Coregularity of Fano varieties

The absolute regularity of a Fano variety, denoted by reĝ(X) reg ^ ( X ) , is the largest dimension of the dual complex of a log Calabi–Yau structure on X . The absolute coregularity is defined to be aligned coreĝ(X):= X - reĝ(X)-1. aligned coreg ^ ( X ) : = dim X - reg ^ ( 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 theorems, introduce the coregularity, and propose several problems regarding this invariant of Fano varieties.