Derived categories of surfaces isogenous to a higher product : 종수가 2이상인 대수곡선 2개의 곱공간을 덮개공간으로 갖는 대수곡면들의 유도된 범주이론

다국어 초록 (Multilingual Abstract)

Let $S=(C \times D)/G$ be a surface isogenous to a higher product of unmixed type with $p_g=q=0$. In this paper, we study exceptional sequences of line bundles on $S$. The orthogonal complements of the admissible subcategories generated by exceptional sequences of maximal length in the derived category of $S$ are quasiphantom categories. We compute the Hochschild cohomologies of the quasiphantom categories and prove that for some exceptional sequences we made the DG algebras of endomorphisms are deformation invariant.

국문 초록 (Abstract)

이 논문에서 우리는 종수가 2이상인 대수곡선 2개의 곱공간을 덮개공간으로 갖는 기하종수가 0인 대수곡면들의 유도된 범주를 탐구한다. 이러한 대수곡면들 은 Bauer, Catanese, Grunewald에 의하여 분류되었다. 이러한 대수곡면의 유 도된 범주는 특이수열로 생성되는 범주와 이 범주와 수직인 범주로 나누어 질 수 있고 우리는 각각의 범주들을 공부함으로써 대수곡면의 유도된 범주이론에 대하여 탐구할 수 있다. 우리는 특이수열들과 수직인 범주들의 Hochschild 호 몰로지가 0이라는 것을 보였고 이를 통해 새로운 quasiphantom 범주들의 예를 만들어냈다. 또한 우리는 이러한 범주들의 Hochschild 코호몰로지와 대수곡면의 Hochschild 코호몰로지와의 관계를 탐구했고 특이수열로 생성되는 범주들 중 어 떤 것들은 곡면의 복소구조의 변형에 대하여 불변하다는 것을 보였다.

목차 (Table of Contents)

• Contents
• Abstract i
• 1 Introduction 1
• 1.1 Derived categories of algebraic varieties . . . . . . . . . . . . . 1
• 1.2 Derived categories of surfaces of general type . . . . . . . . . . 4
• 1.3 Derived categories of surfaces isogenous to a higher product
• with pg = q = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
• I Preliminaries 11
• 2 Representations and cohomologies of finite groups 13
• 2.1 Representation theory of finite group . . . . . . . . . . . . . . . 13
• 2.1.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . 14
• 2.1.2 Character . . . . . . . . . . . . . . . . . . . . . . . . . . 14
• 2.1.3 Induced representations . . . . . . . . . . . . . . . . . . 16
• 2.1.4 Representations of finite abelian groups . . . . . . . . . 16
• 2.1.5 Representations of finite nonabelian groups . . . . . . . 17
• 2.2 Cohomology of groups . . . . . . . . . . . . . . . . . . . . . . . 25
• 2.2.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . 25
• 2.2.2 Galois cohomology . . . . . . . . . . . . . . . . . . . . . 26
• 2.2.3 Schur multiplier . . . . . . . . . . . . . . . . . . . . . . . 29
• 2.3 Projective representation theory of finite group . . . . . . . . . 31
• 2.3.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . 32
• 2.3.2 Projective character . . . . . . . . . . . . . . . . . . . . 34
• 2.3.3 Spin representation . . . . . . . . . . . . . . . . . . . . . 37
• 2.3.4 Spin representations of S_n . . . . . . . . . . . . . . . . . 39
• 2.3.5 Spin representations of A_n . . . . . . . . . . . . . . . . . 43
• 3 Geometry of curves and surfaces 49
• 3.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
• 3.2 Geometry of algebraic curves . . . . . . . . . . . . . . . . . . . 51
• 3.2.1 Linear series . . . . . . . . . . . . . . . . . . . . . . . . . 51
• 3.2.2 Automorphisms of curves . . . . . . . . . . . . . . . . . 53
• 3.2.3 Invariant line bundles . . . . . . . . . . . . . . . . . . . 55
• 3.3 Geometry of algebraic surfaces . . . . . . . . . . . . . . . . . . 57
• 3.3.1 The Enriques classication of complex algebraic surfaces 58
• 3.3.2 Surfaces of general type . . . . . . . . . . . . . . . . . . 59
• 3.3.3 Surfaces of general type with p_g = q = 0 . . . . . . . . 60
• 3.4 Geometry of surfaces isogenous to a higher product with p_g =q = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
• 3.4.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . 61
• 3.4.2 Classication and moduli spaces . . . . . . . . . . . . . 66
• 4 Derived categories of algebraic varieties 69
• 4.1 Homological algebra . . . . . . . . . . . . . . . . . . . . . . . . 69
• 4.1.1 Abelian categories . . . . . . . . . . . . . . . . . . . . . 69
• 4.1.2 Triangulated categories . . . . . . . . . . . . . . . . . . . 72
• 4.1.3 Derived categories . . . . . . . . . . . . . . . . . . . . . 74
• 4.1.4 DG-categories . . . . . . . . . . . . . . . . . . . . . . . . 76
• 4.1.5 A_{\infty}-categories . . . . . . . . . . . . . . . . . . . . . . . . 77
• 4.2 Derived categories of algebraic varieties . . . . . . . . . . . . . 78
• 4.2.1 Fourier-Mukai transform . . . . . . . . . . . . . . . . . . 80
• 4.2.2 Equivariant derived categories . . . . . . . . . . . . . . . 81
• 4.2.3 McKay correspondence . . . . . . . . . . . . . . . . . . . 82
• 4.3 Derived categories of smooth projective curves . . . . . . . . . 83
• 4.3.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . 83
• 4.3.2 Rational curve . . . . . . . . . . . . . . . . . . . . . . . 84
• 4.3.3 Elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . 84
• 4.3.4 Curves with genus greater than or equal to 2 . . . . . . 85
• 4.4 Derived categories of smooth projective surfaces with Kodaira dimension less than 2 . . . . . . . . . . . . . . . . . . . . . . . 85
• 4.4.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . 85
• 4.4.2 Rational surfaces . . . . . . . . . . . . . . . . . . . . . . 86
• 4.4.3 Abelian surfaces . . . . . . . . . . . . . . . . . . . . . . 86
• 4.4.4 Enriques surfaces . . . . . . . . . . . . . . . . . . . . . . 86
• 4.4.5 Elliptic surfaces . . . . . . . . . . . . . . . . . . . . . . . 87
• 4.5 Derived categories of smooth projective surfaces of general type 87
• 4.5.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . 87
• 4.5.2 Classical Godeaux surface . . . . . . . . . . . . . . . . . 88
• 4.5.3 Primary Burniat surfaces . . . . . . . . . . . . . . . . . 89
• 4.5.4 Beauville surface . . . . . . . . . . . . . . . . . . . . . . 89
• 4.5.5 Barlow surfaces . . . . . . . . . . . . . . . . . . . . . . . 90
• 4.5.6 Fake projective planes . . . . . . . . . . . . . . . . . . . 91
• 4.5.7 Deformations of categories generated by exceptional sequences
• . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
• 4.5.8 Hochschild homology and cohomology . . . . . . . . . . 92
• II Derived categories of surfaces isogenous to a higher
• product 97
• 5 Abelian group cases 99
• 5.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
• 5.2 G = (Z_2)^3 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
• 5.3 G = (Z_3)^2 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
• 5.4 G = (Z_2)^4 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
• 6 Nonabelian group cases 131
• 6.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
• 6.2 Double planes of general type . . . . . . . . . . . . . . . . . . . 132
• 6.2.1 G = Z_2 \times D_4 case . . . . . . . . . . . . . . . . . . . . . . 132
• 6.2.2 G = S_4 case . . . . . . . . . . . . . . . . . . . . . . . . . 136
• 6.2.3 G = Z_2 \times S_4 case . . . . . . . . . . . . . . . . . . . . . . 141
• 6.3 G = A_5 cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
• 6.3.1 G = A_5;A = [3; 3; 5]; B = [2; 2; 2; 2; 2] case . . . . . . . . 144
• 6.3.2 G = A_5;A = [2; 5; 5]; B = [3; 3; 3; 3] case . . . . . . . . . 146
• 6.3.3 G = A_5;A = [2; 2; 2; 3]; B = [5; 5; 5] case . . . . . . . . . 149
• 6.4 2-group cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
• 6.4.1 G = G(16) case . . . . . . . . . . . . . . . . . . . . . . . 152
• 6.4.2 G = G(32) case . . . . . . . . . . . . . . . . . . . . . . . 154
• 7 Further questions 157
• 7.1 Derived categories surfaces with pg = q = 0 and whose Kodaira dimensions are less than or equal to 1 . . . . . . . . . . . 157
• 7.2 Bloch's conjecture . . . . . . . . . . . . . . . . . . . . . . . . . 158
• 7.3 Rationality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
• 7.4 Derived categories of surfaces isogenous to a higher product of mixed type with p_g = q = 0 . . . . . . . . . . . . . . . . . . . . 159
• 7.5 Product-quotient surfaces with p_g = q = 0 . . . . . . . . . . . . 159
• 7.6 Deformation invariance of categories generated by exceptional
• sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
• 7.7 Structures of quasiphantom categories . . . . . . . . . . . . . . 160
• 7.7.1 Uniqueness of quasiphantom categories . . . . . . . . . . 161
• 7.7.2 Application to the study of the moduli space of surfaces of general type . . . . . . . . . . . . . . . . . . . . 161
• 7.8 Quasiphantom categories on higher dimensional varieties . . . . 161
• Abstract (in Korean) 175
• Acknowledgement (in Korean) 177