4.3. Gromov-Hausdorff collapse of K3 surfaces [04YP]

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4.3. Gromov-Hausdorff collapse of K3 surfaces

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 ${\it CMet}_{1}$ .

Conjecture 4.3.

The map

\Phi_{\rm alg}\colon\overline{\mathcal{F}_{2d}}^{{\rm Sat}}\to{\it CMet}_{1}

given above is continuous.

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.

Theorem 4.4.

The map $\Phi_{\rm alg}$ is continuous on $\overline{\mathcal{F}_{2d}}^{\rm Sat}\setminus(\bigcup_{p}\mathcal{F}_{2d}(p))$ . It is continuous also when restricted to the boundary $\partial{\overline{\mathcal{F}_{2d}}^{\rm Sat}}=\overline{\mathcal{F}_{2d}}^{\rm Sat}\setminus\mathcal{F}_{2d}$ .

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.