Joins and Hadamard products

I will discuss a procedure of creating new deformation classes of projective varieties by smoothing a join of two known ones. This way starting from two elliptic curves one can obtain various interesting Calabi–Yau threefolds, some are non-simply-connected. Also this explains that Calabi–Yau threefolds of degree $25$, obtained as intersection of two Grassmannians in $\mathbb P^9$, are in fact linear sections of a smooth Fano sixfold. In a sense, this procedure is a generalization of complete intersection for non-hypersurface case. I will explain why quantum periods of such new Calabi–Yau varieties are Hadamard products of the quantum periods of original pieces. Also if the original varieties had mirror-dual functions $f(x)$ and $g(y)$, then a smoothing of a join will have mirror-dual function given by exterior product $f(x) g(y)$.