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학위논문(박사) -- 서울대학교대학원 : 자연과학대학 수리과학부, 2021.8. 김다노.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, based on joint work with Hoseob Seo, we study problems on equisingular approximation. Recently 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. This also generalizes an algebraic result of Blickle to analytic multiplier ideals.다중조화버금함수는 복소해석학 뿐 아니라 복소기하학, 나아가 대수기하학에서 중요한 연구 대상입니다. 다중조화버금함수의 특이점들은 굉장히 복잡하고 어렵고 직접적인 관찰 대신 이를 연구하기 위한 도구로 승수 아이디얼과 근사 정리를 이용하곤합니다. 첫번째 결과로 서울대학교 수학연구소 소속인 서호섭 박사후 연구원과 equisingular 근사 정리에 대해서 소개하려고 합니다. 최근에 Qi’an Guan에 의해 발표된 해석적 특이점을 갖는 decreasing, equisingular 근사 정리라는 주제를 다중조화버금함수가 toric일 때 부분적으로 일반화할 수 있음을 설명합니다. 두번째 결과는 특이 다양체 위에서의 다중조화버금함수입니다. 기존의 다양체에서와 달리 특이 다양체에서 다중조화버금함수 및 승수 아이디얼이 어떻게 정의되는지 소개합니다. 또한 주요 결과로서, toric 다중조화버금함수의 경우, 승수 아이디얼을 계산하는데 주요 공식 중 하나인 Rashkovskii-Guenancia의 일반화를 제시합니다. 이 결과는 Blickle의 대수적 승수 아이디얼 공식을 해석적으로 일반화한 것이기도 합니다.1. Introduction 1 1.1 Equisingular approximations of plurisubharmonic functions 2 1.2 Multiplier ideal sheaves on singular varieties 4 2. Preliminaries 7 2.1 Plurisubharmonic functions 7 2.2 Plurisubharmonic singularities 10 2.2.1 Lelong numbers of psh functions 10 2.2.2 Multiplier ideal sheaves of psh functions 11 2.3 Toric Plurisubharmonic functions 13 3 Equisingular approximations of plurisubharmonic functions 16 3.1 Equisingular approximations 16 3.2 Equisingular approximations of toric psh functions 20 3.2.1 Newton convex bodies for analytic singularities 22 3.2.2 Convex conjugate of analytic singularities 27 3.3 Proof of Theorem 3.2.1 and some examples 34 4 Multiplier ideal sheaves on singular varieties 38 4.1 Singularities of normal varieties 38 4.1.1 Canonical sheaves on normal varieties 39 4.1.2 Singularities of pairs 40 4.2 Toric geometry 42 4.2.1 Convex polyhedral cones 42 4.2.2 Affine toric varieties 45 4.2.3 Singularities in toric geometry 47 4.3 Multiplier ideal sheaves on singular varieties 49 4.4 Multiplier ideal sheaves on toric varieties 54 4.4.1 Newton convex bodies of toric psh functions on C^n 55 4.4.2 Newton convex bodies of toric psh functions on affine toric variety 56 4.4.3 Proof of the Theorem 4.4.1 60 Abstract (in Korean) 67 Acknowledgement (in Korean) 68박

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