Abstract:
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 $f(x) g(y)$.
