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Given a normal crossings degeneration f:(X,w_X)-->D of compact Kahler manifolds, in
recent work with T. Pelka we have shown how to associate a smooth locally trivial
fibration f_A:X_A---> D_log over the real oriented blow up of the disc D. It is
moreover endowed with a closed 2-form w_A giving it the structure of a symplectic
fibration. The restriction of w_A to every fibre of f_A "at positive radius" (that
is over a point of D\{0} is the modification by a potential of the restriction of
w_X to the same fibre. The construction can be regarded as a symplectic realization
of A'Campo model for the monodromy and has found the following applications: (1) We can produce symplectic representatives of the monodromy with very special dynamics, and based on this and on a spectral sequence due to McLean prove the family version of Zariski's multiplicity conjecture. (2) If f is a maximal Calabi-Yau degeneration we can produce Lagrangian torus fibrations over a the complement of a codimension 2 set over the (expanded) essential skeleton of the degeneration, satisfying many of the properties conjectured by Kontsevich and Soibelman. In the talk I will highlight the main aspects of the construction, and present some of the application (2). |