Restriction, subadditivity, and semicontinuity theorems for Hodge ideals
  We prove results concerning the behavior of Hodge ideals under restriction to hypersurfaces or fibers of morphisms, and addition. The main tool is the description of restriction functors for mixed Hodge modules by means of the $V$-filtration. 
Mircea Mustata and Mihnea Popa
math.AG