Since the Strominger-Yau-Zaslow conjecture [SYZ96] was made there has been a considerable interest in geometry of Kähler Ricci-flat half-dimensional torus fibrations. For a nonsingular fibration the metrics which are flat along the fibers were used by Hitchin [Hit97] in relation with mirror symmetry and SYZ conjecture. This, so called, semi-flat case was further studied in [LYZ01] and [Leu00]. The Ricci-flatness condition becomes equivalent to the real Monge-Ampère equation on the base. The mirror symmetry is then provided by the Legendre transform.

This simple case has inspired another, more recent, conjecture of Gross and Wilson [GW00] and Kontsevich and Soibelman [KS01] about existence of an integral Kähler affine structure on the limiting space of the metric collapse. But in order to understand this conjecture for general compact Calabi-Yau manifolds (not just tori) one needs a local description of the metric behavior near the singular fibers.

The first example of such a metric was constructed by Ooguri and Vafa [OV96] in two dimensions via periodic Gibbons-Hawking ansatz and was used later by Gross and Wilson [GW00] to justify the collapse picture for K3. Pedersen and Poon [PP91] have written the (non-linear) differential equations for the GH ansatz in higher dimensions but the solutions they found were not periodic in the remaining number of torus variables, and hence cannot apply to our case of interest. An attempt to apply the generalization of Gibbons-Hawking in the periodic situation was made by Matessi [Mat01], though no explicit solutions were found.

In this paper we do not try to solve Gibbons-Hawking differential equation. Rather the goal is to set up the geometric framework for investigation of limiting behavior of such metrics as the size of tori goes to zero. Unfortunately, the key exponential decay lemma is left unproven. A substantial amount of hard analysis of non-linear elliptic PDE with singularities is required for the proof, and we plan to do it elsewhere.

The conjectural description of the limiting metric space is in agreement with the metric collapse picture of [KS01] and [GW00]. But we suggest a more explicit condition for asymptotics at the singular locus. The coordinates of the Gibbons-Hawking ansatz are related to the affine ones via the partial Legendre transform introduced in the last section of the paper.