Notes On Euclidean Geometry (Math Olympiad) by Kedlaya K

By Kedlaya K

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At 0 lcfi ... lq' 53 II. 3. • xiq), OlPn(d)hol) satisfying the cocycle rule. This data can be expressed as functions f. • Z. • lq lo lq satisfying for i 0 < • . •

H (X, B+l))/( I m(H (X, B. 1) 1 1 0 H (X, B. )) l i- This makes it important to find cohomologically trivial sheaves, that is, sheaves F such that J(x, F) = 0 for i > 0. Flabby sheaves are cohomologically trivial. i,x are non-zero at any A such that stalk, and~$. IP(O off Ui i,x = id at any stalk. A partition of unity argument shows that the sheaf 0diff on any differ -entiable manifold is soft; this will be the most important soft sheaf for us. }, A) as the set of all maps f which to each i+l-tuple at 1 47 II.

III. B. ::. n(r) -> a: be a holomorphic function. _!!. /f / is a local maximum at 0 then f is constant. Given zECn, define f (u) for small UEC by f (u) z z = f(uz). By the one variable maximum principle, fz is constant. The proposition follows from this. Resolution of singularities easily gives an extension of this to analytic spaces. We can also do this by branched coverings. III. C. 2 maximum at XEX. Then f is constant in a neighborhood of X. We may assume that Xis irreducible and that there is 11: X -> b.

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