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Virtual pullbacks, cosection localization, and Donaldson-Thomas theory of Calabi-Yau 4-folds : 가상 당김, 여절단 국소화 및 4차원 칼라비-야우 다양체의 도널드슨-토마스 이론

DC Field Value Language
dc.contributor.advisor김영훈-
dc.contributor.author박현준-
dc.date.accessioned2022-12-29T15:08:08Z-
dc.date.available2022-12-29T15:08:08Z-
dc.date.issued2022-
dc.identifier.other000000173494-
dc.identifier.urihttps://hdl.handle.net/10371/188582-
dc.identifier.urihttps://dcollection.snu.ac.kr/common/orgView/000000173494ko_KR
dc.description학위논문(박사) -- 서울대학교대학원 : 자연과학대학 수리과학부, 2022. 8. 김영훈.-
dc.description.abstractThe main purpose of this dissertation is to generalize Manolache's virtual pullbacks and Kiem-Li's cosection localization to Donaldson-Thomas theory of Calabi-Yau 4-folds. The three main applications are Lefschetz principle, Pairs/Sheaves correspondence, and a foundation of surface counting theory.

A secondary purpose is to revisit virtual pullbacks and cosection localization via the Kimura sequence for Artin stacks, derived algebraic geometry, and algebraic cobordism. We also prove Graber-Pandaripande's torus localization formula in full generality.		-
dc.description.abstract이 학위 논문의 주요 목적은 가상 당김과 여절단 국소화를 칼라비-야우 4차원 다양체의 도널드슨-토마스 이론으로 확장하는 것입니다. 이차적인 목적은 가상 당김과 여절단 국소화를 아틴 스택의 교차이론, 파생 대수 기하학 및 대수적 코보디즘을 이용하여 일반화하는 것입니다.-
dc.description.tableofcontentsAbstract i

Introduction 1

I Virtual intersection theory 17

1 Intersection theory 18
1.1 Intersection theory for schemes 18
1.2 Intersection theory for Artin stacks 25
1.3 Algebraic cobordism 29

2 Virtual pullbacks 35
2.1 Intrinsic normal cones 36
2.2 Perfect obstruction theories 47
2.3 Virtual pullbacks and virtual cycles 52

3 Cosection localization 65
3.1 Cone reduction 66
3.2 Reduced virtual cycles 72
3.3 Cosection-localized virtual pullbacks 75

II Donaldson-Thomas theory of Calabi-Yau 4-folds 83

4 Virtual pullbacks in DT4 theory 84
4.1 Local models 86
4.2 Symmetric obstruction theories 97
4.3 Square root virtual pullbacks 108

5. Cosection localization in DT4 theory 118
5.1 Cone reductions 119
5.2 Reduced virtual cycles 122
5.3 Cosection-localized virtual cycles 125

6 Applications to enumerative geometry 131
6.1 Moduli spaces, virtual cycles, and invariants 131
6.2 Lefschetz principle 140
6.3 Pairs/Sheaves correspondence 142
6.4 Counting surfaces on Calabi-Yau 4-folds 146

III Generalizations 148

7 Torus localization via equivariant virtual pullbacks 149

7.1 Equivariant virtual pullbacks 149
7.2 Localization of virtual cycles 159

8 Cosection localization via (-1) -shifted 1-forms 163
8.1 Three reductions 164
8.2 Localized virtual cycles 167

9 Virtual cycles in algebraic cobordism 170
9.1 Limit algebraic cobordism 170
9.2 Virtual pullbacks 176
9.3 Cosection localization 184
9.4 Torus localization 189

A Kimura sequence for Artin stacks 193
A.1 Kimura sequence for Artin stacks 193
A.2 Chow lemma for Artin stacks 197

Abstract (in Korean) i

Acknowledgement (in Korean) ii		-
dc.format.extent218-
dc.language.isoeng-
dc.publisher서울대학교 대학원-
dc.subjectVirtualpullbacks-
dc.subjectcosectionlocalization-
dc.subjectDonaldson-ThomastheoryofCalabi-Yau4-folds-
dc.subject.ddc510-
dc.titleVirtual pullbacks, cosection localization, and Donaldson-Thomas theory of Calabi-Yau 4-folds-
dc.title.alternative가상 당김, 여절단 국소화 및 4차원 칼라비-야우 다양체의 도널드슨-토마스 이론-
dc.typeThesis-
dc.typeDissertation-
dc.contributor.AlternativeAuthorHyeon Jun PARK-
dc.contributor.department자연과학대학 수리과학부-
dc.description.degree박사-
dc.date.awarded2022-08-
dc.contributor.major대수기하학-
dc.identifier.uciI804:11032-000000173494-
dc.identifier.holdings000000000048▲000000000055▲000000173494▲-
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