Topics in Algebraic and Analytic Geometry. (MN-13): Notes by Phillip A. Griffiths, John Frank Adams

By Phillip A. Griffiths, John Frank Adams

This quantity bargains a scientific remedy of convinced uncomplicated elements of algebraic geometry, offered from the analytic and algebraic issues of view. The notes specialise in comparability theorems among the algebraic, analytic, and non-stop categories.

Contents comprise: 1.1 sheaf conception, ringed areas; 1.2 neighborhood constitution of analytic and algebraic units; 1.3 P
n 2.1 sheaves of modules; 2.2 vector bundles; 2.3 sheaf cohomology and computations on P
n; 3.1 greatest precept and Schwarz lemma on analytic areas; 3.2 Siegel's theorem; 3.3 Chow's theorem; 4.1 GAGA; 5.1 line bundles, divisors, and maps to P
n; 5.2 Grassmanians and vector bundles; 5.3 Chern sessions and curvature; 5.4 analytic cocycles; 6.1 K-theory and Bott periodicity; 6.2 K-theory as a generalized cohomology concept; 7.1 the Chern personality and obstruction concept; 7.2 the Atiyah-Hirzebruch spectral series; 7.3 K-theory on algebraic forms; 8.1 Stein manifold idea; 8.2 holomorphic vector bundles on polydisks; 9.1 concluding feedback; bibliography.

Originally released in 1974.

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Sample text

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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