Motivated by mirror symmetry of one-parameter models, an interesting class of Fuchsian differential operators can be singled out, the so-called Calabi--Yau operators, introduced by Almkvist and Zudilin. They conjecturally determine $Sp(4)$-local systems that underly a $\mathbb{ Q}$-VHS with Hodge numbers \[h^{ 3 0}=h^{ 2 1}=h^{ 1 2}=h^{ 0 3}=1\] and in the best cases they make their appearance as Picard--Fuchs operators of families of Calabi--Yau threefolds with $h^{ 12}=1$ and encode the numbers of rational curves on a mirror manifold with $h^{ 11}=1$. We review some of the striking properties of this rich class of operators.
Categories: math.AG