By Phillip A. Griffiths, John Frank Adams

ISBN-10: 0691081514

ISBN-13: 9780691081519

This quantity deals a scientific therapy of definite easy elements of algebraic geometry, provided from the analytic and algebraic issues of view. The notes concentrate on 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 concept; 7.1 the Chern personality and obstruction thought; 7.2 the Atiyah-Hirzebruch spectral series; 7.3 *K*-theory on algebraic kinds; 8.1 Stein manifold thought; 8.2 holomorphic vector bundles on polydisks; 9.1 concluding comments; bibliography.

Originally released in 1974.

The **Princeton Legacy Library** makes use of the newest print-on-demand expertise to back make on hand formerly out-of-print books from the prestigious backlist of Princeton college Press. those paperback versions shield the unique texts of those vital books whereas featuring them in sturdy paperback variations. The target of the Princeton Legacy Library is to tremendously bring up entry to the wealthy scholarly background present in the millions of books released via Princeton collage Press considering that its founding in 1905.

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

**Sample text**

Nu. luin uj J id II .. n u. n u. l J is nothing but a map of the appropriate type. k l = 11 lj.. n Uk' defining an element l J of H1(X, GL(n,O)). The same sort of analysis as we just went through with vector bundles shows that, H1(X, GL(n,O)) is equivalent to {isomorphism classes of locally free sheaves of rank n}. This is consistent with the previous association from vector bundles to locally free sheaves - so these notions are essentially equivalent. ' (G), etc. , then take the associated bundles.

X «:n (or i J -----' in the obvious way; the condition on triple overlaps allows us to do this consistently. 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

X THEOREM The map 44 II. 12 is an isomorphism. • , zn} An element f of I' (1Pn, 01Pn(m)hol) is a holomorphic function on a:n+l _{O} such that f(A. z) = A. mf(z) for all z. By Hartogs' theorem f is holomorphic in a:n+l. (This shows already that f=O unless m :;:: 0). f =. •. ,1n o n Represent f as a power series io in zo··· zn A. m O! • • 10" •. in series representations. m. by the uniqueness of power This shows that f is a homogeneous polynomial of degree 45 II. 3. l Chapter Two Sheaf cohomology and computations on IPn.

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

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