Varieties of Characters
|
|
- Bethanie Barnett
- 5 years ago
- Views:
Transcription
1 Sean Lawton George Mason University Fall Eastern Sectional Meeting September 25, 2016 Lawton (GMU) (AMS, September 2016)
2 Step 1: Groups 1 Let Γ be a finitely generated group. Lawton (GMU) (AMS, September 2016)
3 Step 1: Groups 1 Let Γ be a finitely generated group. So Γ can be presented as γ 1,..., γ r r i (γ 1,..., γ r ) = 1, i I, Lawton (GMU) (AMS, September 2016)
4 Step 1: Groups 1 Let Γ be a finitely generated group. So Γ can be presented as γ 1,..., γ r r i (γ 1,..., γ r ) = 1, i I, where r i (γ 1,..., γ r ) are words in the letters γ ±1 i. Lawton (GMU) (AMS, September 2016)
5 Step 1: Groups 1 Let Γ be a finitely generated group. So Γ can be presented as γ 1,..., γ r r i (γ 1,..., γ r ) = 1, i I, where r i (γ 1,..., γ r ) are words in the letters γ ±1 i. Ex: Z r or Z r or RAAGs. Lawton (GMU) (AMS, September 2016)
6 Step 1: Groups 1 Let Γ be a finitely generated group. So Γ can be presented as γ 1,..., γ r r i (γ 1,..., γ r ) = 1, i I, where r i (γ 1,..., γ r ) are words in the letters γ ±1 i. Ex: Z r or Z r or RAAGs. Let K be a compact Lie group. Lawton (GMU) (AMS, September 2016)
7 Step 1: Groups 1 Let Γ be a finitely generated group. So Γ can be presented as γ 1,..., γ r r i (γ 1,..., γ r ) = 1, i I, where r i (γ 1,..., γ r ) are words in the letters γ ±1 i. Ex: Z r or Z r or RAAGs. Let K be a compact Lie group. K is real algebraic group (zeros of real polynomials). Lawton (GMU) (AMS, September 2016)
8 Step 1: Groups 1 Let Γ be a finitely generated group. So Γ can be presented as γ 1,..., γ r r i (γ 1,..., γ r ) = 1, i I, where r i (γ 1,..., γ r ) are words in the letters γ ±1 i. Ex: Z r or Z r or RAAGs. Let K be a compact Lie group. K is real algebraic group (zeros of real polynomials). Define G to be the complex zeros of those polynomials. Lawton (GMU) (AMS, September 2016)
9 Step 1: Groups 1 Let Γ be a finitely generated group. So Γ can be presented as γ 1,..., γ r r i (γ 1,..., γ r ) = 1, i I, where r i (γ 1,..., γ r ) are words in the letters γ ±1 i. Ex: Z r or Z r or RAAGs. Let K be a compact Lie group. K is real algebraic group (zeros of real polynomials). Define G to be the complex zeros of those polynomials. Any and every (affine) algebraic C-group G that arises in this fashion is called reductive. Lawton (GMU) (AMS, September 2016)
10 Step 1: Groups 1 Let Γ be a finitely generated group. So Γ can be presented as γ 1,..., γ r r i (γ 1,..., γ r ) = 1, i I, where r i (γ 1,..., γ r ) are words in the letters γ ±1 i. Ex: Z r or Z r or RAAGs. Let K be a compact Lie group. K is real algebraic group (zeros of real polynomials). Define G to be the complex zeros of those polynomials. Any and every (affine) algebraic C-group G that arises in this fashion is called reductive. We will say a Zariski dense subgroup G G is real reductive if G(R) 0 G G(R). Lawton (GMU) (AMS, September 2016)
11 Step 1: Groups 1 Let Γ be a finitely generated group. So Γ can be presented as γ 1,..., γ r r i (γ 1,..., γ r ) = 1, i I, where r i (γ 1,..., γ r ) are words in the letters γ ±1 i. Ex: Z r or Z r or RAAGs. Let K be a compact Lie group. K is real algebraic group (zeros of real polynomials). Define G to be the complex zeros of those polynomials. Any and every (affine) algebraic C-group G that arises in this fashion is called reductive. We will say a Zariski dense subgroup G G is real reductive if G(R) 0 G G(R). Ex: SL(n, C) or SL(n, R) or SU(n) or SO(n) Lawton (GMU) (AMS, September 2016)
12 Step 2: Homomorphisms Let R Γ (G) = Hom(Γ, G), and denote r {}}{ G G G by G r. Lawton (GMU) (AMS, September 2016)
13 Step 2: Homomorphisms r Let R Γ (G) = Hom(Γ, G), and denote {}}{ G G G by G r. Define Ev : Hom(Γ, G) G r by Ev(ρ) = (ρ(γ 1 ),..., ρ(γ r )). Lawton (GMU) (AMS, September 2016)
14 Step 2: Homomorphisms r Let R Γ (G) = Hom(Γ, G), and denote {}}{ G G G by G r. Define Ev : Hom(Γ, G) G r by Ev(ρ) = (ρ(γ 1 ),..., ρ(γ r )). Ev is always injective (since ρ are homomorphisms). Lawton (GMU) (AMS, September 2016)
15 Step 2: Homomorphisms r Let R Γ (G) = Hom(Γ, G), and denote {}}{ G G G by G r. Define Ev : Hom(Γ, G) G r by Ev(ρ) = (ρ(γ 1 ),..., ρ(γ r )). Ev is always injective (since ρ are homomorphisms). It is not generally surjective since not all r-tuples of elements of G will satisfy r i (g 1,..., g r ) = 1 for all i I. Lawton (GMU) (AMS, September 2016)
16 Step 2: Homomorphisms r Let R Γ (G) = Hom(Γ, G), and denote {}}{ G G G by G r. Define Ev : Hom(Γ, G) G r by Ev(ρ) = (ρ(γ 1 ),..., ρ(γ r )). Ev is always injective (since ρ are homomorphisms). It is not generally surjective since not all r-tuples of elements of G will satisfy r i (g 1,..., g r ) = 1 for all i I. One notable exception is the case when there are no relations r i, that is, when Γ is a free group. Lawton (GMU) (AMS, September 2016)
17 Step 2: Homomorphisms r Let R Γ (G) = Hom(Γ, G), and denote {}}{ G G G by G r. Define Ev : Hom(Γ, G) G r by Ev(ρ) = (ρ(γ 1 ),..., ρ(γ r )). Ev is always injective (since ρ are homomorphisms). It is not generally surjective since not all r-tuples of elements of G will satisfy r i (g 1,..., g r ) = 1 for all i I. One notable exception is the case when there are no relations r i, that is, when Γ is a free group. However, this does show that Hom(Γ, G) = {(g 1,..., g r ) G r r i (g 1,..., g r ) = 1, i I } G r is a subvariety of the smooth variety G r. Lawton (GMU) (AMS, September 2016)
18 Step 3: Moduli Space G acts on Hom(Γ, G) by conjugation: (g, ρ) = gρg 1. Lawton (GMU) (AMS, September 2016)
19 Step 3: Moduli Space G acts on Hom(Γ, G) by conjugation: (g, ρ) = gρg 1. As Hom(Γ, G) is a topological space, let Hom(Γ, G) be the subspace of closed orbits. Lawton (GMU) (AMS, September 2016)
20 Step 3: Moduli Space G acts on Hom(Γ, G) by conjugation: (g, ρ) = gρg 1. As Hom(Γ, G) is a topological space, let Hom(Γ, G) be the subspace of closed orbits. The G-character variety of Γ is the conjugation orbit space X Γ (G) := Hom(Γ, G) /G. Lawton (GMU) (AMS, September 2016)
21 Step 3: Moduli Space G acts on Hom(Γ, G) by conjugation: (g, ρ) = gρg 1. As Hom(Γ, G) is a topological space, let Hom(Γ, G) be the subspace of closed orbits. The G-character variety of Γ is the conjugation orbit space X Γ (G) := Hom(Γ, G) /G. It is a non-trivial result of R. W. Richardson, P. J. Slodowy from 1990 that this space is closed and Hausdorff. Lawton (GMU) (AMS, September 2016)
22 Step 3: Moduli Space G acts on Hom(Γ, G) by conjugation: (g, ρ) = gρg 1. As Hom(Γ, G) is a topological space, let Hom(Γ, G) be the subspace of closed orbits. The G-character variety of Γ is the conjugation orbit space X Γ (G) := Hom(Γ, G) /G. It is a non-trivial result of R. W. Richardson, P. J. Slodowy from 1990 that this space is closed and Hausdorff. If G is real algebraic, then the character variety is semi-algebraic. Lawton (GMU) (AMS, September 2016)
23 Step 3: Moduli Space G acts on Hom(Γ, G) by conjugation: (g, ρ) = gρg 1. As Hom(Γ, G) is a topological space, let Hom(Γ, G) be the subspace of closed orbits. The G-character variety of Γ is the conjugation orbit space X Γ (G) := Hom(Γ, G) /G. It is a non-trivial result of R. W. Richardson, P. J. Slodowy from 1990 that this space is closed and Hausdorff. If G is real algebraic, then the character variety is semi-algebraic. If G is compact, then the character variety is compact and is just the usual orbit space of Hom(Γ, G). Lawton (GMU) (AMS, September 2016)
24 Step 3: Moduli Space G acts on Hom(Γ, G) by conjugation: (g, ρ) = gρg 1. As Hom(Γ, G) is a topological space, let Hom(Γ, G) be the subspace of closed orbits. The G-character variety of Γ is the conjugation orbit space X Γ (G) := Hom(Γ, G) /G. It is a non-trivial result of R. W. Richardson, P. J. Slodowy from 1990 that this space is closed and Hausdorff. If G is real algebraic, then the character variety is semi-algebraic. If G is compact, then the character variety is compact and is just the usual orbit space of Hom(Γ, G). If G is complex reductive, then the character variety is identified with the GIT quotient. Lawton (GMU) (AMS, September 2016)
25 Why care? If you Google any of the following key words, you will find that the study of character varieties at least touches the following topics: flat G-bundles, G-Higgs bundles, holomorphic vector bundles, (G, X )-structures, Mirror symmetry, String vacua, Yang-Mills connections, Convex Cocompactness, knot invariants, Geometric Langlands, Quantization, Spin Networks, A-polynomial, hyperbolic manifolds Lawton (GMU) (AMS, September 2016)
26 Examples Figure : X Z (SL(2, R)) Figure : X Z (SU(3)) Lawton (GMU) (AMS, September 2016)
27 Figure : X Z Z (SU(2)) Figure : X Z Z (SL(2, C))(R) with boundary value 2. Lawton (GMU) (AMS, September 2016)
28 X Z 3(SU(2)) is a 3 dimensional orbifold with 8 singularities; each locally C R (RP 2 ). Lawton (GMU) (AMS, September 2016)
29 History The term character variety arose from the seminal work of Peter Shalen and Marc Culler in Varieties of group representations and splittings of 3-manifolds, published in the Annals of Mathematics in Lawton (GMU) (AMS, September 2016)
30 History The term character variety arose from the seminal work of Peter Shalen and Marc Culler in Varieties of group representations and splittings of 3-manifolds, published in the Annals of Mathematics in One of the theorems they showed in this paper is that the space of SL(2, C)-characters was a variety. Lawton (GMU) (AMS, September 2016)
31 History The term character variety arose from the seminal work of Peter Shalen and Marc Culler in Varieties of group representations and splittings of 3-manifolds, published in the Annals of Mathematics in One of the theorems they showed in this paper is that the space of SL(2, C)-characters was a variety...hence the term. Lawton (GMU) (AMS, September 2016)
32 Variety of Characters Fix a connected reductive subgroup G of SL(n, C). Lawton (GMU) (AMS, September 2016)
33 Variety of Characters Fix a connected reductive subgroup G of SL(n, C). The G-trace algebra of Γ is the subalgebra T Γ (G) C[X Γ (G)] generated by trace functions τ γ for γ Γ defined by τ γ (ρ) = tr(ρ(γ)). Lawton (GMU) (AMS, September 2016)
34 Variety of Characters Fix a connected reductive subgroup G of SL(n, C). The G-trace algebra of Γ is the subalgebra T Γ (G) C[X Γ (G)] generated by trace functions τ γ for γ Γ defined by τ γ (ρ) = tr(ρ(γ)). The G-variety of characters of Γ is then Ch Γ (G) := Spec(T Γ (G)). Lawton (GMU) (AMS, September 2016)
35 Variety of Characters Fix a connected reductive subgroup G of SL(n, C). The G-trace algebra of Γ is the subalgebra T Γ (G) C[X Γ (G)] generated by trace functions τ γ for γ Γ defined by τ γ (ρ) = tr(ρ(γ)). The G-variety of characters of Γ is then Ch Γ (G) := Spec(T Γ (G)). By thinking of SO(2, C), one can deduce that character varieties are not generally varieties of characters, Lawton (GMU) (AMS, September 2016)
36 Variety of Characters Fix a connected reductive subgroup G of SL(n, C). The G-trace algebra of Γ is the subalgebra T Γ (G) C[X Γ (G)] generated by trace functions τ γ for γ Γ defined by τ γ (ρ) = tr(ρ(γ)). The G-variety of characters of Γ is then Ch Γ (G) := Spec(T Γ (G)). By thinking of SO(2, C), one can deduce that character varieties are not generally varieties of characters, and by work of Sikora this generalizes to SO(2n, C). Lawton (GMU) (AMS, September 2016)
37 Variety of Characters Fix a connected reductive subgroup G of SL(n, C). The G-trace algebra of Γ is the subalgebra T Γ (G) C[X Γ (G)] generated by trace functions τ γ for γ Γ defined by τ γ (ρ) = tr(ρ(γ)). The G-variety of characters of Γ is then Ch Γ (G) := Spec(T Γ (G)). By thinking of SO(2, C), one can deduce that character varieties are not generally varieties of characters, and by work of Sikora this generalizes to SO(2n, C). For example, X F2 (SO(2, C)) is homotopic to S 1 S 1 while Ch F2 (SO(2, C)) is homotopic to S 2. Lawton (GMU) (AMS, September 2016)
38 Let Y r = gl(n, C) r /SL(n, C). Then in 1976 Procesi proved: Theorem (Procesi) C[Y r ] is generated by the invariants tr(x i1 X i2 X ik ), where X j are generic matrices. Lawton (GMU) (AMS, September 2016)
39 Let Y r = gl(n, C) r /SL(n, C). Then in 1976 Procesi proved: Theorem (Procesi) C[Y r ] is generated by the invariants tr(x i1 X i2 X ik ), where X j are generic matrices. Therefore, C[X Γ (SL(n, C))] is generated by the invariants tr(x i1 X i2 X ik ), where X j are generic unimodular matrices. Lawton (GMU) (AMS, September 2016)
40 Let Y r = gl(n, C) r /SL(n, C). Then in 1976 Procesi proved: Theorem (Procesi) C[Y r ] is generated by the invariants tr(x i1 X i2 X ik ), where X j are generic matrices. Therefore, C[X Γ (SL(n, C))] is generated by the invariants tr(x i1 X i2 X ik ), where X j are generic unimodular matrices. More generally, T Γ (G) coincides with C[X Γ (G)] for any Γ and G equal to SL(m, C), Sp(2m, C), or SO(2m + 1, C); a result of Lawton and also Sikora. Lawton (GMU) (AMS, September 2016)
41 Let Y r = gl(n, C) r /SL(n, C). Then in 1976 Procesi proved: Theorem (Procesi) C[Y r ] is generated by the invariants tr(x i1 X i2 X ik ), where X j are generic matrices. Therefore, C[X Γ (SL(n, C))] is generated by the invariants tr(x i1 X i2 X ik ), where X j are generic unimodular matrices. More generally, T Γ (G) coincides with C[X Γ (G)] for any Γ and G equal to SL(m, C), Sp(2m, C), or SO(2m + 1, C); a result of Lawton and also Sikora. How do they relate in general? Lawton (GMU) (AMS, September 2016)
42 The set of trace-preserving automorphisms of G is Aut T (G) := {α Aut(G) g G, tr(α(g)) = tr(g)}. Lawton (GMU) (AMS, September 2016)
43 The set of trace-preserving automorphisms of G is Aut T (G) := {α Aut(G) g G, tr(α(g)) = tr(g)}. Aut T (G) acts on Hom(Γ, G) by (α, ρ) α ρ and descends to an action of Out T (G) := Aut T (G)/Inn(G) on X Γ (G). Lawton (GMU) (AMS, September 2016)
44 The set of trace-preserving automorphisms of G is Aut T (G) := {α Aut(G) g G, tr(α(g)) = tr(g)}. Aut T (G) acts on Hom(Γ, G) by (α, ρ) α ρ and descends to an action of Out T (G) := Aut T (G)/Inn(G) on X Γ (G). π : N (G) Aut T (G) given by h C h where C h (g) = hgh 1. Lawton (GMU) (AMS, September 2016)
45 The set of trace-preserving automorphisms of G is Aut T (G) := {α Aut(G) g G, tr(α(g)) = tr(g)}. Aut T (G) acts on Hom(Γ, G) by (α, ρ) α ρ and descends to an action of Out T (G) := Aut T (G)/Inn(G) on X Γ (G). π : N (G) Aut T (G) given by h C h where C h (g) = hgh 1. Since Ker(π) contains the center of SL(n, C), π is not one-to-one. Lawton (GMU) (AMS, September 2016)
46 The set of trace-preserving automorphisms of G is Aut T (G) := {α Aut(G) g G, tr(α(g)) = tr(g)}. Aut T (G) acts on Hom(Γ, G) by (α, ρ) α ρ and descends to an action of Out T (G) := Aut T (G)/Inn(G) on X Γ (G). π : N (G) Aut T (G) given by h C h where C h (g) = hgh 1. Since Ker(π) contains the center of SL(n, C), π is not one-to-one. However, π is onto and Out T (G) is finite. Lawton (GMU) (AMS, September 2016)
47 Proof Let α Aut T (G) and consider irreducible ρ : F 2 G, so ρ(f 2 ) is Zariski dense in G. Lawton (GMU) (AMS, September 2016)
48 Proof Let α Aut T (G) and consider irreducible ρ : F 2 G, so ρ(f 2 ) is Zariski dense in G. Since C[X F2 (SL(n, C))] = T F2 (SL(n, C)) and ρ and αρ have the same character, they coincide in X F2 (SL(n, C)). Lawton (GMU) (AMS, September 2016)
49 Proof Let α Aut T (G) and consider irreducible ρ : F 2 G, so ρ(f 2 ) is Zariski dense in G. Since C[X F2 (SL(n, C))] = T F2 (SL(n, C)) and ρ and αρ have the same character, they coincide in X F2 (SL(n, C)). Since G is a reductive subgroup of SL(n, C), ρ and αρ are completely reducible representations in SL(n, C) and, therefore, their SL(n, C)-conjugation orbits are closed in Hom(F 2, G). Lawton (GMU) (AMS, September 2016)
50 Proof Let α Aut T (G) and consider irreducible ρ : F 2 G, so ρ(f 2 ) is Zariski dense in G. Since C[X F2 (SL(n, C))] = T F2 (SL(n, C)) and ρ and αρ have the same character, they coincide in X F2 (SL(n, C)). Since G is a reductive subgroup of SL(n, C), ρ and αρ are completely reducible representations in SL(n, C) and, therefore, their SL(n, C)-conjugation orbits are closed in Hom(F 2, G). Since there is a unique closed orbit in each equivalence class in X F2 (SL(n, C)), ρ and αρ are conjugate in SL(n, C). Lawton (GMU) (AMS, September 2016)
51 Proof Let α Aut T (G) and consider irreducible ρ : F 2 G, so ρ(f 2 ) is Zariski dense in G. Since C[X F2 (SL(n, C))] = T F2 (SL(n, C)) and ρ and αρ have the same character, they coincide in X F2 (SL(n, C)). Since G is a reductive subgroup of SL(n, C), ρ and αρ are completely reducible representations in SL(n, C) and, therefore, their SL(n, C)-conjugation orbits are closed in Hom(F 2, G). Since there is a unique closed orbit in each equivalence class in X F2 (SL(n, C)), ρ and αρ are conjugate in SL(n, C). Thus, since ρ is irreducible, this implies that α : G G coincides with conjugation of G by some element of SL(n, C). Lawton (GMU) (AMS, September 2016)
52 Proof Let α Aut T (G) and consider irreducible ρ : F 2 G, so ρ(f 2 ) is Zariski dense in G. Since C[X F2 (SL(n, C))] = T F2 (SL(n, C)) and ρ and αρ have the same character, they coincide in X F2 (SL(n, C)). Since G is a reductive subgroup of SL(n, C), ρ and αρ are completely reducible representations in SL(n, C) and, therefore, their SL(n, C)-conjugation orbits are closed in Hom(F 2, G). Since there is a unique closed orbit in each equivalence class in X F2 (SL(n, C)), ρ and αρ are conjugate in SL(n, C). Thus, since ρ is irreducible, this implies that α : G G coincides with conjugation of G by some element of SL(n, C). Now, Out T (G) is the epimorphic image of N (G)/G which is finite by a result of Vinberg. Lawton (GMU) (AMS, September 2016)
53 Theorem (Lawton-Sikora, 2016) Let G SL(n, C) be a connected reductive group, and Γ a finitely generated group. Then: 1 Out T (G) acts freely on X i Γ (G). 2 ϕ i : X i Γ (G)/Out T (G) ChΓ i (G) is a finite, birational bijection. 3 If X i Γ (G)/Out T (G) is normal, then ϕ i is a normalization map. Lawton (GMU) (AMS, September 2016)
54 Theorem (Lawton-Sikora, 2016) Let G SL(n, C) be a connected reductive group, and Γ a finitely generated group. Then: 1 Out T (G) acts freely on X i Γ (G). 2 ϕ i : X i Γ (G)/Out T (G) ChΓ i (G) is a finite, birational bijection. 3 If X i Γ (G)/Out T (G) is normal, then ϕ i is a normalization map. So although character varieties are not varieties of characters in general, we do have: [X i Γ (G)/Out T (G)] = [Ch i Γ (G)]. Lawton (GMU) (AMS, September 2016)
55 Corollaries Corollary Under the assumptions of the last Theorem, we have: 1 If X Γ (G)/Out T (G) is irreducible and contains an irreducible representation, then ϕ is a birational morphism. 2 If X Γ (G)/Out T (G) is normal and contains an irreducible representation, then ϕ is a normalization map. 3 ϕ is a normalization map for Γ a free group of rank r 2. 4 ϕ is a normalization map for Γ a genus g 2 surface group and G = SL(m, C) or GL(m, C), m 1. 5 ϕ is a normalization map for Γ a free abelian group of rank r 1 and G = SL(m, C) or Sp(2m, C), m 1. Lawton (GMU) (AMS, September 2016)
56 Corollary Let F r be a free group of rank r 2. If G is a connected reductive subgroup of SL(n, C) such that all simple factors of the Lie algebra of the derived subgroup [G, G] have rank 2 or more, then X sm F r (G) is étale equivalent to ChF sm r (G). Lawton (GMU) (AMS, September 2016)
57 Thank you! References are at I gratefully acknowledge support from: Lawton (GMU) (AMS, September 2016)
arxiv: v2 [math.at] 2 Jul 2014
TOPOLOGY OF MODULI SPACES OF FREE GROUP REPRESENTATIONS IN REAL REDUCTIVE GROUPS A. C. CASIMIRO, C. FLORENTINO, S. LAWTON, AND A. OLIVEIRA arxiv:1403.3603v2 [math.at] 2 Jul 2014 Abstract. Let G be a real
More informationThe Geometry of Character Varieties
The Geometry of Character Varieties Kate Petersen Florida State December 10, 2011 Kate Petersen ( Florida State) Character Varieties December 10, 2011 1 / 22 The character variety, X (M), of a finite volume
More informationAn introduction to arithmetic groups. Lizhen Ji CMS, Zhejiang University Hangzhou , China & Dept of Math, Univ of Michigan Ann Arbor, MI 48109
An introduction to arithmetic groups Lizhen Ji CMS, Zhejiang University Hangzhou 310027, China & Dept of Math, Univ of Michigan Ann Arbor, MI 48109 June 27, 2006 Plan. 1. Examples of arithmetic groups
More informationTHE MAPPING CLASS GROUP ACTS REDUCIBLY ON SU(n)-CHARACTER VARIETIES
THE MAPPING CLASS GROUP ACTS REDUCIBLY ON SU(n)-CHARACTER VARIETIES WILLIAM M. GOLDMAN Abstract. When G is a connected compact Lie group, and π is a closed surface group, then Hom(π, G)/G contains an open
More informationHiggs Bundles and Character Varieties
Higgs Bundles and Character Varieties David Baraglia The University of Adelaide Adelaide, Australia 29 May 2014 GEAR Junior Retreat, University of Michigan David Baraglia (ADL) Higgs Bundles and Character
More informationLinear connections on Lie groups
Linear connections on Lie groups The affine space of linear connections on a compact Lie group G contains a distinguished line segment with endpoints the connections L and R which make left (resp. right)
More informationEXCERPT FROM ON SOME ACTIONS OF STABLY ELEMENTARY MATRICES ON ALTERNATING MATRICES
EXCERPT FROM ON SOME ACTIONS OF STABLY ELEMENTARY MATRICES ON ALTERNATING MATRICES RAVI A.RAO AND RICHARD G. SWAN Abstract. This is an excerpt from a paper still in preparation. We show that there are
More informationHodge Structures. October 8, A few examples of symmetric spaces
Hodge Structures October 8, 2013 1 A few examples of symmetric spaces The upper half-plane H is the quotient of SL 2 (R) by its maximal compact subgroup SO(2). More generally, Siegel upper-half space H
More information1: Lie groups Matix groups, Lie algebras
Lie Groups and Bundles 2014/15 BGSMath 1: Lie groups Matix groups, Lie algebras 1. Prove that O(n) is Lie group and that its tangent space at I O(n) is isomorphic to the space so(n) of skew-symmetric matrices
More informationAutomorphic Galois representations and Langlands correspondences
Automorphic Galois representations and Langlands correspondences II. Attaching Galois representations to automorphic forms, and vice versa: recent progress Bowen Lectures, Berkeley, February 2017 Outline
More informationGeometric Structure and the Local Langlands Conjecture
Geometric Structure and the Local Langlands Conjecture Paul Baum Penn State Representations of Reductive Groups University of Utah, Salt Lake City July 9, 2013 Paul Baum (Penn State) Geometric Structure
More informationReductive group actions and some problems concerning their quotients
Reductive group actions and some problems concerning their quotients Brandeis University January 2014 Linear Algebraic Groups A complex linear algebraic group G is an affine variety such that the mappings
More informationTHE CREMONA GROUP: LECTURE 1
THE CREMONA GROUP: LECTURE 1 Birational maps of P n. A birational map from P n to P n is specified by an (n + 1)-tuple (f 0,..., f n ) of homogeneous polynomials of the same degree, which can be assumed
More informationBranching rules of unitary representations: Examples and applications to automorphic forms.
Branching rules of unitary representations: Examples and applications to automorphic forms. Basic Notions: Jerusalem, March 2010 Birgit Speh Cornell University 1 Let G be a group and V a vector space.
More informationAlgebraic group actions and quotients
23rd Autumn School in Algebraic Geometry Algebraic group actions and quotients Wykno (Poland), September 3-10, 2000 LUNA S SLICE THEOREM AND APPLICATIONS JEAN MARC DRÉZET Contents 1. Introduction 1 2.
More informationMath 210C. The representation ring
Math 210C. The representation ring 1. Introduction Let G be a nontrivial connected compact Lie group that is semisimple and simply connected (e.g., SU(n) for n 2, Sp(n) for n 1, or Spin(n) for n 3). Let
More informationTHE HITCHIN FIBRATION
THE HITCHIN FIBRATION Seminar talk based on part of Ngô Bao Châu s preprint: Le lemme fondamental pour les algèbres de Lie [2]. Here X is a smooth connected projective curve over a field k whose genus
More informationTHE EULER CHARACTERISTIC OF A LIE GROUP
THE EULER CHARACTERISTIC OF A LIE GROUP JAY TAYLOR 1 Examples of Lie Groups The following is adapted from [2] We begin with the basic definition and some core examples Definition A Lie group is a smooth
More informationLECTURE 14: LIE GROUP ACTIONS
LECTURE 14: LIE GROUP ACTIONS 1. Smooth actions Let M be a smooth manifold, Diff(M) the group of diffeomorphisms on M. Definition 1.1. An action of a Lie group G on M is a homomorphism of groups τ : G
More informationBirational geometry and deformations of nilpotent orbits
arxiv:math/0611129v1 [math.ag] 6 Nov 2006 Birational geometry and deformations of nilpotent orbits Yoshinori Namikawa In order to explain what we want to do in this paper, let us begin with an explicit
More informationMATRIX LIE GROUPS AND LIE GROUPS
MATRIX LIE GROUPS AND LIE GROUPS Steven Sy December 7, 2005 I MATRIX LIE GROUPS Definition: A matrix Lie group is a closed subgroup of Thus if is any sequence of matrices in, and for some, then either
More informationVarieties of characters and knot symmetries (I)
Varieties of characters and knot symmetries (I) Joan Porti Universitat Autònoma de Barcelona June 2017 INdAM Meeting Geometric Topology in Cortona Organization Day 1 Character Varieties. Definition and
More informationRings and groups. Ya. Sysak
Rings and groups. Ya. Sysak 1 Noetherian rings Let R be a ring. A (right) R -module M is called noetherian if it satisfies the maximum condition for its submodules. In other words, if M 1... M i M i+1...
More informationRepresentations and Linear Actions
Representations and Linear Actions Definition 0.1. Let G be an S-group. A representation of G is a morphism of S-groups φ G GL(n, S) for some n. We say φ is faithful if it is a monomorphism (in the category
More informationMAPPING CLASS ACTIONS ON MODULI SPACES. Int. J. Pure Appl. Math 9 (2003),
MAPPING CLASS ACTIONS ON MODULI SPACES RICHARD J. BROWN Abstract. It is known that the mapping class group of a compact surface S, MCG(S), acts ergodically with respect to symplectic measure on each symplectic
More informationWeyl Group Representations and Unitarity of Spherical Representations.
Weyl Group Representations and Unitarity of Spherical Representations. Alessandra Pantano, University of California, Irvine Windsor, October 23, 2008 β ν 1 = ν 2 S α S β ν S β ν S α ν S α S β S α S β ν
More informationGelfand Pairs and Invariant Distributions
Gelfand Pairs and Invariant Distributions A. Aizenbud Massachusetts Institute of Technology http://math.mit.edu/~aizenr Examples Example (Fourier Series) Examples Example (Fourier Series) L 2 (S 1 ) =
More informationThe Grothendieck-Katz Conjecture for certain locally symmetric varieties
The Grothendieck-Katz Conjecture for certain locally symmetric varieties Benson Farb and Mark Kisin August 20, 2008 Abstract Using Margulis results on lattices in semi-simple Lie groups, we prove the Grothendieck-
More informationCharacteristic classes in the Chow ring
arxiv:alg-geom/9412008v1 10 Dec 1994 Characteristic classes in the Chow ring Dan Edidin and William Graham Department of Mathematics University of Chicago Chicago IL 60637 Let G be a reductive algebraic
More informationMath 210C. A non-closed commutator subgroup
Math 210C. A non-closed commutator subgroup 1. Introduction In Exercise 3(i) of HW7 we saw that every element of SU(2) is a commutator (i.e., has the form xyx 1 y 1 for x, y SU(2)), so the same holds for
More informationMODULI OF ALGEBRAIC SL 3 -VECTOR BUNDLES OVER ADJOINT REPRESENTATION
Masuda, K. Osaka J. Math. 38 (200), 50 506 MODULI OF ALGEBRAIC SL 3 -VECTOR BUNDLES OVER ADJOINT REPRESENTATION KAYO MASUDA (Received June 2, 999). Introduction and result Let be a reductive complex algebraic
More informationLECTURE 15-16: PROPER ACTIONS AND ORBIT SPACES
LECTURE 15-16: PROPER ACTIONS AND ORBIT SPACES 1. Proper actions Suppose G acts on M smoothly, and m M. Then the orbit of G through m is G m = {g m g G}. If m, m lies in the same orbit, i.e. m = g m for
More informationClassification of discretely decomposable A q (λ) with respect to reductive symmetric pairs UNIVERSITY OF TOKYO
UTMS 2011 8 April 22, 2011 Classification of discretely decomposable A q (λ) with respect to reductive symmetric pairs by Toshiyuki Kobayashi and Yoshiki Oshima T UNIVERSITY OF TOKYO GRADUATE SCHOOL OF
More informationLECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI. 1. Maximal Tori
LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI 1. Maximal Tori By a torus we mean a compact connected abelian Lie group, so a torus is a Lie group that is isomorphic to T n = R n /Z n. Definition 1.1.
More informationDelzant s Garden. A one-hour tour to symplectic toric geometry
Delzant s Garden A one-hour tour to symplectic toric geometry Tour Guide: Zuoqin Wang Travel Plan: The earth America MIT Main building Math. dept. The moon Toric world Symplectic toric Delzant s theorem
More informationwhere m is the maximal ideal of O X,p. Note that m/m 2 is a vector space. Suppose that we are given a morphism
8. Smoothness and the Zariski tangent space We want to give an algebraic notion of the tangent space. In differential geometry, tangent vectors are equivalence classes of maps of intervals in R into the
More informationRepresentations Are Everywhere
Representations Are Everywhere Nanghua Xi Member of Chinese Academy of Sciences 1 What is Representation theory Representation is reappearance of some properties or structures of one object on another.
More informationAn Introduction to Kuga Fiber Varieties
An Introduction to Kuga Fiber Varieties Dylan Attwell-Duval Department of Mathematics and Statistics McGill University Montreal, Quebec attwellduval@math.mcgill.ca April 28, 2012 Notation G a Q-simple
More informationThese notes are incomplete they will be updated regularly.
These notes are incomplete they will be updated regularly. LIE GROUPS, LIE ALGEBRAS, AND REPRESENTATIONS SPRING SEMESTER 2008 RICHARD A. WENTWORTH Contents 1. Lie groups and Lie algebras 2 1.1. Definition
More informationInverse Galois Problem for C(t)
Inverse Galois Problem for C(t) Padmavathi Srinivasan PuMaGraSS March 2, 2012 Outline 1 The problem 2 Compact Riemann Surfaces 3 Covering Spaces 4 Connection to field theory 5 Proof of the Main theorem
More informationSymplectic varieties and Poisson deformations
Symplectic varieties and Poisson deformations Yoshinori Namikawa A symplectic variety X is a normal algebraic variety (defined over C) which admits an everywhere non-degenerate d-closed 2-form ω on the
More informationTrace fields of knots
JT Lyczak, February 2016 Trace fields of knots These are the knotes from the seminar on knot theory in Leiden in the spring of 2016 The website and all the knotes for this seminar can be found at http://pubmathleidenunivnl/
More informationMathematics 7800 Quantum Kitchen Sink Spring 2002
Mathematics 7800 Quantum Kitchen Sink Spring 2002 4. Quotients via GIT. Most interesting moduli spaces arise as quotients of schemes by group actions. We will first analyze such quotients with geometric
More informationA Numerical Criterion for Lower bounds on K-energy maps of Algebraic manifolds
A Numerical Criterion for Lower bounds on K-energy maps of Algebraic manifolds Sean Timothy Paul University of Wisconsin, Madison stpaul@math.wisc.edu Outline Formulation of the problem: To bound the Mabuchi
More informationCombinatorics and geometry of E 7
Combinatorics and geometry of E 7 Steven Sam University of California, Berkeley September 19, 2012 1/24 Outline Macdonald representations Vinberg representations Root system Weyl group 7 points in P 2
More informationTopics in Representation Theory: Roots and Weights
Topics in Representation Theory: Roots and Weights 1 The Representation Ring Last time we defined the maximal torus T and Weyl group W (G, T ) for a compact, connected Lie group G and explained that our
More informationAn inclusion between sets of orbits and surjectivity of the restriction map of rings of invariants
Hokkaido Mathematical Journal Vol. 37 (2008) p. 437 454 An inclusion between sets of orbits and surjectivity of the restriction map of rings of invariants Takuya Ohta (Received February 28, 2007; Revised
More informationCurtis Heberle MTH 189 Final Paper 12/14/2010. Algebraic Groups
Algebraic Groups Curtis Heberle MTH 189 Final Paper 12/14/2010 The primary objects of study in algebraic geometry are varieties. Having become acquainted with these objects, it is interesting to consider
More informationBetti numbers of abelian covers
Betti numbers of abelian covers Alex Suciu Northeastern University Geometry and Topology Seminar University of Wisconsin May 6, 2011 Alex Suciu (Northeastern University) Betti numbers of abelian covers
More informationSpace of surjective morphisms between projective varieties
Space of surjective morphisms between projective varieties -Talk at AMC2005, Singapore- Jun-Muk Hwang Korea Institute for Advanced Study 207-43 Cheongryangri-dong Seoul, 130-722, Korea jmhwang@kias.re.kr
More informationREGULAR TRIPLETS IN COMPACT SYMMETRIC SPACES
REGULAR TRIPLETS IN COMPACT SYMMETRIC SPACES MAKIKO SUMI TANAKA 1. Introduction This article is based on the collaboration with Tadashi Nagano. In the first part of this article we briefly review basic
More information4. The concept of a Lie group
4. The concept of a Lie group 4.1. The category of manifolds and the definition of Lie groups. We have discussed the concept of a manifold-c r, k-analytic, algebraic. To define the category corresponding
More informationNotation. For any Lie group G, we set G 0 to be the connected component of the identity.
Notation. For any Lie group G, we set G 0 to be the connected component of the identity. Problem 1 Prove that GL(n, R) is homotopic to O(n, R). (Hint: Gram-Schmidt Orthogonalization.) Here is a sequence
More informationPOLYNOMIAL IDENTITY RINGS AS RINGS OF FUNCTIONS
POLYNOMIAL IDENTITY RINGS AS RINGS OF FUNCTIONS Z. REICHSTEIN AND N. VONESSEN Abstract. We generalize the usual relationship between irreducible Zariski closed subsets of the affine space, their defining
More informationThe Geometry of Conjugacy Classes of Nilpotent Matrices
The Geometry of Conjugacy Classes of Nilpotent Matrices A 5 A 3 A 1 A1 Alessandra Pantano A 2 A 2 a 2 a 2 Oliver Club Talk, Cornell April 14, 2005 a 1 a 1 a 3 a 5 References: H. Kraft and C. Procesi, Minimal
More informationMorse theory and stable pairs
Richard A. SCGAS 2010 Joint with Introduction Georgios Daskalopoulos (Brown University) Jonathan Weitsman (Northeastern University) Graeme Wilkin (University of Colorado) Outline Introduction 1 Introduction
More informationAffine Geometry and Hyperbolic Geometry
Affine Geometry and Hyperbolic Geometry William Goldman University of Maryland Lie Groups: Dynamics, Rigidity, Arithmetic A conference in honor of Gregory Margulis s 60th birthday February 24, 2006 Discrete
More informationAZUMAYA ALGEBRAS AND CANONICAL COMPONENTS
AZUMAYA ALGEBRAS AND CANONICAL COMPONENTS TED CHINBURG, ALAN W. REID, AND MATTHEW STOVER Abstract. Let M be a compact 3-manifold and Γ = π 1 (M). The work of Thurston and Culler Shalen established the
More informationGeometry of moduli spaces
Geometry of moduli spaces 20. November 2009 1 / 45 (1) Examples: C: compact Riemann surface C = P 1 (C) = C { } (Riemann sphere) E = C / Z + Zτ (torus, elliptic curve) 2 / 45 (2) Theorem (Riemann existence
More informationK-stability and Kähler metrics, I
K-stability and Kähler metrics, I Gang Tian Beijing University and Princeton University Let M be a Kähler manifold. This means that M be a complex manifold together with a Kähler metric ω. In local coordinates
More informationIVAN LOSEV. Lemma 1.1. (λ) O. Proof. (λ) is generated by a single vector, v λ. We have the weight decomposition (λ) =
LECTURE 7: CATEGORY O AND REPRESENTATIONS OF ALGEBRAIC GROUPS IVAN LOSEV Introduction We continue our study of the representation theory of a finite dimensional semisimple Lie algebra g by introducing
More informationIsogeny invariance of the BSD conjecture
Isogeny invariance of the BSD conjecture Akshay Venkatesh October 30, 2015 1 Examples The BSD conjecture predicts that for an elliptic curve E over Q with E(Q) of rank r 0, where L (r) (1, E) r! = ( p
More informationLecture 1. Toric Varieties: Basics
Lecture 1. Toric Varieties: Basics Taras Panov Lomonosov Moscow State University Summer School Current Developments in Geometry Novosibirsk, 27 August1 September 2018 Taras Panov (Moscow University) Lecture
More informationH(G(Q p )//G(Z p )) = C c (SL n (Z p )\ SL n (Q p )/ SL n (Z p )).
92 19. Perverse sheaves on the affine Grassmannian 19.1. Spherical Hecke algebra. The Hecke algebra H(G(Q p )//G(Z p )) resp. H(G(F q ((T ))//G(F q [[T ]])) etc. of locally constant compactly supported
More informationIN POSITIVE CHARACTERISTICS: 3. Modular varieties with Hecke symmetries. 7. Foliation and a conjecture of Oort
FINE STRUCTURES OF MODULI SPACES IN POSITIVE CHARACTERISTICS: HECKE SYMMETRIES AND OORT FOLIATION 1. Elliptic curves and their moduli 2. Moduli of abelian varieties 3. Modular varieties with Hecke symmetries
More informationNotes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop. Eric Sommers
Notes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop Eric Sommers 17 July 2009 2 Contents 1 Background 5 1.1 Linear algebra......................................... 5 1.1.1
More informationALGEBRAIC GROUPS J. WARNER
ALGEBRAIC GROUPS J. WARNER Let k be an algebraically closed field. varieties unless otherwise stated. 1. Definitions and Examples For simplicity we will work strictly with affine Definition 1.1. An algebraic
More informationINSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD
INSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD () Instanton (definition) (2) ADHM construction (3) Compactification. Instantons.. Notation. Throughout this talk, we will use the following notation:
More informationMargulis Superrigidity I & II
Margulis Superrigidity I & II Alastair Litterick 1,2 and Yuri Santos Rego 1 Universität Bielefeld 1 and Ruhr-Universität Bochum 2 Block seminar on arithmetic groups and rigidity Universität Bielefeld 22nd
More informationLECTURES ON SYMPLECTIC REFLECTION ALGEBRAS
LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS IVAN LOSEV 16. Symplectic resolutions of µ 1 (0)//G and their deformations 16.1. GIT quotients. We will need to produce a resolution of singularities for C 2n
More informationSpring 2018 CIS 610. Advanced Geometric Methods in Computer Science Jean Gallier Homework 3
Spring 2018 CIS 610 Advanced Geometric Methods in Computer Science Jean Gallier Homework 3 March 20; Due April 5, 2018 Problem B1 (80). This problem is from Knapp, Lie Groups Beyond an Introduction, Introduction,
More informationSPHERICAL UNITARY REPRESENTATIONS FOR REDUCTIVE GROUPS
SPHERICAL UNITARY REPRESENTATIONS FOR REDUCTIVE GROUPS DAN CIUBOTARU 1. Classical motivation: spherical functions 1.1. Spherical harmonics. Let S n 1 R n be the (n 1)-dimensional sphere, C (S n 1 ) the
More informationQUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday 10 February 2004 (Day 1)
Tuesday 10 February 2004 (Day 1) 1a. Prove the following theorem of Banach and Saks: Theorem. Given in L 2 a sequence {f n } which weakly converges to 0, we can select a subsequence {f nk } such that the
More informationOn a question of B.H. Neumann
On a question of B.H. Neumann Robert Guralnick Department of Mathematics University of Southern California E-mail: guralnic@math.usc.edu Igor Pak Department of Mathematics Massachusetts Institute of Technology
More informationNotes 10: Consequences of Eli Cartan s theorem.
Notes 10: Consequences of Eli Cartan s theorem. Version 0.00 with misprints, The are a few obvious, but important consequences of the theorem of Eli Cartan on the maximal tori. The first one is the observation
More information1 Smooth manifolds and Lie groups
An undergraduate approach to Lie theory Slide 1 Andrew Baker, Glasgow Glasgow, 12/11/1999 1 Smooth manifolds and Lie groups A continuous g : V 1 V 2 with V k R m k open is called smooth if it is infinitely
More informationTopics in Representation Theory: SU(n), Weyl Chambers and the Diagram of a Group
Topics in Representation Theory: SU(n), Weyl hambers and the Diagram of a Group 1 Another Example: G = SU(n) Last time we began analyzing how the maximal torus T of G acts on the adjoint representation,
More informationThe local geometry of compact homogeneous Lorentz spaces
The local geometry of compact homogeneous Lorentz spaces Felix Günther Abstract In 1995, S. Adams and G. Stuck as well as A. Zeghib independently provided a classification of non-compact Lie groups which
More informationLemma 1.3. The element [X, X] is nonzero.
Math 210C. The remarkable SU(2) Let G be a non-commutative connected compact Lie group, and assume that its rank (i.e., dimension of maximal tori) is 1; equivalently, G is a compact connected Lie group
More informationMany of the exercises are taken from the books referred at the end of the document.
Exercises in Geometry I University of Bonn, Winter semester 2014/15 Prof. Christian Blohmann Assistant: Néstor León Delgado The collection of exercises here presented corresponds to the exercises for the
More informationMirror symmetry, Langlands duality and the Hitchin system I
Mirror symmetry, Langlands duality and the Hitchin system I Tamás Hausel Royal Society URF at University of Oxford http://www.maths.ox.ac.uk/ hausel/talks.html April 200 Simons lecture series Stony Brook
More informationRepresentation Theory and Orbital Varieties. Thomas Pietraho Bowdoin College
Representation Theory and Orbital Varieties Thomas Pietraho Bowdoin College 1 Unitary Dual Problem Definition. A representation (π, V ) of a group G is a homomorphism ρ from G to GL(V ), the set of invertible
More information16.2. Definition. Let N be the set of all nilpotent elements in g. Define N
74 16. Lecture 16: Springer Representations 16.1. The flag manifold. Let G = SL n (C). It acts transitively on the set F of complete flags 0 F 1 F n 1 C n and the stabilizer of the standard flag is the
More informationLECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS. 1. Lie groups
LECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS 1. Lie groups A Lie group is a special smooth manifold on which there is a group structure, and moreover, the two structures are compatible. Lie groups are
More informationCurves with many symmetries
ETH Zürich Bachelor s thesis Curves with many symmetries by Nicolas Müller supervised by Prof. Dr. Richard Pink September 30, 015 Contents 1. Properties of curves 4 1.1. Smoothness, irreducibility and
More informationDefinition 9.1. The scheme µ 1 (O)/G is called the Hamiltonian reduction of M with respect to G along O. We will denote by R(M, G, O).
9. Calogero-Moser spaces 9.1. Hamiltonian reduction along an orbit. Let M be an affine algebraic variety and G a reductive algebraic group. Suppose M is Poisson and the action of G preserves the Poisson
More informationStable bundles on CP 3 and special holonomies
Stable bundles on CP 3 and special holonomies Misha Verbitsky Géométrie des variétés complexes IV CIRM, Luminy, Oct 26, 2010 1 Hyperkähler manifolds DEFINITION: A hyperkähler structure on a manifold M
More informationarxiv: v2 [math.ag] 5 Jun 2018
A Brief Survey of Higgs Bundles 1 21/03/2018 Ronald A. Zúñiga-Rojas 2 Centro de Investigaciones Matemáticas y Metamatemáticas CIMM Universidad de Costa Rica UCR San José 11501, Costa Rica e-mail: ronald.zunigarojas@ucr.ac.cr
More informationOn some smooth projective two-orbit varieties with Picard number 1
On some smooth projective two-orbit varieties with Picard number 1 Boris Pasquier March 3, 2009 Abstract We classify all smooth projective horospherical varieties with Picard number 1. We prove that the
More information15 Elliptic curves and Fermat s last theorem
15 Elliptic curves and Fermat s last theorem Let q > 3 be a prime (and later p will be a prime which has no relation which q). Suppose that there exists a non-trivial integral solution to the Diophantine
More informationMathematical Research Letters 3, (1996) EQUIVARIANT AFFINE LINE BUNDLES AND LINEARIZATION. Hanspeter Kraft and Frank Kutzschebauch
Mathematical Research Letters 3, 619 627 (1996) EQUIVARIANT AFFINE LINE BUNDLES AND LINEARIZATION Hanspeter Kraft and Frank Kutzschebauch Abstract. We show that every algebraic action of a linearly reductive
More informationFiniteness of the Moderate Rational Points of Once-punctured Elliptic Curves. Yuichiro Hoshi
Hokkaido Mathematical Journal ol. 45 (2016) p. 271 291 Finiteness of the Moderate Rational Points of Once-punctured Elliptic Curves uichiro Hoshi (Received February 28, 2014; Revised June 12, 2014) Abstract.
More informationLeafwise Holomorphic Functions
Leafwise Holomorphic Functions R. Feres and A. Zeghib June 20, 2001 Abstract It is a well-known and elementary fact that a holomorphic function on a compact complex manifold without boundary is necessarily
More informationSubgroups of Linear Algebraic Groups
Subgroups of Linear Algebraic Groups Subgroups of Linear Algebraic Groups Contents Introduction 1 Acknowledgements 4 1. Basic definitions and examples 5 1.1. Introduction to Linear Algebraic Groups 5 1.2.
More informationTransparent connections
The abelian case A definition (M, g) is a closed Riemannian manifold, d = dim M. E M is a rank n complex vector bundle with a Hermitian metric (i.e. a U(n)-bundle). is a Hermitian (i.e. metric) connection
More informationz, w = z 1 w 1 + z 2 w 2 z, w 2 z 2 w 2. d([z], [w]) = 2 φ : P(C 2 ) \ [1 : 0] C ; [z 1 : z 2 ] z 1 z 2 ψ : P(C 2 ) \ [0 : 1] C ; [z 1 : z 2 ] z 2 z 1
3 3 THE RIEMANN SPHERE 31 Models for the Riemann Sphere One dimensional projective complex space P(C ) is the set of all one-dimensional subspaces of C If z = (z 1, z ) C \ 0 then we will denote by [z]
More informationThe Spinor Representation
The Spinor Representation Math G4344, Spring 2012 As we have seen, the groups Spin(n) have a representation on R n given by identifying v R n as an element of the Clifford algebra C(n) and having g Spin(n)
More informationHomogeneous Coordinate Ring
Students: Kaiserslautern University Algebraic Group June 14, 2013 Outline Quotients in Algebraic Geometry 1 Quotients in Algebraic Geometry 2 3 4 Outline Quotients in Algebraic Geometry 1 Quotients in
More informationMath 594. Solutions 5
Math 594. Solutions 5 Book problems 6.1: 7. Prove that subgroups and quotient groups of nilpotent groups are nilpotent (your proof should work for infinite groups). Give an example of a group G which possesses
More informationThe Moduli Space of Rank 2 Vector Bundles on Projective Curves
The Moduli Space of Rank Vector Bundles on Projective urves T. Nakamura, A. Simonetti 1 Introduction Let be a smooth projective curve over and let L be a line bundle on. onsider the moduli functor VB (,
More information