By Phillip A. Griffiths, John Frank Adams

This quantity deals a scientific remedy of yes easy 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 contain: 1.1 sheaf concept, 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 idea; 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 comments; bibliography.

Originally released in 1974.

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**Extra info for Topics in Algebraic and Analytic Geometry. (MN-13): Notes From a Course of Phillip Griffiths**

**Example text**

We call the set of all such maps {11 Q .. } with respect to the. } of X according to which 1 both Y and Y' are defined and such that the map U. x CCn -> U. x a::n l l . (v)) 1 2 U. -> i a:n = Matnxn(O::). Note that the maps cp i must satisfy 36 II. 2. 4 (All maps will be required to be differentiable, continuous, holomorphic, or algebraic, according to context. ) Conversely, from a collection of maps {cp i} satisfying II ij

For a proof, see Narasimhan [ 26]. Now we'll give some applications of this theorem. 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.

2. 8 This is also a locally free sheaf of rank one, so we have defined a multiplication of line bundles. The inverse of a locally free sheaf of rank one, LI' is Hom JLI'O) because Hom (L1, 0) ® L1 -o 0 -=-> Because of the existence of inverse, locally 0. free sheaves of rank one are called invertible sheaves. } of X and for each i an f. , and so that 1 II .. f. = lJ J 1 f. such that f. , 1 1 J II .. ,O). • g. 1 up Wj, 1 J cp ij a unit in r (UP Wj, 0). J lJ J J = f. 1 on Now from an effective Cartier divisor f.