Coregularity of Fano varieties

The regularity of a Fano variety, denoted by $\mathrm{r}\mathrm{e}\mathrm{g}(X)$, is the largest dimension of the dual complex of a log Calabi–Yau structure on $X$. The coregularity is defined to be $$\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{g}(X):=\dim X-\mathrm{r}\mathrm{e}\mathrm{g}(X)-1.$$ The coregularity is the complementary dimension of the regularity. We expect that the coregularity of a Fano variety governs, to a large extent, the geometry of $X$. In this note, we review the history of Fano varieties, give some examples, survey some important theorems, introduce the coregularity, and propose several problems regarding this invariant of Fano varieties.