Equivariant Embeddings of Algebraic Groups

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Clearly Γ(Y, y 0 ) = ⋂ h∈G hΓ(Y, y 0)h −1 ⊂ ⋂ h∈G hΓ(X, x 0)h −1 = Γ(X, x 0 ) G ⊂ Γ(X, x 0 ), so that A Γ(X,x0 ) ⊂ A Γ(X,x0 ) G ⊂ A Γ(Y,y 0 ), as indicated in the diagram below. Moreover, as Γ(X, x 0 ) G is G-stable, applying Corollary 3 again shows that A Γ(X,x0 ) G is a uniquely determined subalgebra of k[G] which is independent of the choice of base point x 0 ∈ X made above. Thus we have the following diagram of algebras: k[Y ] ϕ ◦ k[X] k[G] ⊂ ∼= A ψy ◦ Γ(Y,y0 ) 0 ⊂ ⊂ ⊂ ∼= A ψx ◦ Γ(X,x0 ) 0 A Γ(X,x0 ) G = k[X G] The morphism ϕ (XG ,x 0 ) : Y → X G corresponds to the restriction of the homomorphism (ψ ◦ y 0 ) −1 from k[X G ] = A Γ(X,x0 ) G → k[Y ], while β (X,x 0 ) : X G → X is dual to the composition of ψ ◦ x 0 : k[X] → A Γ(X,x0 ) followed by the canonical inclusion A Γ(X,x0 ) ⊂ A Γ(X,x0 ) G. Then it is clear that ϕ factors as ϕ = β (X,x0 ) ◦ ϕ (XG ,x 0 ). Now suppose that x 1 = h · x 0 is another base point in X. Then h · y 0 is the unique element of Y such that ϕ(y) = x 1 , for ϕ(h · y 0 ) = h · ϕ(y 0 ) = h · x 0 . Then ψ ◦ x 1 is an isomorphism from k[X] to A Γ(X,h·x0 ) = A hΓ(X,x0 )h −1 = r h(A Γ(X,x0 )) ⊂ k[G]. Yet Γ(Y, h · y 0 ) = Γ(Y, y 0 ) ⊂ Γ(X, x 0 ) G = Γ(X, h · x 0 ) G ⊂ Γ(X, h · x 0 ), so we have ϕ ◦ = (ψh·y ◦ 0 ) −1 ◦ ψh·x ◦ 0 : k[X] → k[Y ]. As above, this factors through the inclusions A Γ(X,h·x0 ) ⊂ k[X G ] = A Γ(X,h·x0 ) G ⊂ A Γ(Y,h·y 0 ), giving morphisms ϕ (XG ,h·x 0 ) : Y → X G and β (X,h·x0 ) : X G → X corresponding to (ψh·y ◦ 0 ) −1 = [r h (ψy ◦ 0 )] −1 = r h −1[(ψy ◦ 0 ) −1 ] and ψh·x ◦ 0 = r h (ψx ◦ 0 ), respectively. Hence ϕ (XG ,h·x 0 ) = r h −1(ϕ (XG ,x 0 )) and β (X,h·x0 ) = r h (β (X,x0 )). 66
References [1] A. Bialynicki-Birula, Some theorems on actions of algebraic groups, Ann. of Math. (2) 98 (1973), 480–497. [2] F. Bien and M. Brion, Automorphisms and local rigidity of regular varieties, Compositio Mathematica 104 (1996), 1–26. [3] E. Bifet, C. De Concini and C. Procesi, Cohomology of regular embeddings, Adv. Math. 82:1 (1990), 1–34. [4] A. Borel, Linear Algebraic Groups, 2nd Edition, Graduate Texts in Mathematics 126, Springer- Verlag, 1991. [5] M. Brion, The behaviour at infinity of the Bruhat decomposition, Comm. Math. Helvetici 73 (1998), 137–174. [6] M. Brion, Group completions via Hilbert schemes, J. Algebraic Geom. 12 (2003), no. 4, 605– 626. [7] C. De Concini and C. Procesi, Complete symmetric varieties in Invariant Theory (Montecantini 1982), 1–44, Lecture Notes in Math 996, Springer, 1983. [8] C. De Concini and C. Procesi, Complete symmetric varieties, II Intersection theory in Algebraic Groups and Related Topics (Kyoto/Nagoya, 1983), 481–513, Adv. Stud. Pure Math. 6, North- Holland, 1985. [9] C. De Concini and C. Procesi, Cohomology and compactifications of algebraic groups, Duke Math. J. 53:3 (1986), 585–594. 67
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c○ Copyright by David Charles Mur
Page 3 and 4:
Abstract We classify embeddings of
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Acknowledgements This thesis would
Page 7 and 8:
Chapter 1 Introduction A basic prob
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x i ≠ 0} for i = 0, 1, 2, the one
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complement of U in G/H. Remark 1 ([
Page 13 and 14:
G. The wonderful compactification o
Page 15 and 16:
X is to use one-parameter subgroups
Page 17 and 18:
Chapter 2 Toric varieties and group
Page 19 and 20:
Proposition 3 ([28], Proposition I.
Page 21 and 22: and every toric variety for T arise
Page 23 and 24: To see this, suppose D is a T -stab
Page 25 and 26: Conversely, every subdivision of C
Page 27 and 28: 0 : 0 : 1 : 1], [0 : 0 : 0 : 1 : 1]
Page 29 and 30: Mumford’s embedding is constructe
Page 31 and 32: Moreover, Ind(X) is a G-embedding,
Page 33 and 34: the T -action with respect to the c
Page 35 and 36: from X with a line bundle from the
Page 37 and 38: G/B × G/B − corresponding to the
Page 39 and 40: since G (and hence G × G) is semi-
Page 41 and 42: 1. There is a finite-dimensional G-
Page 43 and 44: the following (obviously) commutati
Page 45 and 46: Now, to prove the claim, by Lemma 7
Page 47 and 48: ational polyhedral cone in X ∗ (T
Page 49 and 50: ⋃ P ⊃T P • (σ ∩ ∆ P (T )
Page 51 and 52: Section 3.2.3, is a uniqueness theo
Page 53 and 54: Therefore, the classification of af
Page 55 and 56: For each k-point g of G, we have ma
Page 57 and 58: is over a finite set of integers J.
Page 59 and 60: i ≥ 0, 〈χ ij , γ (0,1) 〉 =
Page 61 and 62: Lemma 12. If γ 1 , γ 2 ∈ X ∗
Page 63 and 64: an integrally closed left-invariant
Page 65 and 66: of the subalgebras A Γ(X,x0 ) ⊂
Page 67 and 68: on varieties discussed in this sect
Page 69 and 70: Proof. Let Γ be a strongly convex
Page 71: that Γ(X, [I 2 , (1, 1)]) GL 2 = {
Page 75 and 76: [24] B. Iversen, Cohomology of Shea
Page 77: Vita David Charles Murphy was born
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Equivariant Embeddings of Algebraic Groups

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