New PDF release: Positivity in algebraic geometry

By Lazarsfeld R.

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There is a canonical Gal(F /F )-equivariant embedding Z(G) −→ Z(M). 1) is trivial if F is local, and in Ker1 (F, Z(G)) if F is global. The G-triple (M , sM , ηM,0 ) is called elliptic if it is elliptic as an endoscopic triple for M. Let (M1 , s1 , η1,0 ) and (M2 , s2 , η2,0 ) be endoscopic G-triples for M. 5) such that the images of s1 and α(s2 ) in Z(M1 )/Z(G) are equal. Let (M , sM , ηM,0 ) be an endoscopic G-triple for M. Then there is an isomorphism class of endoscopic triples for G associated to (M , sM , ηM,0 ) in the following way (cf.

On the other hand, there exists an odd integer C ∈ Z such that, for every z ∈ C× = WC , ϕ∞ (z) = ((1, (B1 (z), . . , Br (z))), z), with  − − z Cni /2 z −Cni /2 In+i Bi (z) =  0  0 + + z −Cni /2 z Cni /2 In−i . We finish this section by a calculation of Tamagawa numbers. 3 (i) Let n1 , . . , nr ∈ N∗ and G = G(U(a1 , b1 ) × · · · × U(ar , br )), C× × with ai + bi = ni . Then Ker1 (Q, G) = {1}, and Z(G)Gal(E/Q) {( 1 , . . , r ) ∈ {±1}r | 1n1 . . rnr = 1}. Hence the Tamagawa number of G is τ (G) = 2r 2 r−1 if all the ni are even, otherwise.

Hence (Z(H)Gal(E/Q) )0 = C× × {1} ⊂ Z(G), and (H, s, η0 ) is elliptic. chapter02 October 9, 2009 38 CHAPTER 2 We want to calculate the group of outer automorphisms of (H, s, η0 ). It is the same to calculate the group of outer automorphisms of the endoscopic data (s, ρ) associated to (H, s, η0 ) (cf. 6). Let − I = {i ∈ {1, . . , r}|n+ i = ni }. Let g ∈ G be such that Int(g)(η0 (H)) = η0 (H). Let a, b ∈ N be such that a + b = n > 0, and G = GLa 0 0 GLb ⊂ GLn . If a = b, then the normalizer of G in GLn is G .

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Positivity in algebraic geometry by Lazarsfeld R.


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