Let $X=(X_{t})_{t\to 0}$ be a maximally degenerate K3 surface over the field ${{\bf C}}_{t}^{mer}$ (see Section 5.1). We denote by $\Lambda_{X}$ the quotient group $[\gamma_{0}]^{\perp}/{{\bf Z}}[\gamma_{0}]$ where $[\gamma_{0}]\in H_{2}(X_{t},{\bf Z})$ is the vanishing cycle. Then $\Lambda_{X}\simeq\Lambda_{2,18}$. Let us assume that the monodromy acts trivially on $\Lambda_{X}$.

We define a natural homomorphism $\rho_{X}:\Lambda_{X}\to({{\bf C}}_{t}^{mer})^{\times}$ by the formula

One can give a more abstract definition of $\rho_{X}$ in terms of the variation of Hodge structure. It is easy to see that $\left(val_{{{\bf C}}_{t}^{mer}}\circ\rho_{X}\right)([\gamma])=(v_{X},[\gamma])$ where $v_{X}\in\Lambda_{X}$ is a vector such that $(v_{X},v_{X})>0$, and $val_{{{\bf C}}_{t}^{mer}}$ is the standard valuation on the field ${{\bf C}}_{t}^{mer}\subset{{\bf C}}((t))$.

Let $X^{an}$ be the corresponding analytic K3 surface over the field $K={{\bf C}}((t))$. We have an analytic torus fibration over $S^{2}\setminus\{x_{1},...,x_{24}\}$ which can be extended to a continuous map $X^{an}\to S^{2}$. Let us call such an extension a singular analytic torus fibration with standard singularities.

Notice that for K3 surfaces any birational automorphism is biregular. Hence the group of birational automorphisms $Aut^{brt}(X)$ acts by a ZPL-transformations of the sphere $S^{2}$ which is equipped with a singular ${\bf Z}$-affine structure (see Section 6.6), i.e. we have a homomorphism