Gromov-Hausdorff limits of flat Riemmannian surfaces

Let X be a holomorphic family of smooth compact complex curves of genus >=1 over a punctured disc, and let \Omega be a relative holomorphic 1-form on X. Consider the pseudo-Kahler metric on the fibres X_t with the Kahler form i/2 \Omega_t \wedge \bar\Omega_t and further rescale it so that the diameter of X_t is constantly 1.  In this talk I will describe the Gromov-Hausdorff limit of X_t as t tends to 0, which is isometric either to a metric graph or to a collection of smooth complex curves with flat metrics, glued at finitely many points. The description makes use of the weight function, which depends on the form \Omega, on the dual intersection complex of the snc boundary divisor of a partial compactification of the family X, introduced by Kontsevich and Soibelman and further studied by Mustata, Nicaise, Xu, and Temkin.