Let $\mathcal{F}_{2d}$ be the moduli space of polarized K3 surfaces of degree $2d$ possibly with ADE singularities. Its structure is known as follows. Let $\Lambda_{\rm K3}:=E_{8}(-1)^{\oplus 2}\oplus U^{\oplus 3}$ be the K3 lattice and fix a primitive vector $\lambda_{2d}$ with $(\lambda_{2d},\lambda_{2d})=2d$ and $\Lambda_{2d}:=\lambda_{2d}^{\perp}$. The complex manifold

$$\Omega(\Lambda_{2d}):=\{[w]\in\mathbb{P}(\Lambda_{2d}\otimes\mathbb{C})\mid(w,w)=0,\ (w,\bar{w})>0\}.$$

has two connected components. We choose one component and denote by $\mathcal{D}_{\Lambda_{2d}}$. Let $O(\Lambda_{\rm K3})$ denote the isomorphism group of the lattice $\Lambda_{\rm K3}$ preserving the bilinear form and set

$\displaystyle\tilde{O}(\Lambda_{2d}):=\{g|_{\Lambda_{2d}}:g\in O(\Lambda_{\rm K3}),\,g(\lambda_{2d})=\lambda_{2d}\}.$

The group $\tilde{O}(\Lambda_{2d})$ naturally acts on $\Omega(\Lambda_{2d})$. We define $\tilde{O}^{+}(\Lambda_{2d})$ to be the index two subgroup of $\tilde{O}(\Lambda_{2d})$ consisting of the elements preserving each connected component of $\Omega(\Lambda_{2d})$. Then it is well-known that

$\displaystyle\mathcal{F}_{2d}\simeq\tilde{O}^{+}(\Lambda_{2d})\backslash\mathcal{D}_{\Lambda_{2d}}\simeq\tilde{O}(\Lambda_{2d})\backslash\Omega(\Lambda_{2d}).$

Let $\overline{\mathcal{F}_{2d}}^{{\rm Sat},\tau_{\rm ad}}$ (or simply $\overline{\mathcal{F}_{2d}}^{{\rm Sat}}$ in our papers) be the Satake compactification of $\mathcal{F}_{2d}$ corresponding to the adjoint representation of $O(2,19)$. It decomposes as

$$\overline{\mathcal{F}_{2d}}^{{\rm Sat}}=\mathcal{F}_{2d}\sqcup\bigcup_{l}\mathcal{F}_{2d}(l)\sqcup\bigcup_{p}\mathcal{F}_{2d}(p),$$

where $l$ runs over one-dimensional isotropic subspaces of $\Lambda_{2d}\otimes\mathbb{Q}$, and $p$ runs over two-dimensional isotropic subspaces of $\Lambda_{2d}\otimes\mathbb{Q}$. Also, we simply define the tropical geometric compactification of $\mathcal{F}_{2d}$ as this $\overline{\mathcal{F}_{2d}}^{\rm Sat}$. The boundary component $\mathcal{F}_{2d}(l)$ is given as

$$\mathcal{F}_{2d}(l)=\{v\in(l^{\perp}/l)\otimes\mathbb{R}\mid(v,v)>0\}/\sim.$$

Here $v\sim v^{\prime}$ if $g\cdot v=cv^{\prime}$ for some $g\in\tilde{O}^{+}(\Lambda_{2d})$ and $c\in\mathbb{R}^{\times}$. We have $\mathcal{F}_{2d}(l)=\mathcal{F}_{2d}(l^{\prime})$ if $g\cdot l=l^{\prime}$ for some $g\in\tilde{O}^{+}(\Lambda_{2d})$ and $\mathcal{F}_{2d}(l)\cap\mathcal{F}_{2d}(l^{\prime})=\emptyset$ if otherwise. Since $(l^{\perp}/l)\otimes\mathbb{R}$ has signature $(1,18)$, there is an isomorphism

$$\{v\in(l^{\perp}/l)\otimes\mathbb{R}\mid(v,v)>0\}/\mathbb{R}^{\times}\\ \simeq O(1,18)/O(1)\times O(18)$$

and hence $\mathcal{F}_{2d}(l)$ is an arithmetic quotient of $O(1,18)/O(1)\times O(18)$. The other component $\mathcal{F}_{2d}(p)$ is a point and $\mathcal{F}_{2d}(p)=\mathcal{F}_{2d}(p^{\prime})$ if and only if $g\cdot p=p^{\prime}$ for some $g\in\tilde{O}^{+}(\Lambda_{2d})$. Therefore, if we take representatives of $l$ and $p$ from each equivalence class, we get a finite decomposition:

$$\overline{\mathcal{F}_{2d}}^{{\rm Sat}}=\mathcal{F}_{2d}\sqcup\bigsqcup_{l}\mathcal{F}_{2d}(l)\sqcup\bigsqcup_{p}\mathcal{F}_{2d}(p).$$

In our paper, what we mean by tropical polarized K3 surface is a topological space $B$ homeomorphic to the sphere $S^{2}$, with an affine structure away from certain finite points ${\rm Sing}(B)$, with a metric which is Mongé-Ampere metric $g$ with respect to the affine structure on $B\setminus{\rm Sing}(B)$. Studies of such object as tropical version of K3 surfaces are pioneered in well-known papers of Gross-Wilson [GW00] and Kontsevich-Soibelman [KS04].

Here we assign such tropical K3 surface to each point in the boundary component $\mathcal{F}_{2d}(l)$ as follows. Let $l$ be an oriented one-dimensional isotropic subspace of $\Lambda_{2d}\otimes\mathbb{Q}$. Write $e$ for the primitive element of $l$ such that $\mathbb{R}_{>0}e$ agrees with the orientation of $l$. Take a vector $v\in(l^{\perp}/l)\otimes\mathbb{R}$ such that $(v,v)>0$. Write $[e,v]$ for the corresponding point in $\mathcal{F}_{2d}(l)$. Then there exists a (not necessarily projective) K3 surface $X$ and a marking $\alpha_{X}\colon H^{2}(X,\mathbb{Z})\to\Lambda$ with

• •

  $\alpha_{X}(H^{2,0})\subset\mathbb{R}\lambda+\sqrt{-1}\mathbb{R}v$,

• •

  $\alpha_{X}^{-1}(e)$ is in the closure of Kähler cone.

The pair $(X,\alpha_{X})$ is unique up to isomorphisms.

Let $L$ be a line bundle on $X$ such that $\alpha_{X}([L])=e$. Then we get an elliptic fibration $f:X\to B(\simeq\mathbb{P}^{1})$. Take a holomorphic volume form $\Omega$ on $X$ such that $\alpha_{X}([\mathop{\mathrm{Re}}\nolimits\Omega])=\lambda$. The map $f$ is a Lagrangian fibration with respect to the symplectic form $\mathop{\mathrm{Re}}\nolimits\Omega$. Hence it gives an affine manifold structure on $B\setminus\Delta$, where $\Delta$ denotes the finite set of singular points. Similarly, the imaginary part $\mathop{\mathrm{Im}}\nolimits\Omega$ gives another affine manifold structure on $B\setminus\Delta$.

We endow the base space $B$ with the McLean metric on the base $B$ ([ML98]), where we regard $f$ as special Lagrangian fibration after hyperKähler rotation. A straightforward calculation shows that this coincides with the “special Kähler metric” $g_{\it sp}$ introduced and studied in [DW96, Hit96, Freed99] and appears as the metric on $\mathbb{P}^{1}$ in [GTZ16]. We rescale the metric to make its diameter $1$ and denote this obtained tropical K3 surface by $\Phi_{\rm alg}([e,v])$.

For a point in $\mathcal{F}_{2d}$ we have a corresponding polarized K3 surface $(X,L)$, equipped with a natural Ricci-flat metric. For $[e,v]\in\mathcal{F}_{2d}(l)$ we defined in a previous section $\Phi_{\rm alg}([e,v])$. For a point in $\mathcal{F}_{2d}(p)$ we assign a (one-dimensional) segment, which we denote by $\Phi_{\rm alg}(\mathcal{F}_{2d}(p))$. Let us normalize these metric spaces so that their diameters are one. We thus obtained a map $\Phi_{\rm alg}\colon\overline{\mathcal{F}_{2d}}^{{\rm Sat}}\to\{\text{compact metric spaces with diameter one}\}$. Here, we associate Gromov-Hausdorff distance to the right hand side (target space) and denote it by ${\it CMet}_{1}$.

We would like to simply set the tropical geometric compactification of $\mathcal{F}_{2d}$ as $\overline{\mathcal{F}_{2d}}^{{\rm T}}:=\overline{\mathcal{F}_{2d}}^{{\rm Sat}}$. Indeed, if Conjecture 4.3 holds, we get a continuous map $\overline{\mathcal{F}_{2d}}^{{\rm Sat}}\to\overline{\mathcal{F}_{2d}}^{\rm GH}$ and we also observe that each $\mathcal{F}_{2d}(l)$ encodes affine structure of the limit tropical K3 surface as well. (This answers a question of Prof. B. Siebert in 2016 to the first author, regarding if one can associate tropical affine structure to limit of any collapsing sequence). So far, we have partially confirmed the conjecture. The case of ($A_{1}$-singular flat) Kummer surfaces, with $3$-dimensional moduli, are easily reduced to [Od16]. More generally, we have proved the following. In particular, the conjecture 4.3 holds at least away from finite points.

The proof of the former half of the statements involves some symmetric space theory, hyperKähler geometry, algebraic geometry of moduli, and a priori analytic estimates. The estimates heavily depends on [Tos10, GW00, GTZ13, GTZ16, TZ17] and their extensions. One nontrivial part of the extension is, for instance, to make many of the $C^{2}$-estimations in op.cit following methods of [Yau78] locally uniform with respect to a family of elliptic K3 surfaces even along degenerations to orbifolds.

During our work, we learnt that Kenji Hashimoto, Yuichi Nohara, Kazushi Ueda [HNU] also studied the Gromov-Hausdorff collapses along certain $2$-dimensional subvariety of $\mathcal{F}_{2d}$, i.e., the moduli of $E_{8}^{\oplus 2}\oplus U(\oplus\langle-2\rangle)$-polarized K3 surfaces. Moereover, a result of Hashimoto and Ueda [HU] implies that the restriction of $\Phi_{\rm alg}$ to the boundary is a generically two-to-one map. We appreciate their gentle discussion with us.

Theorem 4.4 (resp., Conjecture 4.3) combined with Proposition 2.2 determines the Gromov-Hausdorff limits of Type III (resp., Type II) one parameter family of Ricci-flat algebraic K3 surfaces, which solves a conjecture of Kontsevich-Soibelman [KS04, Conjecture 1], Todorov, and Gross-Wilson (cf., e.g., [Gross12, Conjecture 6.2]) in the K3 surfaces case.

In the next section, we discuss collapsing of general Kähler K3 surfaces, which are not necessarily algebraic.