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Plurisubharmonic functions are fundamental objects in complex analysis with many applications in complex geometry and even in algebraic geometry. Their singularities can be extremely complicated : some of the most important tools one can use to study the singularities include multiplier ideals and approximation theorems.
In the first part, we study problems on equisingular approximation. Recently Qi'an Guan gave a criterion for the existence of decreasing equisingular approximations with analytic singularities, in the case of diagonal type plurisubharmonic functions. We generalize a weaker version of this to
arbitrary toric plurisubharmonic functions.
In the second part, we study plurisubharmonic singularities on singular varieties. Our main result in this part is a generalization of the Rashkovskii-Guenancia theorem on multiplier ideals of toric plurisubharmonic functions to the normal Q-Gorenstein case.