Hamiltonian reductions in derived algebraic geometry = 유도 대수기하학 내의 해밀토니안 축소

다국어 초록 (Multilingual Abstract)

We study Hamiltonian reductions of shifted symplectic structures and their applications to
enumerative geometry. These reductions are modified quotients for shifted symplectic structures
under group actions. We take two viewpoints to understand and use this concept.
First, we take the viewpoint that Hamiltonian reductions produce new shifted symplectic
structures by “modding out” certain symmetries of the given ones. For example, the moduli
stacks of perfect complexes on Calabi–Yau varieties–central examples of shifted symplectic de-
rived moduli spaces–admit BGm-symmetries coming from the scalar multiplication Gm. We
show that it is possible to take Hamiltonian reductions by these symmetries when the dimen-
sions of the ambient varieties are at least 2. More generally, we develop a detailed theory of
BGm-actions on shifted symplectic structures which includes explicit and precise obstructions
to taking Hamiltonian BGm-reductions.
The resulting Hamiltonian reductions–referred to symplectic rigidifications–provide shifted
symplectic structures on the moduli schemes of stable sheaves. These structures are the first ones
found on the components parametrizing torsion sheaves. We demonstrate the usefulness of sym-
plectic rigidifications by establishing new reduced pairs/sheaves correspondences in Donaldson–
Thomas theory of Calabi–Yau fourfolds using these.
Second, we take the viewpoint that Hamiltonian reductions reveal additional information
about shifted symplectic structures by providing local presentations. Precisely, we show that
under mild assumptions–such as reductive stabilizers–that 0-shifted symplectic derived alge-
braic stacks are formally locally Hamiltonian reductions of formally smooth symplectic formal
schemes by actions of the stabilizer groups. We also obtain a similar result for almost symplectic
structures, by which we mean non-degenerate 2-forms (no closedness required).
As future directions, we discuss extensions, applications and improvements of our results.

국문 초록 (Abstract)

우리는 이동 사교 구조의 해밀토니안 축소 및 그것들의 셈 기하학으로의 응용을 공부한다. 이들은
군 작용을 갖는 이동 사교 구조들을 위한 보완된 몫 구조들이다. 우리는 두 가지 관점을 통해 이러한
개념을 이해하고 사용한다.
첫째, 우리는 해밀토니안 축소가 주어진 이동 사교 구조의 일부 대칭을 “제거하여” 새로운 구조
를 생산한다는 관점을 취한다. 예시로, 칼라비–야우다양체 위의 완벽 복합체들의 모듈리 공간–이동
사교구조를 갖는 유도 모듈리 공간의 대표 예시–는 스칼라 곱 Gm 에서 비롯된 BGm-대칭을 갖는
다. 우리는 배경 다양체의 차원이 2 이상인 경우 위 대칭에 대한 해밀토니안 축소를 취할 수 있음을
보인다. 더 일반적으로, 이동 사교구조에의 BGm-작용에 대한 자세한 이론을 개발해 해밀토니안
축소의 취함에 대한 구체적이고 정확한 방해를 알아낸다.
이렇게 얻어진 해밀토니안 축소–사교 강성화–는, 안정 쉬프의 모듈리 스킴에 이동 사교 구조를
준다. 위 구조들은 꼬임 쉬프들을 매개화하는 성분들에서 찾아진 첫 이동 사교 구조다. 우리는 사교
강성화의 유용성을 입증하기 위해, 이들을 사용해 칼라비–야우 4-다양체의 도널슨–토마스이론에서
새로운 축소 쌍/쉬프 대응을 수립한다.
둘째, 우리는 해밀토니안 축소가 국소적인 표현을 통해 이동 사교 구조들에 대한 추가적인 정
보를 드러낸다는 관점을 취한다. 구체적으로, 가벼운 조건–안정자 군의 축약성 등–하에서, 우리는
0-이동 사교 유도 대수적 스택들이 형식적 국소적으로 형식적으로 매끈한 사교 형식 스킴의 안정자
군들의 작용에 의한 해밀토니안 축소임을 보인다. 또한 거의 사교 구조들, 이들은 비퇴화 2-형식을
뜻하는데 (닫힘을 요구하지 않는다), 에 대해서도 유사한 결과를 얻는다.
향후 연구 방향으로 우리 결과들의 확장, 응용 및 개선을 논의한다.

목차 (Table of Contents)

• Abstract i
• 1 Introduction 1
• 1.1 Shifted symplectic structures on derived moduli spaces 1
• 1.2 Hamiltonian reductions of shifted symplectic structures 2
• 1.3 Applications 7
• 1.4 Contents of the Thesis 9
• I Background 11
• 2 ∞-categories 12
• 2.1 Introduction 12
• 2.1.1 Motivation 12
• 2.1.2 Approach 13
• 2.1.3 Basic notions and facts 15
• 2.2 Straightening equivalences17
• 2.2.1 Straighetening equivalences 17
• 2.2.2 Yoneda lemma 19
• 2.3 Adjunction 20
• 2.3.1 Adjunction 20
• 2.3.2 Adjoint functor theorem 21
• 2.3.3 Kan extensions 22
• 2.4 Monoid, group and groupoid 23
• 2.4.1 Monoids, groups and groupoids 23
• 2.4.2 Groupoids and ∞-topoi 25
• 2.5 Span categories 26
• 2.5.1 Definition 27
• 2.5.2 Fuctoriality 29
• 2.5.3 Adjunction 30
• 3 Derived algebraic geometry 31
• 3.1 Introduction 31
• 3.1.1 Motivation 31
• 3.1.2 Approach 32
• 3.1.3 Higher algebra background 33
• 3.2 Derived stacks 37
• 3.2.1 Derived stacks 37
• 3.2.2 Mapping stacks 39
• 3.2.3 Classical truncations 39
• 3.2.4 Geometricity 40
• 3.2.5 Finiteness conditions 42
• 3.3 Quasi-coherent complexes 43
• 3.3.1 Quasi-coherent complexes 43
• 3.3.2 t-structures 44
• 3.3.3 Pushforwards 44
• 3.3.4 Natural substacks 44
• 3.3.5 Relative spectra 46
• 3.3.6 Base-change and projection formulas 46
• 3.3.7 Cotangent complexes 47
• 3.3.8 Total spaces 50
• 3.4 Differential forms 51
• 3.4.1 Filtered complexes 51
• 3.4.2 De Rham complex 54
• 3.4.3 Differential forms 58
• 3.4.4 K¨unneth formulas 59
• 3.5 Negatively shifted closed forms are exact 61
• 3.5.1 Statement 61
• 3.5.2 Proper hypercovers 61
• 3.5.3 Proof of Theorem 3.86 62
• 3.5.4 Non-example 63
• 3.6 Formal completions 63
• 3.6.1 Reduced substacks and de Rham stacks 63
• 3.6.2 Formal completions 65
• 4 Shifted symplectic structures 68
• 4.1 Shifted symplectic structures 68
• 4.1.1 Shifted symplectic structures 68
• 4.1.2 BG 69
• 4.1.3 Perf 70
• 4.2 Lagrangians 71
• 4.2.1 Lagrangian structures 71
• 4.2.2 Extension Lagrangian correspondence 72
• 4.2.3 Lagrangian intersections 73
• 4.2.4 Symplectic categories 74
• 4.3 Shifted cotangent bundles 75
• 4.3.1 Symplectic structures 75
• 4.3.2 Lagrangian structures 75
• 4.3.3 Functoriality 77
• 4.4 Mapping stacks 79
• 4.4.1 Integration along fibers 79
• 4.4.2 Symplectic and Lagrangian structures 80
• II Hamiltonian Reductions in Derived Algebraic Geometry 83
• 5 Hamiltonian reductions 84
• 5.1 Hamiltonian actions 84
• 5.1.1 Invariant and equivariant forms 84
• 5.1.2 Hamiltonian actions 87
• 5.2 Hamiltonian categories 90
• 5.2.1 Equivariant symplectic categories 90
• 5.2.2 Hamiltonian categories 92
• 5.2.3 Hamiltonian reduction via adjunction 94
• 5.2.4 Generalizations 95
• 5.3 Exact variants 99
• 5.3.1 Exact Hamiltonian actions 99
• 5.3.2 Exact Hamiltonian categories 100
• 6 Shifted symplectic rigidification 103
• 6.1 The group BGm 103
• 6.1.1 E2 groups 103
• 6.1.2 Monoid stacks 104
• 6.1.3 Equivalence BGm ≃Pic 106
• 6.2 BGm-actions 106
• 6.2.1 BGm action on Perf(X) and Exact(X) 107
• 6.2.2 BGm-quotient is rigidification 109
• 6.3 Symplectic rigidification 110
• 6.3.1 BG×rm -actions are symplectic 110
• 6.3.2 Contractions 112
• 6.3.3 Contraction of chern character 114
• 6.3.4 Obstructions to Hamiltonian actions 116
• 6.3.5 Previous symplectic structures on rigidified moduli 119
• 6.3.6 Non-example I 120
• 6.3.7 Non-example II 120
• 6.4 Functoriality 121
• 6.4.1 Functoriality 121
• 6.4.2 Rigidified extension correspondence 122
• 6.4.3 Symplectic rigidification as partial reduction 125
• 7 Local structure theorems for 0-shifted symplectic stacks 127
• 7.1 Introduction 127
• 7.1.1 Local structure theorems in the negative shifts 127
• 7.1.2 The shift 0 127
• 7.1.3 Contents of this chapter 128
• 7.2 Stacks geometry 128
• 7.2.1 Inertia stacks and stabilizer groups 128
• 7.2.2 Residual gerbes 128
• 7.2.3 Zero locus towers 129
• 7.3 Formal local structures of 0-shifted symplectic stacks 130
• 7.3.1 Statement 130
• 7.3.2 Exactness via formal completion 131
• 7.3.3 Extending exact 2-forms 131
• 7.3.4 Extending non-degeneracy 134
• 7.3.5 Proof of Theorem 7.6 137
• 7.3.6 Comments on ”formal” 138
• 7.4 'Etale local structures of 0-shifted almost symplectic stacks 139
• 7.4.1 Almost Hamiltonian reductions 139
• 7.4.2 Statement 139
• 7.4.3 Extending ordinary 2-forms 140
• 7.4.4 Extending non-degeneracy 141
• 7.4.5 Proof of Theorem 7.19 143
• III Applications 145
• 8 Pairs/Sheaves Lagrangian correspondences in DT4 theory 146
• 8.1 Introduction 146
• 8.1.1 DT4 theory 146
• 8.1.2 Pairs/sheaves correspondences 146
• 8.1.3 Shifted symplectic geometry and DT4 theory 147
• 8.1.4 Contents of this chapter 147
• 8.2 Virtual classes in DT4 theory 147
• 8.2.1 Symmetric complex 147
• 8.2.2 Symmetric obstruction theory 148
• 8.2.3 Virtual classes 149
• 8.3 Virtual Lagrangian classes 151
• 8.3.1 Orientations 151
• 8.3.2 Induced SOT 152
• 8.3.3 Isotropic condition 152
• 8.3.4 Functoriality 153
• 8.4 Moduli spaces 153
• 8.4.1 Stable sheaves 153
• 8.4.2 Stable pairs 155
• 8.5 Pairs/Sheaves Lagrangian correspondences 157
• 8.5.1 Lagrangian correspondence without stability 158
• 8.5.2 Lagrangian correspondence 158
• 8.5.3 Virtual pullback formula 159
• 8.5.4 Comparison 160
• 8.5.5 Comment on orientations 160
• 9 Reduced Pairs/Sheaves Lagrangian correspondences in DT4 theory 161
• 9.1 Introduction 161
• 9.1.1 Reduced DT4 theory 161
• 9.1.2 Shifted symplectic geometry and reduced DT4 theory 161
• 9.1.3 Contents of this chapter 161
• 9.2 Reduced virtual classes in DT4 theory 162
• 9.2.1 Cosections 162
• 9.2.2 Isotropic cosections 163
• 9.2.3 Non-degenerate cosections 164
• 9.3 Reductions by shifted functions 165
• 9.3.1 Reductions by shifted functions 165
• 9.3.2 Reduced DT4 classes as virtual Lagrangian classes 167
• 9.3.3 Shifted functions from Hochschild homology 168
• 9.3.4 Proof of Proposition 9.22 169
• 9.4 Reduced Pairs/Sheaves Lagrangian correspondences I 174
• 9.4.1 Lagrangian correspondence without stability 174
• 9.4.2 Lagrangian correspondence 176
• 9.4.3 Virtual pullback formulas 177
• 9.4.4 Comparison 178
• 9.5 Reduced Pairs/Sheaves correspondences II 179
• 9.5.1 Lagrangian correspondence without stability 179
• 9.5.2 Lagrangian correspondence 182
• 9.5.3 Virtual pullback formulas 183
• 9.5.4 Comparison 183
• 10 Future directions 186
• 10.1 Reduced DT4 theory of abelian 4-folds 186
• 10.2 Formality for 0-shifted symplectic derived Artin stacks 187
• 10.3 Analytic-local structures of 0-shifted symplectic derived Artin stacks 189
• Bibliography 190
• Abstract (in Korean) 201
• Acknowledgement (in Korean) 202