\mathfrak{t}^{\mathbb{R}^{m}}$, which is constructed by gluing together coordinate charts modeled on $\mathfrak{T}^{m}_{A_{k}}$ where $A_{k}$ are the integral affine cones corresponding to the strata of $P$.

Explicitly, suppose that the point $\mathfrak{t}^{a^{k}}\in\mathfrak{t}^{\mathbb{R}^{m}}$ is inside the $k$th strata of $P$, and $A_{k}$ is the integral affine cone so that locally near $\mathfrak{t}^{a^{k}}$, $P$ is equal to $A_{k}$ shifted by $\mathfrak{t}^{a^{k}}$.

$$\text{near }\mathfrak{t}^{a^{k}},\ \ \ \ P=\{\mathfrak{t}^{a}\in\mathfrak{t}^{\mathbb{R}^{m}}\text{ so that }\mathfrak{t}^{a-a^{k}}\in A_{k}\}$$

There are coordinates $\tilde{z}_{1},\dotsc,\tilde{z}_{m}\in\mathcal{E}^{\times}\left(\mathfrak{T}^{m}_{P}\right)$ so that the set of points in $\mathfrak{T}^{m}_{P}$ are given by specifying values $(\tilde{z}_{1}(p),\dotsc,\tilde{z}_{m}(p))\in\left(\mathbb{C}^{*}\mathfrak{t}^{\mathbb{R}}\right)^{m}$ so that

$$(\lfloor\tilde{z}_{1}(p)\rfloor,\dotsc,\lfloor\tilde{z}_{m}(p)\rfloor)\in P$$

Denote by $P_{k}\subset P$ the subset of $P$ which is the union of all strata of $P$ whose closure contains the $k$th strata of $P$. There is a corresponding subset $\mathfrak{T}^{m}_{P_{k}}$ of $\mathfrak{T}^{m}_{P}$:

$$\mathfrak{T}^{m}_{P_{k}}:=\left\{\tilde{z}\text{ so that }\lfloor\tilde{z}\rfloor\in P_{k}\right\}$$

We put an exploded structure on $\mathfrak{T}^{m}_{P_{k}}$ so it is isomorphic to the corresponding subset of $\mathfrak{T}^{m}_{A_{k}}$ with the standard coordinates given by $(\mathfrak{t}^{-a^{k}_{1}}\tilde{z}_{1},\dotsc,\mathfrak{t}^{-a^{k}_{m}}\tilde{z}_{m})$. It is an easy exercise to check that the restriction of the exploded structure this gives to $\mathfrak{T}^{m}_{P_{k}}\cap\mathfrak{T}^{m}_{P_{k^{\prime}}}$ is compatible with the restriction of the one coming from $\mathfrak{T}^{m}_{P_{k^{\prime}}}$, so this gives a globally well defined exploded structure.

An alternative way to describe this structure is as follows: Consider the collection of functions $\tilde{w}:=\mathfrak{t}^{c}\tilde{z}^{\alpha}$ so that on $P$, $\lfloor\tilde{w}\rfloor\leq\mathfrak{t}^{0}$. Choose some finite generating set $\{\tilde{w}_{1},\dotsc,\tilde{w}_{n}\}$ of these functions so that any other function of this type $\tilde{w}$ can be written as $\tilde{w}=\mathfrak{t}^{c}\tilde{w}_{1}^{\beta_{1}}\dotsb\tilde{w}_{n}^{\beta_{n}}$ where $\beta_{i}\in\mathbb{N}$ and $c\in[0,\infty)$. Exploded functions on $\mathfrak{T}^{m}_{P}$ can then be described as functions of the form $f(\lceil\tilde{w}_{1}\rceil,\dotsc,\lceil\tilde{w}_{n}\rceil)\mathfrak{t}^{c}\tilde{z}^{\alpha}$, where $f:\mathbb{C}^{n}\longrightarrow\mathbb{C}^{*}$ is smooth. The underlying topological space $[\mathfrak{T}^{m}_{P}]$ is equal to the closure of the set

$$\{\lceil\tilde{w}_{1}\rceil,\dotsc,\lceil\tilde{w}_{n}\rceil,\lfloor\tilde{z}_{1}\rfloor,\dotsc,\lfloor\tilde{z}_{m}\rfloor\}\subset\mathbb{C}^{n}\times P$$

The tropical part of $\mathfrak{T}^{m}_{P}$ is given by

$$\lfloor\mathfrak{T}^{m}_{P}\rfloor=P=\{\lfloor\tilde{z}_{1}\rfloor,\dotsc,\lfloor\tilde{z}_{m}\rfloor\}\subset\mathfrak{t}^{\mathbb{R}^{m}}$$

The smooth part of $\mathfrak{T}^{m}_{P}$ is given by

$$\lceil\mathfrak{T}^{m}_{P}\rceil:=\{\lceil\tilde{w}_{1}\rceil,\dotsc,\lceil\tilde{w}_{n}\rceil\}\subset\mathbb{C}^{n}$$

Simple examples of the above construction are $\mathfrak{T}^{1}_{(a,b)}$ which is equal to the subset of $\mathfrak{T}$ where $\mathfrak{t}^{a}>\lfloor\tilde{z}\rfloor>\mathfrak{t}^{b}$, and $\mathfrak{T}^{1}_{[0,b)}$ which is equal to the subset of $\mathfrak{T}^{1}_{1}$ where $\lfloor\tilde{z}\rfloor>\mathfrak{t}^{b}$. (Strictly speaking, we should be writing $\mathfrak{T}^{1}_{\mathfrak{t}^{(a,b)}}$, as the above confuses the interval $(a,b)\subset\mathbb{R}$ with $\mathfrak{t}^{(a,b)}\subset\mathfrak{t}^{\mathbb{R}}$.)

Similarly, we can construct $\mathbb{R}^{k}\times\mathfrak{T}^{m}_{P}$. This has coordinates $(x,\tilde{z})\in\mathbb{R}^{k}\times\mathfrak{T}^{m}_{P}$, has underlying topological space $[\mathbb{R}^{k}\times\mathfrak{T}^{m}_{P}]=\mathbb{R}^{k}\times[\mathfrak{T}^{m}_{P}]$, so that $[(x,\tilde{z})]=(x,[\tilde{z}])$, and has exploded functions equal to functions of the form $f(x,\lceil\tilde{w}_{1}\rceil,\dotsc,\lceil\tilde{w}_{n}\rceil)\mathfrak{t}^{c}\tilde{z}^{\alpha}$, where $f:\mathbb{R}^{k}\times\mathbb{C}^{n}\longrightarrow\mathbb{C}^{*}$ is smooth.

Given a smooth manifold $M$ and $m$ complex line bundles on $M$, we can construct the exploded fibration $M\rtimes\mathfrak{T}^{m}_{P}$ as follows: Denote by $E$ the corresponding space of $m$ $\mathbb{C}^{*}$ bundles over $M$. This has a smooth free $\left(\mathbb{C}^{*}\right)^{m}$ action. The exploded fibration $\mathfrak{T}^{m}_{P}$ also has a $\left(\mathbb{C}^{*}\right)^{m}$ action given by multiplying the coordinates $\tilde{z}_{1},\dotsc,\tilde{z}_{m}$ by the coordinates of $\left(\mathbb{C}^{*}\right)^{m}$. $M\rtimes\mathfrak{T}^{m}_{P}$ is the exploded fibration constructed by taking the quotient of $E\times\mathfrak{T}^{m}_{P}$ by the action of $\left(\mathbb{C}^{*}\right)^{m}$ by $(c,c^{-1})$. As the action of $\left(\mathbb{C}^{*}\right)^{m}$ is trivial on $\lfloor\mathfrak{T}^{m}_{P}\rfloor$, the tropical part of $M\rtimes\mathfrak{T}^{m}_{P}$ is still defined, and is equal to $P$.

Alternately, choose coordinate charts on $E$ equal to $U\times\left(\mathbb{C}^{*}\right)^{m}\subset\mathbb{R}^{n}\times\mathbb{C}^{n}$. The transition maps are of the form $(u,z_{1},\dotsc,z_{m})\mapsto(\phi(u),f_{1}(u)z_{1},\dotsc,f_{m}(u)z_{m})$. Replace these coordinate charts with $U\times\mathfrak{T}^{m}_{P}\subset\mathbb{R}^{n}\times\mathfrak{T}^{m}_{P}$, and replace the above transition maps with maps of the form $(u,\tilde{z}_{1},\dotsc,\tilde{z}_{m})\mapsto(\phi(u),f_{1}(u)\tilde{z}_{1},\dotsc,f_{m}(u)\tilde{z}_{m})$. The map to the tropical part $\lfloor M\rtimes\mathfrak{T}^{m}_{P}\rfloor:=P$ in these coordinates is given by

$$\lfloor(u,\tilde{z}_{1},\dotsc,\tilde{z}_{n})\rfloor=(\lfloor\tilde{z}_{1}\rfloor,\dotsc,\lfloor\tilde{z}_{n}\rfloor)\in P$$

It is an easy exercise to check that if $\mathfrak{B}$ is basic, the smooth part $\lceil\mathfrak{B}\rceil$ is also a stratified space, with strata equal to the manifolds $\lceil\mathfrak{B}_{i}\rceil$. The stratification of $\lceil\mathfrak{B}\rceil$ is in some sense dual to the stratification of $\lfloor\mathfrak{B}\rfloor$ in that $\lfloor\mathfrak{B}_{i}\rfloor$ is in the closure of $\lfloor\mathfrak{B}_{j}\rfloor$ if and only if $\lceil\mathfrak{B}_{j}\rceil$ is in the closure of $\lceil\mathfrak{B}_{i}\rceil$.

All our examples up until now apart from those introduced in example 2.15 have been basic. Another example of an exploded fibration that is not basic is given by quotienting $\mathbb{R}\times\mathfrak{T}$ by $(x,\tilde{z})=(x+1,\tilde{z}^{-1})$. A less pathological example of an exploded fibration that is not basic is given by the quotient of $\mathfrak{T}$ by $\tilde{z}=\mathfrak{t}^{1}\tilde{z}$. Any exploded fibration is locally basic.

To summarize the topological spaces we have involved in the case where $\mathfrak{B}$ is basic, we have the following topological spaces.

$$\begin{split}\{p\rightarrow\mathfrak{B}\}\longrightarrow&[\mathfrak{B}]\longrightarrow\lceil\mathfrak{B}\rceil\\ &\downarrow\\ &\lfloor\mathfrak{B}\rfloor\end{split}$$

1. 1.

   The smooth or topological part $\lceil\mathfrak{B}\rceil$. Any continuos morphism of $\mathfrak{B}$ to a Hausdorff topological space factors through $\lceil\mathfrak{B}\rceil$. The smooth part of a smooth manifold $M$ is $M$. The smooth part of $\mathfrak{T}^{n}$ is a single point.

2. 2.

   The tropical part $\lfloor\mathfrak{B}\rfloor$. This is a stratified integral affine space, which can be thought of as what $\mathfrak{B}$ looks like on the large scale. The tropical part of a connected smooth manifold is a single point. The tropical part of $\mathfrak{T}^{n}$ is $\mathfrak{t}^{\mathbb{R}^{n}}$.

3. 3.

   The underlying topological space $[\mathfrak{B}]$. This strange topological space is a mixture of $\lceil\mathfrak{B}\rceil$ and $\lfloor\mathfrak{B}\rfloor$. It is the topological space on which $\mathcal{E}^{\times}$ is a sheaf, and this is the topology in which $\mathfrak{B}$ is locally modeled on $\mathbb{R}^{n}\times\mathfrak{T}^{m}_{A}$.

4. 4.

   In the case that $\mathfrak{B}$ is not a manifold, the set of points $\{p\rightarrow\mathfrak{B}\}$ is not Hausdorff in any of the above topologies. We can put a topology on this set by choosing the strongest topology in which all continuous morphisms from manifolds into $\mathfrak{B}$ are continuous maps to ${\{p\rightarrow\mathfrak{B}\}}$. For example $\{p\rightarrow\mathfrak{T}^{1}_{1}\}$ is homeomorphic to the disjoint union of $[0,\infty)$ copies of $\mathbb{C}^{*}$. (When we have defined tangent spaces and metrics on exploded fibrations, this would be the topology coming from any metric.)

The following is a picture of an ‘exploded curve’ built up using coordinate charts modeled on open subsets of $\mathfrak{T}^{1}_{1}$. Note that $\lceil\mathfrak{C}\rceil$ is what we would see if we mapped $\mathfrak{C}$ to a smooth manifold. On the other hand, we would see $\lfloor\mathfrak{C}\rfloor$ if we mapped $\mathfrak{C}$ to $\mathfrak{T}^{n}$.

\psfrag{ETF1math}{$\lfloor\mathfrak{T}^{1}_{1}\rfloor:=\mathfrak{t}^{\mathbb{R}^{+}}$}\psfrag{ETF1mathC}{$\lceil\mathfrak{T}^{1}_{1}\rceil:=\mathbb{C}$}\psfrag{ETF1mathtot}{A picture of $[\mathfrak{T}^{1}_{1}]$}\psfrag{ETF1Cs}{$\mathbb{C}^{*}$}\psfrag{ETF3tot}{$[\mathfrak{C}]$}\psfrag{ETF3totl}{$\lceil\mathfrak{C}\rceil$}\psfrag{ETF3totb}{$\lfloor\mathfrak{C}\rfloor$}\includegraphics{ETF3}

The notation $\mathfrak{T}^{1}_{(0,l)}$ is that introduced in example 2.16. An equivalent way of stating the second condition is that the map $[g]:(0,l)\longrightarrow[\mathfrak{B}]$ extends to a continuous map $[0,l)\longrightarrow[\mathfrak{B}]$ if and only if $[f\circ g]:(0,l)\longrightarrow[\mathfrak{C}]$ extends to a continuous map $[0,1)\longrightarrow[\mathfrak{C}]$

Complete maps should be thought of as being analogous to proper maps. A smooth manifold is complete if and only if it is compact. An example of a non-compact exploded fibration which is complete is $\mathfrak{T}^{n}$. In the case that $\mathfrak{B}$ is basic, completeness means that the topological part $\lceil\mathfrak{B}\rceil$ is compact, and the tropical part $\lfloor\mathfrak{B}\rfloor$ is a complete stratified integral affine space.