4.2. Tropical K3 surfaces [04YL]

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4.2. Tropical K3 surfaces

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 $\Phi_{\rm alg}([e,v])$ .

Remark 4.1.

Recall the concepts of the class of metric (metric class) and the radiance obstruction of Mongé-Ampére manifolds $B$ with singularities. They are introduced in [KS04] and discussed in [GS06] in more details. We denote them by $k(B)\in H^{1}(B,i_{*}\tilde{\Lambda}^{\vee}\otimes\mathbb{R})$ and $c(B)\in H^{1}(B,i_{*}\Lambda)$ , respectively. Here, $\Lambda$ is the affine structure as a $\mathbb{Z}^{\it dim(B)}$ -local system in tangent bundle $T(B\setminus\Delta)$ , $-^{\vee}$ denotes $-$ ’s dual local system, $\tilde{\Lambda}^{\vee}$ is local system of affine functions. In particular, we naturally have a morphism of local systems $f\colon\tilde{\Lambda}^{\vee}\to\Lambda^{\vee}$ which induces $f_{*}\colon H^{1}(B,i_{*}\tilde{\Lambda}^{\vee})\to H^{1}(B,i_{*}\Lambda^{\vee})$ . It is also easy to see that, if we slightly change the definition of the metric class, to extract its “linear” part as $f_{*}k(B)$ . Then, it naturally recovers the data $\overline{v}\in(e^{\perp}\otimes\mathbb{R}/\mathbb{R}e)$ i.e., we have $f_{*}k(\Phi_{\rm alg}([e,v]))=[v],$ under the natural identification $H^{1}(\Phi_{\rm alg}([e,v]),i_{*}\Lambda^{\vee}\otimes\mathbb{R})\hookrightarrow(e^{\perp}\otimes\mathbb{R}/\mathbb{R}e)$ which comes from the Leray spectral sequence applied to the elliptic fibration $X\twoheadrightarrow\Phi_{\rm alg}([e,v])$ in §4.2. Our results in [Od16] and Theorem 3.1 for $A_{g}$ can be re-interpretted similarly (but with weight $1$ ).

Remark 4.2.

Yuto Yamamoto [Yam] has some ongoing interesting work which seems to be related to our works, where he constructs a sphere with an integral affine structure from the tropicalization of an anticanonical hypersurface in a toric Fano 3-fold, and computes its radiance obstruction.