3.3.4 Collision of singularities [04PZ]

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3.3.4 Collision of singularities

In Section 3.3.2, the retraction $\rho^{\prime}$ depends on the choice of the points $a_{e}$ ; the same holds for the induced integral affine structure, whose singular locus consists indeed of the points $a_{e}$ . Moving a point $a_{e}$ in the interior of the edge $e$ affects the location of the singular points, but it does not change the monodromy around the point (see Section 3.3.3). We observe now, in two examples, what happens if we let a point $a_{e}$ move to a vertex of $\Sk(\mathscr{X})$ . We write $\rho=\rho_{a_{e}}$ to emphasize the dependency on the choice of singular points.

When all the points $a_{e}$ lie in the interior of the respective edges as in Section 3.3.2, $\rho_{a_{e}}=\rho_{\mathscr{X}_{ikh}}$ on $\Star(v_{j})^{\prime}$ , the induced integral affine structure is smooth at $v_{j}$ and such that

	$\displaystyle T_{\rho}(\gamma_{a_{e_{24}}})$	$\displaystyle=\left(\begin{matrix}1&0\\ b_{4,\mathscr{X}_{123}}-b_{4,\mathscr{X}_{134}}&1\end{matrix}\right)=\left(\begin{matrix}1&0\\ 4&1\end{matrix}\right)$
		$\displaystyle\text{in the basis $(v_{D_{k}},v_{D_{h}})$ and origin $v_{D_{j}}$},$
	$\displaystyle T_{\rho}(\gamma_{a_{e_{23}}})$	$\displaystyle=\left(\begin{matrix}1&0\\ b_{3,\mathscr{X}_{124}}-b_{3,\mathscr{X}_{134}}&1\end{matrix}\right)^{-1}=\left(\begin{matrix}1&0\\ 4&1\end{matrix}\right)^{-1}=\left(\begin{matrix}1&0\\ -4&1\end{matrix}\right)$
		$\displaystyle\text{in the basis $(v_{D_{h}},v_{D_{k}})$ and origin $v_{D_{j}}$}.$

When $a_{e_{24}}$ collides with the vertex $v_{j}$ , $\rho_{a_{e}}$ on $\Star(v_{j})^{\prime}$ is equal to $\rho_{\mathscr{X}_{ikj}}$ , the integral affine structure is singular at $v_{j}$ with

	$\displaystyle T_{\rho_{\mathscr{X}_{ikj}}}(\gamma_{j})$	$\displaystyle=\left(\begin{matrix}1&0\\ 4&1\end{matrix}\right)=T_{\rho}(\gamma_{a_{e_{24}}})$
		$\displaystyle\text{in the basis $(v_{D_{k}},v_{D_{h}})$ and origin $v_{D_{j}}$}.$

When both $a_{e_{24}}$ and $a_{e_{23}}$ collide with $v_{j}$ , $\rho_{a_{e}}$ on $\Star(v_{j})^{\prime}$ is equal to $\rho_{\mathscr{X}_{ijk}}$ , the integral affine structure is singular at $v_{j}$ with

	$\displaystyle T_{\rho_{\mathscr{X}_{ijk}}}(\gamma_{j})$	$\displaystyle=\left(\begin{matrix}-15&-4\\ 4&1\end{matrix}\right)$
		$\displaystyle=\left(\begin{matrix}1&-4\\ 0&1\end{matrix}\right)\left(\begin{matrix}1&0\\ 4&1\end{matrix}\right)=T_{\rho}(\gamma_{a_{e_{23}}})T_{\rho}(\gamma_{a_{e_{24}}})$
		$\displaystyle\text{in the basis $(v_{D_{k}},v_{D_{h}})$ and origin $v_{D_{j}}$}.$

The computations above suggest that the singularities and the monodromy representation induced by the non-archimedean SYZ fibration $\rho_{\mathscr{X}_{ijk}}$ can be viewed respectively as a collision of singular points and a product of monodromies induced by the $\rho_{a_{e}}$ when the $a_{e}$ ’s collide. The affine structure induced by $\rho$ turns out to be more symmetric and simpler, as all the singular points have the same monodromy and are of focus-focus type.