By K. Ueno

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**Example text**

N = p+q V ~ C vP,q e vp,q We choose a polarization a morphism polarizable then i and write the Weil operator. ) C2 acts as V (D(-n) for (C Cvp,q = is reductive. is often called i-P+q vp,q , and so is the weight of V. In H H~, ~ e v,v' V • then the real-form of a a (complex conjugation) on 0 Hover of Ha(~) • We are a connected algebraic is reductive if it has a compact real-form ]R H there is an Ha < u,v > V °= J Ha < hu,hv > dh where V. If < is any , > W is an Ha-stable V, then its orthogonal complement is also subspace of Ha-stable.

E ~xlz GI IR Let such that z I}. ·h(~x) = G :m it follows that GI GO = Gl Then and be the smallest I GlR GI contains C G , and h(U I ) = , and therefore is connected. Since Its square C c2 = h(i) acts as acts as (_l)n i'f on 1 on v and therefore lies in the 46 centre of GO(IR). defined by C is therefore an involution. 7. u,v e v~ = tP(gu,CC -1 gCv) = $ (gu,Cg*v) 1lJ(gu,gCv) -1- g* Gm. we have lIJ(U,CV) where For of on ad C V , <9> • Thus the positive definite form m is invariant under the real-form of and so the real-form is compact.

In one special case this is easy. 5. and let A d A be an abelian variety of dimension 2" O AO @lI! E. ) (~) E Let = consists of absolute Hodge cycles. 15», and for this we need to consider polarized abelian varieties. e. (The Rosati involution be such that f -f, and let lJJ There exists a unique E-Hermitian such that W(x,y) V and = TrE/W(f$(x,y» • W be finite-dimensional vector E, and let $ :V x W -+~(ev,w) E-bilinear form V $' = a$. We first need: spaces over (V 9 on is determined by the condition A.