Dual complexes of log Calabi-Yau pairs (after Kollár and Xu)

Log Calabi–Yau varieties form a broad and useful class of varieties. They include, for example, both the “classical” Calabi–Yau varieties and log Fano varieties. By definition, a variety X is called log Calabi–Yau if for some boundary B the divisor K_X + B is numerically trivial. Combinatorial part of the geometry of the boundary divisor B can be described using the notion of a dual complex. A well-known hypothesis states that D(B) is a quotient of a sphere by a finite group. Another hypothesis (related to Mirror Symmetry) says that for a maximal degeneration of “classical” Calabi–Yau varieties the dual complex of the special fiber is a sphere. An affirmative answer to the latter hypothesis in dimension 2 is given by the famous Kulikov's theorem. In our talk we give the answers to the hypotheses in lower dimensions following the work by Kollár and Xu).