Hodge level of weighted complete intersections

Hodge level, first introduced by Rappoport in 1972, is the maximal width of Hodge diamond of an algebraic variety. Roughly speaking, it measures how complicated the variety can be from homological point of view. The well known method going back to Griffiths enables one to compute Hodge numbers for weighted complete intersections. Using it we classify all smooth Fano weighted complete intersections with small Hodge level. It turns out that Hodge levels of all of these varieties are small by categorical reasons. On the contrary, we expect that Hodge levels of small weighted complete intersections of general type are maximally possible. We confirm this expectation in some cases. We also show that this maximality may fail for the quasi-smooth case.