The elusive nature of Calabi-Yau manifolds

I am going to talk about the phenomenon that sometimes the objects interesting for physics are the most difficult for algebraic geometry. More precisely, one of the geometric objects of particular interest to physics are Calabi-Yau manifolds. I will start by discussing the Kodaira embedding theorem. It gives a precise condition when complex manifolds in general are projective varieties, that is, when they can be identified with a subset of the projective space defined by polynomials. I will then explain that projective Calabi-Yau manifolds (Calabi-Yau varieties) happen to be also one of the 5 building blocks of projective varieties in general, according to the classification theory of projective varieties. In fact, at this point it seems that Calabi-Yau varieties are the only building blocks that are of particular interest for physics. The main point of the talk is to explain how these building blocks are the most elusive for algebraic geometers. I will present some conjectures that are well understood for the other building blocks, but not for Calabi-Yau's, including one on which I work also with my group.