Algebraic Geometry. Proc. conf. Tokyo, Kyoto, 1982 by M. Raynaud, T. Shioda

By M. Raynaud, T. Shioda

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Example text

And 7r : N x 111 of the equivalence relation. Given y = 7(x,t) E Mjet V = N x (t 1/2,t + 1/2) and U = r(V). It is easily verified that ir I (U) = U„(z V,„ where V, = gn (V) and 7r : Vn U is a homeomorphism for every n E Z . One concludes then that Ir : N X 1R M is a covering map. On the other hand, for every n E Z , one has that g n = (7r! V) -I o 7r I V : V Va is a Cr diffeomorphism. Hence it is possible to induce on M a e. manifold structure such that 7r is a local e. diffeomorphism and dim ( M) = dim (N) + 1 (see example 6 in Chapter I).

Given u E Tg M, we can write u = + u2 where E P(q) and u 2 = A g (u) E (q) are unique. If X is a continuous vector field on M, it is easy to see that Y( q) = A g ( X (q)) is also a continuous field. Moreover if F C TM is a plane complementary to P(q) then the restric tion A g j F: F P I (q) is an isomorphism. , A q 1 ( 14, * k ) 1. Therefore 0 is a continuous orientation of (P" if and only if A* (0), defined by A* (0)(q) = A:( (9 ) is a continuous ) orientation of /5. • Theorem 5. true: Let P be a Cr k-plane field on M.

Suppose now that M has an atlas g which defines a codimension s foliation, according to the definition in §1. Consider a cover C = U i E /I of M by domains of local charts of g , as in Lemma 1. Given an open set U, E e, i E I, we have defined so : U rRn x Ile on g such that io,(U,) x U', where U 1 and U`2 are open balls in JR' and 1R respectively. Let /3 2 : 111' x IR IRS be the projection on the second factor. 1 0, U, U Uj is contained in the domain of a chart ( ) E a with q, ( V) = V I X V2 .