On smooth fillings of Calabi-Yau  manifolds

For a family of projective manifolds over a punctured disc, it is not always possible to extend it smoothly. One obstruction to these extensions is given by monodromy. Famous good reductions theorems state that for abelian varieties trivial monodromy implies the possibility to fill smoothly. For general Calabi-Yau families there are however examples of families of trivial monodromy with no smooth fillings. Motivated by our work in mirror symmetry we introduce an invariant that give obstructions to the existence of smooth fillings in these cases, even after finite base extensions and birational equivalence. In this talk I will overview known results in the area and explain how the new invariant generalizes previously known cases. This is joint work with Gerard Freixas.