Inversion of Adjunction in High Codimension.

Abstract

The inversion of adjunction theorems study what happens when singularities of a pair (X, Delta) are restricted to a special subvarieties Z of X, such as hypersurfaces or Q-Gorenstein subvarieties. We study how singularities behave under restriction to arbitrary subvarieties by connecting the question to subadjunction for non-exceptional log-canonical centers. We prove a theorem that computes the multiplier ideal of the boundary Delta_Z that appears in subadjunction in terms of an adjoint ideal on the ambient variety. We also provide a proof with characteristic zero methods of work of S. Takagi on the restriction theorem when Z is Q-Gorenstein. We investigate in detail a construction of C. Hacon of an ideal closely related to the asymptotic multiplier ideal. We call this ideal the restricted multiplier ideal and we discuss its basic properties. Finally, we use everything we have developed to prove an extension theorem for pluri-canonical forms from an exceptional center.