## An introduction to real and complex manifolds. by Giuliano Sorani

By Giuliano Sorani

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Theorem: For any measure Jl E M(IPn - 1 (k)), the stabilizer PGL(n, k)JL has a normal subgroup of finite index which is k-almost algebraic. In particular, for k = IR, PGL(n, IR)p is itself the real points of an IR-group . 4 Proof: As it should cause no confusion, for this proof we shall refer to the k-points of a k-group simply as an algebraic group. For any J1E M( IPn - 1 (k)) we can find a countable family of (positive) measures { u;} on IP n - 1 (k) of total measure at most 1 such that: (a) for i # j, Jl; l_ Jli; (bl LJli i = jl; (c) if V; is of minimal dimension among all linear subspaces with Jl;( [ V] ) > 0, then supp(Jl;) c [ V;]; 41 Algebraic groups and measure theory (d) with [ V;] a s i n (c), if i # j, then [ V;] # [ ViJ As in 3.

Since n is a rational representation, n: Gk/ker(n) ---+ n(Gk) is an isomorphism (of topological groups) onto a topologically closed subgroup of GL(n, k) and since G is almost k-simple, ker(n) is finite. u is compact if and only if its 48 Ergodic theory and semisimple groups image in PGL(n, k) is compact If [n(Gk)]11 is not compact, we argue as in 3 . 2. 1 5 Namely, choose Vi as in Corollary 3 . 2. 2, let M be the k-group with Mk = {gE GL(n, k) l g(u Vi) = u Vi } , and L = n - 1 (M).. By construction (Gk)11 c Lk, and hence it suffices to see that dim L < dim G.

Thus the class of quasi-projective varieties contains both the affine and projective varieties Regular functions into K and regular maps between varieties are defined to be continuous functions that are regular when restricted to mappings between open affine subvarieties . A rational map is one that is regular on an open set Clearly, all of these notions can be defined over k if the varieties are, and if f is defined over k, we still have I ( Vk) c wk The group GL(n, K) is a principal open set in K n 2, and hence is a variety.