The K3 surface. [04IQ]

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The K3 surface.

Theorem 3.22.

Let $(B,\Delta,\mathscr{A})$ be the affine manifold with singularities in Example 3.16 and let $X(B_{0},\mathscr{A})$ be the associated $T^{2}$ bundle with symplectic structure $\omega_{0}$ and projection $f_{0}$ induced by the standard ones in $T_{B_{0}}^{\ast}$ . There exists a compact symplectic manifold $(X,\omega)$ , a Lagrangian fibration $f:X\rightarrow B$ and an embedding $\iota:X(B_{0},\mathscr{A})\rightarrow X$ such that $\iota^{\ast}\omega=\omega_{0}$ and $f\circ\iota=f_{0}$ . Moreover $X$ is diffeomorphic to a smooth $K3$ surface.

Proof.

Let $f_{V}:X_{V}\rightarrow V$ be a focus-focus fibration over a small open neighborhood $V$ of its node $0\in V$ . Let $V^{\ast}=V-\{0\}$ and denote by $(V^{\ast},\mathscr{A}_{V})$ the integral affine manifold induced by $f_{V}$ . Let $X(V^{\ast},\mathscr{A}_{V})$ be the associated Lagrangian $T^{2}$ bundle over $V^{\ast}$ . It can be shown that $f_{V}$ has a Lagrangian section $s:V\rightarrow X_{V}$ such that $s(V)\cap\Crit(f_{V})=\varnothing$ . Then from Theorem 3.3 it follows that $f^{-1}_{V}(V^{\ast})\subset X_{V}$ is symplectically conjugate to $X(V^{\ast},\mathscr{A}_{V})$ .

Now let $P\in\Delta$ and let $U\subset B$ be a small neighborhood of $P$ . Denote by $U^{\ast}=U-P$ and by $X(U^{\ast},\mathscr{A})$ the Lagrangian $T^{2}$ bundle over $U^{\ast}$ given by the restriction of $X(B_{0},\mathscr{A})$ to $U^{\ast}$ . Recall that both $U$ and $V$ are simple affine manifold with singularities. Then, after taking $U$ and $V$ small enough, there exists an integral affine isomorphism $V^{\ast}\cong U^{\ast}$ . From Corollary 3.4, the latter isomorphism induces is a symplectic conjugation,

f^{-1}_{V}(V^{\ast})\cong X(V^{\ast},\mathscr{A}_{V})\cong X(U^{\ast},\mathscr{A}),

which can be used to symplectically glue $X_{V}$ to $X(B_{0})$ . Define $(X,\omega)$ to be the symplectic manifold obtained after applying this gluing over all points $P\in\Delta$ and $f:X\rightarrow B$ the resulting fibration. It is clear that $(X,\omega)$ is a semi-stable compactification of $(X(B_{0},\mathscr{A}),\omega_{0})$ such that $\iota^{\ast}\omega=\omega_{0}$ . It is easy to check that $(X,\omega,f,B)$ is topologically conjugate to a simply connected elliptic fibration with 24 singular fibres of type $I_{1}$ . It follows that $X$ is diffeomorphic to a K3 surface. ∎

Corollary 3.23.

In view of Remark 3.21, given $(B,\Delta,\mathscr{A})$ as in Example 3.16, a compactification $X(B_{0},\mathscr{A})\hookrightarrow(X,\omega)$ as above is uniquely determined up to symplectic conjugation by a choice of 24 formal power series in two variables:

\mathfrak{q}_{1},\ldots,\mathfrak{q}_{24}\in\mathbb{R}[\![x,y]\!]

corresponding to germs of focus-focus fibrations $\mathcal{F}_{1},\ldots\mathcal{F}_{24}$ . In particular, there are infinitely many Lagrangian fibrations of a symplectic K3 surface, fibering over $(B,\Delta,\mathscr{A})$ , which are all topologically conjugate but not symplectically conjugate.

The space $\mathbb{R}[[x,y]]$ being contractible, implies that every two focus-focus fibrations can be connected with a path in $\mathbb{R}[[x,y]]$ . The standard Moser’s argument implies that the corresponding total spaces are symplectomorphic. Similarly, any two symplectic structures obtained using Theorem 3.22 can be connected with a path in $\mathbb{R}[[x,y]]^{24}$ . Moser’s argument implies that all such manifolds are symplectomorphic.

Following an alternative approach, Zung obtained a Lagrangian fibration of a symplectic 4-manifold which is also diffeomorphic to a K3 surface (cf. [35]Example 4.19). Leung and Symington [24] use affine geometry as starting point to construct and classify –up to diffeomorphism– the so-called almost toric symplectic 4-manifolds. The fibration we obtained in Theorem 3.22 coincides with one of the list in [24].

Other ways of constructing affine manifolds with singularities have been proposed by Gross and Siebert [12, 13], Hasse and Zharkov [16, 17, 18]. In [8], Gross finds a combinatorial method to obtain simple affine manifolds with singularities out of the geometry of the polytopes which Batyrev and Borisov use to construct pairs of Calabi-Yau varieties as complete intersections inside Fano toric varieties. From Theorem 0.1 of [8] (proved by Gross and Siebert in [11]) it follows that these affine manifolds give rise to topological semi-stable compactifications homeomorphic to the two Batyrev-Borisov’s Calabi-Yau varieties. We shall see in this paper that similar compactifications can be carried out in the symplectic category.