3.2 Integral affine structure induced by combining several models [04PK]

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3.2 Integral affine structure induced by combining several models

We start with a general definition.

Definition 3.2.1.

Let $\tau$ be a simplex of dimension $m$ and consider the first barycentric subdivision $\tau^{\prime}$ of $\tau$ . For each vertex $v$ of $\tau$ , we denote the star of $v$ in $\tau^{\prime}$ by $\Star(v)^{\prime}$ and define $\Gamma_{m-1}$ to be the polyhedral complex of dimension $m-1$ given by

\Gamma_{m-1}\coloneqq\tau\setminus\bigcup_{v\in\tau}\Star(v)^{\prime}\subset\tau.

For instance, if $m=2$ , $\Gamma_{1}$ is the union of the three line segments joining the barycenter of the triangle to the barycenters of the edges.

We return to the setting of Section 3, that is, let $X/K$ be a smooth $n$ -dimensional maximally degenerate Calabi–Yau variety. Assume we are given two minimal models $\mathscr{X}$ , $\mathscr{X}^{\prime}$ of $X$ such that $\mathcal{D}(\mathscr{X}_{k})=\mathcal{D}(\mathscr{X}^{\prime}_{k})$ , so that $\Sk(\mathscr{X})=\Sk(\mathscr{X}^{\prime})=\Sk(X)$ , not only as sets but also with the same triangulation. We fix an ordered labelling $(1,\ldots,s)$ of the vertices of $\Sk(X)$ , equivalently of the irreducible components of the special fiber of $\mathscr{X}$ (resp. $\mathscr{X}^{\prime}$ ).
Fix a codimension 1 face $\tau$ of $\mathcal{D}(\mathscr{X}_{k})=\mathcal{D}(\mathscr{X}^{\prime}_{k})$ , with vertices $v_{i_{1}},\ldots,v_{i_{n}}$ . We write $C$ (resp. $C^{\prime}$ ) for the corresponding strata curves of $\mathscr{X}$ (resp. $\mathscr{X}^{\prime}$ ), and $D_{i_{l}}$ (resp. $D^{\prime}_{i_{l}}$ ), $l=1,\ldots,n$ the corresponding components. We then have $C=D_{i_{1}}\cap\ldots\cap D_{i_{n}}$ , and similarly for $C^{\prime}$ . We write

b_{i_{l}}=-(C\cdot D_{i_{l}})_{\mathscr{X}}

the intersection number computed inside $\mathscr{X}$ , and similarly:

b^{\prime}_{i_{l}}=-(C^{\prime}\cdot D^{\prime}_{i_{l}})_{\mathscr{X}^{\prime}}

for $l=1,\ldots,n$ . The $(n-1)$ -dimensional face $\tau$ is contained in two maximal faces $\tau_{p_{0}}$ and $\tau_{p_{\infty}}$ of $\mathcal{D}(\mathscr{X}_{k})$ , since the boundary of $C$ in $(\mathscr{X},\mathscr{X}_{k})$ consists of two strata points $p_{0}=C\cap D_{i_{0}}$ and $p_{\infty}=C\cap D_{i_{\infty}}$ ; we assume $i_{0}<i_{\infty}$ . We set $\mathcal{S}\coloneqq\tau_{p_{0}}\cup\tau_{p_{\infty}}\subset\Sk(X)$ , and $\Gamma\coloneqq\Gamma_{n-2}\subset\tau_{C}$ as in Definition 3.2.1.
Given two vertices $v_{i_{m}},v_{i_{m^{\prime}}}$ of $\tau_{C}$ , with corresponding components $D_{i_{m}}$ and $D_{i_{m^{\prime}}}$ of $\mathscr{X}_{k}$ containing $C$ , we assume $i_{m}<i_{m^{\prime}}$ and construct a loop $\gamma$ as follows:

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$\gamma$ is contained in $\mathcal{S}\setminus\Gamma$ ;
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$\gamma$ goes around the segment joining the barycenter of $\tau_{C}$ with the barycenter of the edge between $v_{i_{m}}$ and $v_{i_{m^{\prime}}}$ ;
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$\gamma$ has an orientation induced by the fixed ordered labelling on the vertices of $\Sk(X)$ in the following way: in $\mathcal{S}\setminus\Gamma$ , $\gamma$ is homotopy equivalent to the closed path given by the edges which connect in order

$v_{D_{i_{0}}},v_{D_{i_{m}}},v_{D_{i_{\infty}}},v_{D_{i_{m^{\prime}}}}.$

Figure 1: *

Two examples of loops $\gamma$ in the case $n=3$ and $C=D_{2}\cap D_{3}\cap D_{4}$

Suppose we are given a retraction $\rho:X^{\an}\longrightarrow\Sk(X)$ such that

\rho=\begin{cases}\rho_{\mathscr{X}}&\textrm{ over }U_{i_{m}}\coloneqq\textrm{Int}(\tau_{p_{0}})\cup\textrm{Int}(\tau_{p_{\infty}})\cup\Star(v_{i_{m}})^{\prime}\\ \rho_{\mathscr{X}^{\prime}}&\textrm{ over }U_{i_{m^{\prime}}}\coloneqq\textrm{Int}(\tau_{p_{0}})\cup\textrm{Int}(\tau_{p_{\infty}})\cup\Star(v_{i_{m^{\prime}}})^{\prime}.\end{cases}

Proposition 3.2.2.

The monodromy along the loop $\gamma$ , of the $\mathbb{Z}$ -affine structure induced by $\rho$ on $U_{i_{m}}\cup U_{i_{m^{\prime}}}$ is

(3.2.3)

T_{\rho}(\gamma)=\left(\begin{matrix}1&0&0&\ldots&0\\ b_{i_{1}}-b^{\prime}_{i_{1}}&1&0&\ldots&0\\ b_{i_{2}}-b^{\prime}_{i_{2}}&0&1&\ldots&0\\ \vdots&\vdots&&\ddots&\\ b_{i_{n-1}}-b^{\prime}_{i_{n-1}}&0&0&\ldots&1\end{matrix}\right)

with respect to the basis $(v_{D_{i_{0}}},v_{D_{i_{1}}},\ldots,v_{D_{i_{n-1}}})$ and origin $v_{D_{i_{n}}}$ .

Proof.

We need to compute the parallel transport of the vectors

(u_{0},\ldots,u_{n-1})\coloneqq(v_{D_{i_{0}}}-v_{D_{i_{n}}},v_{D_{i_{1}}}-v_{D_{i_{n}}},\ldots,v_{D_{i_{n-1}}}-v_{D_{i_{n}}})

along the loop $\gamma$ . By Proposition 3.1.1 the $\mathbb{Z}$ -affine structure on $\Star(\tau_{C})$ induced by $\rho_{\mathscr{X}}$ is described by the chart which has the following vertices:

$v_{0}=(1,0,\ldots,0)$ , $v_{1}=(0,1,\ldots,0),\,\ldots,\,v_{n}=0$ and $v_{\infty}=(-1,b_{i_{1}},\ldots,b_{i_{n-1}});$

while the $\mathbb{Z}$ -affine structure induced by $\rho_{\mathscr{X}^{\prime}}$ is given by:

$v^{\prime}_{0}=(1,0,\ldots,0)$ , $v^{\prime}_{1}=(0,1,\ldots,0),\,\ldots,\,v^{\prime}_{n}=0$ and $v^{\prime}_{\infty}=(-1,b^{\prime}_{i_{1}},\ldots,b^{\prime}_{i_{n-1}}).$

Moreover, the vectors $u_{l}$ correspond to the vectors $v_{l}$ (resp. $v^{\prime}_{l}$ ) in the chart for $\rho_{\mathscr{X}}$ (resp. $\rho_{\mathscr{X}^{\prime}}$ ). We now have $v_{0}=-v_{\infty}+\sum_{l=1}^{n-1}b_{i_{l}}v_{l}$ , so that the vectors we are transporting are written on $\tau_{p_{\infty}}$

(-v_{\infty}+\sum_{l=1}^{n-1}b_{i_{l}}v_{l}\,,v_{1},\ldots,v_{n-1})

in the chart for $\rho_{\mathscr{X}}$ . These are thus mapped to the tuple $(-v^{\prime}_{\infty}+\sum_{l=1}^{n-1}b_{i_{l}}v^{\prime}_{l},v^{\prime}_{1},\ldots,v^{\prime}_{n-1})$ by the chart for $\rho_{\mathscr{X}^{\prime}}$ . We now transport back across $\tau_{C}$ in the chart for $\rho_{\mathscr{X}^{\prime}}$ , to get the tuple of vectors

(-v^{\prime}_{0}-\sum_{l=1}^{n-1}b^{\prime}_{i_{l}}v^{\prime}_{l}+\sum_{l=1}^{n-1}b_{i_{l}}v^{\prime}_{l}\,,v^{\prime}_{1},\ldots,v^{\prime}_{n-1})

according to the relation $-v^{\prime}_{\infty}=-v^{\prime}_{0}-\sum_{l=1}^{n-1}b^{\prime}_{i_{l}}v^{\prime}_{l}$ . We now see that after parallel transport the vectors $(u_{0},\ldots,u_{n-1})$ have changed to

(u_{0}+\sum_{l=1}^{n-1}(b_{i_{l}}-b^{\prime}_{i_{l}})u_{l}\,,u_{1},\ldots,u_{n-1}),

hence the formula Eq. 3.2.3 for the monodromy matrix. ∎

3.2.1 Case of K3 surfaces

We focus on the case of a maximally degenerate $K3$ surface $X/K$ . We have $C=D_{i_{m}}\cap D_{i_{m^{\prime}}}$ , $\Gamma=\{a\}$ is a point in the interior of $\tau_{C}$ and $\gamma$ is a loop around $a$ , oriented as the path joining in order $v_{D_{i_{0}}},v_{D_{i_{m}}},v_{D_{i_{\infty}}},v_{D_{i_{m^{\prime}}}}$ . We assume we have a retraction $\rho:X^{\an}\longrightarrow\Sk(X)$ such that

\rho=\begin{cases}\rho_{\mathscr{X}}&\textrm{ over }\textrm{Int}(\tau_{p_{0}})\cup\textrm{Int}(\tau_{p_{\infty}})\cup[v_{i_{m}},a)\\ \rho_{\mathscr{X}^{\prime}}&\textrm{ over }\textrm{Int}(\tau_{p_{0}})\cup\textrm{Int}(\tau_{p_{\infty}})\cup[v_{i_{m^{\prime}}},a)\end{cases}

where $[v_{i_{\cdot}},a)$ is the part of the edge $\tau_{C}$ joining the vertex to $a$ , but not including $a$ . Then Proposition 3.2.2 may be rewritten as follows.

Corollary 3.2.4.

The monodromy along the loop $\gamma$ , of the $\mathbb{Z}$ -affine structure induced by $\rho$ on $\Star(\tau_{C})\setminus\{a\}$ , is

(3.2.5)

T_{\rho}(\gamma)=\left(\begin{matrix}1&0\\ b_{i_{m}}-b^{\prime}_{i_{m}}&1\end{matrix}\right)

with respect to the basis $(v_{D_{i_{0}}},v_{D_{i_{m}}})$ and origin $v_{D_{i_{m^{\prime}}}}$ .