3.1 Integral affine structure induced by a model [04PA]

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3.1 Integral affine structure induced by a model

Let $\mathscr{X}/R$ be a minimal model of $X$ ; we assume that the special fiber $\mathscr{X}_{k}=\sum_{i\in I}D_{i}$ is reduced. We consider a one-dimensional stratum $C=D_{1}\cap\ldots\cap D_{n}$ of $\mathscr{X}_{k}$ , which is therefore a smooth rational curve, and is such that $(\mathscr{X},\mathscr{X}_{k})$ is an snc pair in a formal neighbourhood of $C$ by [NXY19, Corollary 4.6]. Since $(C,\Delta_{C})$ is log Calabi–Yau, we may write its boundary as $\Delta_{C}=p_{0}+p_{\infty}$ , where $p_{0}=C\cap D_{0}$ and $p_{\infty}=C\cap D_{\infty}$ for two irreducible components $D_{0},D_{\infty}$ of $\mathscr{X}_{k}$ meeting $C$ transversally.

Following [NXY19], we write $b_{i}=-(C\cdot D_{i})$ for $i=1,\ldots,n$ ; from $C\cdot\mathscr{X}_{k}=0$ we infer $\sum_{i=1}^{n}b_{i}=2$ . The $\Star(\tau_{C})$ consists on the union of two maximal faces corresponding to the zero-dimensional strata $p_{0},p_{\infty}$ , meeting along $\tau_{C}$ . The goal of this section is to describe the integral affine structure on $\Star(\tau_{C})$ in terms of the intersection numbers $b_{i}$ ’s, with no assumption on their positivity.

Proposition 3.1.1.

Let $\rho_{\mathscr{X}}$ be the retraction associated with the model $\mathscr{X}$ , and endow $\emph{Sk}(X)$ with the $\mathbb{Z}$ -affine structure induced by $\rho_{\mathscr{X}}$ away from the codimension 2 faces of $\Sk(\mathscr{X})$ . Then $\Star(\tau_{C})$ is $\mathbb{Z}$ -affine isomorphic to the union of the simplices $<v_{0},v_{1},\ldots,v_{n}>$ and $<v_{1},\ldots,v_{n},v_{\infty}>$ in $\mathbb{R}^{n}$ where

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

Proof.

We write $b=\min_{i\leqslant n}b_{i}$ ; we assume $b$ to be negative or zero by the condition $b_{1}+\ldots+b_{n}=2$ , as the case $n=2$ and $b_{1}=b_{2}=1$ is already treated in the proof of [NXY19, prop. 5.4].

The blow-up $\mathscr{X}_{1}$ of the point $p_{\infty}$ in $\mathscr{X}$ yields a new irreducible component $D_{\infty,1}$ (we denote the strict transforms by the same letters for notational simplicity) with multiplicity $N_{\infty,1}=n+1$ , the point $p_{\infty,1}=C\cap D_{\infty,1}$ and the intersection numbers $b_{i,1}\coloneqq-(C\cdot D_{i})_{\mathscr{X}_{1}}=b_{i}+1$ . If we repeat the process $s$ times, we obtain the models $\mathscr{X}_{s}$ , the exceptional divisors $D_{\infty,s}$ with multiplicity $N_{\infty,s}=ns+1$ , the points $p_{\infty,s}=C\cap D_{\infty,s}$ and the intersection numbers $b_{i,s}\coloneqq-(C\cdot D_{i})_{\mathscr{X}_{s}}=b_{i}+s$ .

For $s=1-b$ , we have $\min_{i\leqslant n}\{b_{i,1-b}\}>0$ , and by [NXY19] the integral affine structure induced by $\mathscr{X}_{1-b}$ on $\Star(\tau_{C})$ is given by $v_{0},\ldots,v_{n}$ and

(3.1.2)

v_{\infty,1-b}=\frac{1}{n(1-b)+1}(-1,b_{1}+1-b,\ldots,b_{n-1}+1-b).

The sequence of blow-ups $\mathscr{X}_{s+1}\rightarrow\mathscr{X}_{s}$ induces (weighted) barycentric subdivisions of the faces $\tau_{p_{\infty,s}}$ with vertices such that

(3.1.3)

N_{\infty,s+1}v_{\infty,s+1}=N_{\infty,s}v_{\infty,s}+\sum_{i=1}^{n}v_{i}.

Combining Eq. 3.1.2 and Eq. 3.1.3, at each step we obtain that

v_{\infty,s}=\frac{1}{ns+1}(-1,b_{1}+s,\ldots,b_{n-1}+s),

and in particular $v_{\infty}=(-1,b_{1},\ldots,b_{n-1})$ . The proposition follows from the following lemma. ∎

Lemma 3.1.4.

Let $B=\tau_{1}\cup\tau_{2}$ be the union of two $n$ -dimensional simplices along a face of codimension one. Assume we are given a $\mathbb{Z}$ -affine structure on $B$ , compatible with those on the $\tau_{i}$ ’s.

Suppose there exists a sequence of (weighted) star subdivisions of $\tau_{1}$ such that $B^{\prime}\coloneqq\Star(\tau_{1}\cap\tau_{2})$ (with respect to this subdivision) can be embedded in $\mathbb{R}^{n}$ compatibly with the $\mathbb{Z}$ -affine structure. Then this embedding extends to $B$ , and the $\mathbb{Z}$ -affine structure on $B$ is uniquely recovered by this embedding.

Proof.

The assumptions yield two charts for the $\mathbb{Z}$ -affine structure on $B$ : the $\mathbb{Z}$ -affine subsets $B^{\prime}$ and $\tau_{1}$ . These two charts are glued along $B^{\prime}\cap\tau_{1}$ which is a simplex and thus has no non-trivial $\mathbb{Z}$ -automorphisms preserving the vertices, hence the affine structure on $B$ is uniquely determined. The set $B^{\prime}\subset\mathbb{R}^{n}$ can be obtained as the result of the same star subdivisions of a subset $\tilde{B}\subset\mathbb{R}^{n}$ , and uniqueness of the affine structure ensures a $\mathbb{Z}$ -affine isomorphism $B\simeq\tilde{B}$ . ∎

Remark 3.1.5.

Consider an irreducible component $D_{i}$ of $\mathscr{X}_{k}$ and write $\Delta_{D_{i}}=\sum_{j\neq i}D_{j}\cap D_{i}$ . By adjunction, the pair $(D_{i},\Delta_{D_{i}})$ is log Calabi–Yau, i.e. $D_{i}$ is a smooth projective variety over $k$ and $\Delta_{D_{i}}$ is a divisor such that $K_{D_{i}}+\Delta_{D_{i}}$ is trivial. By [EM21, Theorem 6.14] there exists a Lagrangian torus fibration

\phi:\mathcal{U}\rightarrow B\subseteq\Star(v_{D_{i}})\setminus W

where $\mathcal{U}$ is a symplectic tubular neighborhood of the $1$ -dimensional strata of $\Delta_{D_{i}}$ , $B$ is a retract of $\Star(v_{D_{i}})\setminus W$ , and $W$ is the union of cells of codimension $\geqslant 2$ in $\Sk(\mathscr{X})$ . The fibration $\phi$ is constructed gluing toric moment maps defined in the neighborhood of each stratum curve of $\Delta_{D_{i}}$ . Evans and Mauri compare the monodromy $T_{\phi}$ induced by $\phi$ on $B$ to the monodromy $T_{\rho_{\mathscr{X}}}$ induced by the affinoid torus fibration

{\rho_{\mathscr{X}}}:\rho_{\mathscr{X}}^{-1}(\Star(v_{D_{i}})\setminus W)\rightarrow\Star(v_{D_{i}})\setminus W

and conclude that they are dual. This means that given a loop $\gamma\in\pi_{1}(B)\simeq\pi_{1}(\Star(v_{D_{i}})\setminus W)$ , we have $T_{\rho_{\mathscr{X}}}(\gamma)=(T_{\phi}(\gamma)^{-1})^{T}$ . Thus the affine structure constructed in [NXY19] has a symplectic topological analog. The duality is due to the fact that the image of the moment maps is $M_{\mathbb{R}}$ , while the image of the tropicalization map $\val$ is in $N_{\mathbb{R}}$ .

3.1.1 Case of K3 surfaces

Let $X/K$ be a maximally degenerate $K3$ surface and let $\mathscr{X}/R$ be a minimal model of $X$ with reduced special fiber $\mathscr{X}_{k}=\sum_{i\in I}D_{i}$ . The dual complex $\mathcal{D}(\mathscr{X}_{k})$ is well-known to be a triangulated sphere, whose vertices correspond to the irreducible components of $\mathscr{X}_{k}$ .
We focus our attention to such a vertex $v_{D}$ , and hence to the corresponding irreducible component $D$ of $\mathscr{X}_{k}$ , which has boundary $\Delta_{D}\coloneqq\sum_{i=1}^{r}(D_{i}\cap D)=\sum_{i=1}^{r}C_{i}$ . Since the simple normal crossing curve $\Delta_{D}\in\lvert-K_{D}\rvert$ is an anticanonical curve by adjunction, it follows from general surface theory that $\Delta_{D}$ is a cycle of rational curves $(C_{i})_{i\leq r}$ , whose geometry is encoded by the $b_{i}=-(C_{i}\cdot D)=-(C_{i}^{2})_{D}$ . We label the curves so that for $i\leq r$ , $C_{i}\cap C_{i+1}\neq\varnothing$ , with convention $C_{r+1}=C_{1}$ .

One can associate to the pair $(D,\Delta_{D})$ a pseudo-fan, which is a singular affine structure on $\mathbb{R}^{2}$ , singular at most at $0$ . The singularity at $0$ is a way to measure the defect of $(D,\Delta_{D})$ of being toric: the affine structure affine extends smoothly at $0$ if and only $(D,\Delta_{D})$ is a toric pair [Eng18, Proposition 3.9].
The construction, as explained in [GHK15, §1.2], is the following. For each node $p_{i}=C_{i}\cap C_{i+1}$ , consider a cone $\sigma_{i}\coloneqq\mathbb{R}_{\geq 0}v_{i}+\mathbb{R}_{\geq 0}v_{i+1}\subset\mathbb{R}^{2}$ , $(v_{i},v_{i+1})$ being a basis of the lattice $\mathbb{Z}^{2}$ . The cones $\sigma_{i}$ and $\sigma_{i+1}$ are then glued to each other along $\mathbb{R}_{\geq 0}v_{i+1}$ , and the affine structure is extended through the edge by pretending that the pair $(D,\Delta_{D})$ is toric. If the pair was toric, the $\sigma_{i}$ ’s would be the maximal cones of its fan, and the relation

v_{i+2}+v_{i}=b_{i+1}v_{i+1}

would hold by Eq. 1.2.4, so that the chart $\psi_{i}:\sigma_{i}\cup\sigma_{i+1}$ that defines the $\mathbb{Z}$ -affine structure satisfies $\psi_{i}(0)=0$ , $\psi_{i}(v_{i})=(1,0)$ , $\psi_{i}(v_{i+1})=(0,1)$ and $\psi_{i}(v_{i+2})=(-1,b_{i+1})$ , and is extended by dilatation. The unions of the $\sigma_{i}$ ’s glued along the successive edge is homeomorphic to $\mathbb{R}^{2}$ , and we obtain this way an $\mathbb{Z}$ -affine structure away from the origin, extending to $0$ if and only the pair is toric.

It follows from Proposition 3.1.1 that the singular $\mathbb{Z}$ -affine structure induced by the Berkovich retraction $\rho_{\mathscr{X}}$ coincides with the one described above. We now determine the monodromy around the singularities.

Corollary 3.1.6.

Let $D$ be a component of $\mathscr{X}_{k}$ , with boundary $\Delta_{D}=\sum_{i=1}^{r}C_{i}$ . Writing $b_{i}=-(C_{i}^{2})_{D}$ , the monodromy $T_{\rho_{\mathscr{X}}}$ of the $\mathbb{Z}$ -affine structure induced by $\rho_{\mathscr{X}}$ around $v_{D}$ is given by

T_{\rho_{\mathscr{X}}}=\left(\begin{matrix}b_{r}&1\\ -1&0\end{matrix}\right)\cdot\ldots\cdot\left(\begin{matrix}b_{2}&1\\ -1&0\end{matrix}\right)\cdot\left(\begin{matrix}b_{1}&1\\ -1&0\end{matrix}\right)

with respect to the basis $(v_{D_{r}},v_{D_{1}})$ and origin $v_{D}$ .

Proof.

By Proposition 3.1.1 the integral affine structure on $\Star(\tau_{C_{i}})$ identifies $(v_{D_{i-1}},v_{D_{i}},v_{D},v_{D_{i+1}})$ with

(v_{0}=(1,0),v_{1}=(0,1),v_{2}=(0,0),v_{\infty}=(-1,b_{i})),

while on $\Star(\tau_{C_{i+1}})$ identifies $(v_{D_{i}},v_{D_{i}+1},v_{D},v_{D_{i+2}})$ with

(v_{0}=(1,0),v_{1}=(0,1),v_{2}=(0,0),v_{\infty}=(-1,b_{i+1})).

It follows that the transition map from the chart $\Star(\tau_{C_{i}})$ to $\Star(\tau_{C_{i+1}})$ of the integral affine structure on $\Star(\tau_{C_{i}})\cap\Star(\tau_{C_{i+1}})$ is given by the matrix $\left(\begin{matrix}b_{i}&1\\ -1&0\end{matrix}\right)$ . Thus, the composition of such matrices gives the monodromy around $v_{D}$ , along a loop oriented as the path connecting $v_{D_{1}},v_{D_{2}},\ldots,v_{D_{r}},v_{D_{1}}$ . ∎

Remark 3.1.7.

It is well-known (see for instance [GHK15]) that $T_{\rho_{\mathscr{X}}}=\Id$ if and only the pair $(D,\Delta_{D}=\sum_{i=1}^{r}C_{i})$ is toric, or if and only if the charge $Q$ vanishes, where

Q=\chi_{\text{top}}(D\setminus\Delta_{D})=12+\sum_{i=1}^{r}(b_{i}-3).