We define an affine structure with singularities on $B=\mathbb{R}^{2}$. Let $\Delta=\{0\}$ and let $(x_{1},x_{2})$ be the standard coordinates on $B$. As the covering $\{U_{i}\}$ of $B_{0}=\mathbb{R}^{2}-\Delta$ we take the following two sets

$$U_{1}=\mathbb{R}^{2}-\{x_{2}=0\ \text{and}\ x_{1}\geq 0\},$$

$$U_{2}=\mathbb{R}^{2}-\{x_{2}=0\ \text{and}\ x_{1}\leq 0\}.$$

Denote by $H^{+}$ the set $\{x_{2}>0\}$ and by $H^{-}$ the set $\{x_{2}<0\}$. Let $T$ be the matrix

$$T=\left(\begin{array}[]{cc}1&0\\ 1&1\end{array}\right).$$

(6)

The coordinate maps $\phi_{1}$ and $\phi_{2}$ on $U_{1}$ and $U_{2}$ are defined as follows

	$\displaystyle\phi_{1}$	$\displaystyle=$	$\displaystyle\mathrm{Id}$
	$\displaystyle\phi_{2}$	$\displaystyle=$	$\displaystyle\left\{\begin{array}[]{ll}\mathrm{Id}&\text{on}\ \bar{H}^{+}\cap U_{2},\\ (T^{-1})^{t}&\text{on}\ H^{-}\end{array}\right.$

The atlas $\mathscr{A}=\{U_{i},\phi_{i}\}_{i=1,2}$ is clearly an affine structure on $B_{0}$. It is easy to check that given a point $b\in B_{0}$, we can chose a basis of $T^{\ast}_{b}B_{0}$ with respect to which the holonomy representation $\rho^{\ast}$ sends the anti-clockwise oriented generator of $\pi_{1}(B_{0})$ to the matrix $T$.

Let $I\subseteq\mathbb{R}$ be an open interval. Consider $B=\mathbb{R}^{2}\times I$ and $\Delta=\{0\}\times I$. On $B_{0}=(\mathbb{R}^{2}-0)\times I$ we take the product affine structure between the affine structure on $\mathbb{R}^{2}-0$ described in the previous example and the standard affine structure on $I$.

In the previous example the discriminant locus $\Delta$ was a straight line. We can slightly perturb $\Delta$ so that it becomes a smooth curve. More precisely, let $B=\mathbb{R}^{2}\times I$ as before and consider a smooth function $\tau:I\rightarrow\mathbb{R}$. Let

$$\Delta_{\tau}=\{(\tau(s),0,s),s\in I\}\subset B$$

and define a covering $\{U_{i}\}$ of $B_{0}=B-\Delta_{\tau}$ to be

$$U_{1}=(\mathbb{R}^{2}\times I)-\{(x_{1},0,s)\ |\ x_{1}\geq\tau(s)\},$$

$$U_{2}=(\mathbb{R}^{2}\times I)-\{(x_{1},0,s)\ |\ x_{1}\leq\tau(s)\}.$$

Now let $H^{+}=\{x_{2}>0\}$ and $H^{-}=\{x_{2}<0\}$. Take the following matrix

$$T=\left(\begin{array}[]{ccc}1&0&0\\ 1&1&0\\ 0&0&1\end{array}\right)$$

and define maps $\phi_{j}$ on $U_{j}$ to be

	$\displaystyle\phi_{1}$	$\displaystyle=$	$\displaystyle\mathrm{Id}$
	$\displaystyle\phi_{2}$	$\displaystyle=$	$\displaystyle\left\{\begin{array}[]{ll}\mathrm{Id}&\text{on}\ \bar{H}^{+}\cap U_{2},\\ (T^{-1})^{t}&\text{on}\ H^{-}.\end{array}\right.$

Clearly $\mathscr{A}=\{U_{i},\phi_{i}\}_{i=1,2}$ defines an affine structure on $B_{0}=B-\Delta_{\tau}$. When $\tau=0$, this example coincides with the previous one. Notice that the curve $(\tau(s),0,s)$ is contained inside the $2$-plane $\{x_{2}=0\}$, which can be viewed as an integral surface of the distribution spanned by the vectors in $TB_{0}$ which are invariant with respect to the holonomy representation $\rho$ on $TB_{0}$. Two different curves give non-isomorphic singular affine structures, unless the curves can be taken one into the other via an integral affine transformation.

Take $B=\mathbb{R}\times\mathbb{R}^{2}$, with coordinates $(x_{1},x_{2},x_{3})$ and identify $\mathbb{R}^{2}$ with $\{0\}\times\mathbb{R}^{2}$. Inside $\mathbb{R}^{2}$ consider the cone over three points:

$$\Delta=\{x_{2}=0,\,x_{3}\leq 0\}\cup\{x_{3}=0,\,x_{2}\leq 0\}\cup\{x_{2}=x_{3},\,x_{3}\geq 0\}.$$

Now define closed sets in $B$

	$\displaystyle R$	$\displaystyle=$	$\displaystyle\mathbb{R}\times\Delta,$
	$\displaystyle R^{+}$	$\displaystyle=$	$\displaystyle\mathbb{R}_{\geq 0}\times\Delta,$
	$\displaystyle R^{-}$	$\displaystyle=$	$\displaystyle\mathbb{R}_{\leq 0}\times\Delta,$

and consider the following cover $\{U_{i}\}$ of $\mathbb{R}^{3}-\Delta$:

	$\displaystyle U_{1}$	$\displaystyle=$	$\displaystyle\mathbb{R}^{3}-R^{+},$
	$\displaystyle U_{2}$	$\displaystyle=$	$\displaystyle\mathbb{R}^{3}-R^{-}.$

It is clear that $U_{1}\cap U_{2}$ has the following three connected components

	$\displaystyle V_{1}$	$\displaystyle=$	$\displaystyle\{x_{2}<0,\ x_{3}<0\},$
	$\displaystyle V_{2}$	$\displaystyle=$	$\displaystyle\{x_{2}>0,\ x_{2}>x_{3}\},$
	$\displaystyle V_{3}$	$\displaystyle=$	$\displaystyle\{x_{3}>0,\ x_{3}>x_{2}\}.$

Take two matrices

$$T_{1}=\left(\begin{array}[]{ccc}1&1&0\\ 0&1&0\\ 0&0&1\end{array}\right),\ \ \ T_{2}=\left(\begin{array}[]{ccc}1&0&-1\\ 0&1&0\\ 0&0&1\end{array}\right).$$

(9)

Now on $U_{1},U_{2}$ we define coordinate maps $\phi_{1}$, $\phi_{2}$ as follows

	$\displaystyle\phi_{1}$	$\displaystyle=$	$\displaystyle\I,$
	$\displaystyle\phi_{2}$	$\displaystyle=$	$\displaystyle\left\{\begin{array}[]{ll}\I&\text{on}\ \bar{V}_{1}\cap U_{2},\\ T_{1}^{-1}&\text{on}\ \bar{V}_{2}\cap U_{2}\\ T_{2}&\text{on}\ \bar{V}_{3}\cap U_{2}\end{array}\right.$

Again we see that $\mathscr{A}=\{U_{i},\phi_{i}\}_{i=1,2}$ gives an affine structure on $B_{0}=\mathbb{R}^{3}-\Delta$. One can compute that given a point $b\in B_{0}$ and closed paths $g_{1}$, $g_{2}$ and $g_{3}$ as in Figure 3, we can choose a basis of $T^{\ast}_{b}B_{0}$ with respect to which the holonomy matrices satisfy $\rho^{\ast}(g_{j})=(T_{j}^{-1})^{t}$ for $j=1,2,3$.

In the previous example, $\Delta$ was a graph with three edges meeting in one vertex. All three edges were straight lines. In the spirit of Example 3.9 we can perturb each edge of $\Delta$ to a smooth curve starting at the vertex. Each straight edge of the previous example is contained in a $2$-plane which is an integral plane of the distribution spanned by the vectors which are invariant with respect to the holonomy around that edge. For example, consider the edge $E_{1}=\{x_{1}=x_{2}=0,x_{3}\leq 0\}$ of $\Delta$. Then $E_{1}$ is contained inside the half plane, $P_{1}=\{x_{2}=0,x_{3}\leq 0\}$, whose tangent vectors are $T_{1}$ invariant, where $T_{1}=\rho(g_{1})$ is the holonomy of $T_{B_{0}}$ with respect to $E_{1}$. An analogous thing happens with the other two edges. The union of all three half planes gives $R$. The new perturbed edges, $E_{j}^{\prime}$, must be curves inside the half planes $P_{j}$. More precisely, let $\tau$ be a function on $\Delta$ which is the restriction of a smooth function defined on an open neighborhood of $\Delta$, such that $\tau(0)=0$. If we let $R$ be as in the previous example, define

	$\displaystyle\Delta_{\tau}$	$\displaystyle=$	$\displaystyle\{(\tau(q),q)\in\mathbb{R}\times\Delta\}$
	$\displaystyle R^{+}$	$\displaystyle=$	$\displaystyle\{(x_{1},q)\in\mathbb{R}\times\Delta\ |x_{1}\geq\tau(q)\}$
	$\displaystyle R^{-}$	$\displaystyle=$	$\displaystyle\{(x_{1},q)\in\mathbb{R}\times\Delta\ |x_{1}\leq\tau(q)\}$

Now charts $\mathscr{A}=\{U_{i},\phi_{i}\}_{i=1,2}$ on $B-\Delta_{\tau}$ can be defined like in the previous example, but with these new definitions of $R^{+}$ and $R^{-}$. It is clear that $(B,\Delta_{\tau},\mathscr{A})$ defines an affine manifold with singularities. Two different choices of functions $\tau$ define non-isomorphic integral affine manifolds with singularities, unless their graphs inside $R$ can be mapped one to the other via an integral affine map.

Let $B$ and $\Delta$ be as in Example 3.10. Clearly, $\mathbb{R}^{2}-\Delta$ has three connected components, which we denote $C_{1},C_{2}$ and $C_{3}$. Let $\bar{C}_{j}=C_{j}\cup\partial C_{j}$. Viewing $\mathbb{R}^{2}$ embedded in $B$ as $\{0\}\times\mathbb{R}^{2}$, consider the following three open subsets of $B_{0}$:

	$\displaystyle U_{1}$	$\displaystyle=$	$\displaystyle\mathbb{R}^{3}-(\bar{C}_{2}\cup\bar{C}_{3}),$
	$\displaystyle U_{2}$	$\displaystyle=$	$\displaystyle\mathbb{R}^{3}-(\bar{C}_{1}\cup\bar{C}_{3}),$
	$\displaystyle U_{3}$	$\displaystyle=$	$\displaystyle\mathbb{R}^{3}-(\bar{C}_{1}\cup\bar{C}_{2}).$

Let

	$\displaystyle V^{+}$	$\displaystyle=$	$\displaystyle\{x_{1}>0\},$
	$\displaystyle V^{-}$	$\displaystyle=$	$\displaystyle\{x_{1}<0\}.$

Clearly $U_{i}\cap U_{j}=V^{+}\cup V^{-}$ when $i\neq j$. If $T_{1}$ and $T_{2}$ are as in (9), define the following coordinate charts on $U_{1}$, $U_{2}$, $U_{3}$ respectively:

	$\displaystyle\phi_{1}$	$\displaystyle=$	$\displaystyle\I,$
	$\displaystyle\phi_{2}$	$\displaystyle=$	$\displaystyle\left\{\begin{array}[]{ll}(T_{1}^{-1})^{t}&\text{on}\ \bar{V}^{+}\cap U_{2}\\ \I&\text{on}\ \bar{V}^{-}\cap U_{2}\end{array}\right.$
	$\displaystyle\phi_{3}$	$\displaystyle=$	$\displaystyle\left\{\begin{array}[]{ll}\I&\text{on}\ \bar{V}^{+}\cap U_{3}\\ (T_{2}^{-1})^{t}&\text{on}\ \bar{V}^{-}\cap U_{3}\end{array}\right.$

We can check that the affine structure defined by these charts is such that, for fixed $b\in B_{0}$, there exists a basis of $T^{\ast}_{b}B_{0}$ with respect to which the holonomy representation is such that $\rho^{\ast}(g_{j})=T_{j}$, where $g_{j}$ are as in Figure 3. In particular, the holonomy is given by the inverse transpose matrices of the holonomy in the previous example.

Again, we can perturb the above example by replacing the straight edges of $\Delta$ with smooth curves starting at the origin. This time these curves have to be contained inside $\{0\}\times\mathbb{R}^{2}$, which is the integral surface (containing $\Delta$) of the distribution spanned by the $\rho$-holonomy invariant vectors in $TB_{0}$. The perturbed $\Delta$, which we could denote $\Delta_{\tau}$, still separates $\mathbb{R}^{2}$ in three connected components $C_{1},C_{2}$ and $C_{3}$. Then the definition of the affine structure carries through just like in the previous example and we denote it by $\mathscr{A}_{\tau}$.