  Using inversion of adjunction, we deduce from Nadel's theorem a vanishing
property for ideals sheaves on projective varieties, a special case of which
recovers a result due to Bertram--Ein--Lazarsfeld. This enables us to
generalize to a large class of projective schemes certain bounds on
Castelnuovo--Mumford regularity previously obtained by Bertram--Ein--Lazarsfeld
in the smooth case and by Chardin--Ulrich for locally complete intersection
varieties with rational singularities. Our results are tested on several
examples.
