A note on multiplier ideal sheaves on complex spaces with singularities

Abstract

The goal of this note is to present some recent results of our research concerning multiplier ideal sheaves on complex spaces and singularities of plurisubharmonic functions. We firstly introduce multiplier ideal sheaves on complex spaces (not necessarily normal) via Ohsawa's extension measure, as a special case of which, it turns out to be the so-called Mather-Jacobian multiplier ideals in the algebro-geometric setting. As applications, we obtain a reasonable generalization of (algebraic) adjoint ideal sheaves to the analytic setting and establish some extension theorems on Kähler manifolds from singular hypersurfaces. Relying on our multiplier and adjoint ideals, we also give characterizations for several important classes of singularities of pairs associated to plurisubharmonic functions.Moreover, we also investigate the local structure of singularities of log canonical locus of plurisubharmonic functions. Especially, in the three-dimensional case, we show that for any plurisubharmonic function with log canonical singularities, its associated multiplier ideal subscheme is weakly normal, by which we give a complete classification of multiplier ideal subschemes with log canonical singularities.

On a question of Kollár

“…Following the same argument as Remark 2.1 in [21], one can check that the measure dV S [Ψ] is locally the direct image of measures defined upstairs by…”

“…For the convenience of readers, we state the following result on Nadel-Ohsawa multiplier ideal sheaves established in [21] for smooth ambient spaces (see also [22] for the case of divisors), relying on the L 2 extension theorem and strong openness of multiplier ideal sheaves.…”

“…In this appendix, analogous to the case of divisors, we introduce Nadel-Ohsawa multiplier and adjoint ideal sheaves along (closed) complex subspaces of higher codimension for log pairs, by which we establish an analytic inversion of adjunction in the higher codimensional case. When the ambient space is smooth, one can refer to [21].…”

“…(2) If X is smooth, the above result has been established in [21,22]; refer to [28,7] and [13,14,10] for related topics.…”

“…Remark 1.5. When X is smooth, Question 1.1 has been answered implicitly by the adjunction exact sequence (⋆) established in [21] or [22]; see also Theorem 1.4 in [15] for an explicit presentation based on an analogous argument.…”