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Unformatted text preview: Lectures on algebraic geometry Jacobs 2010 Notes of lectures 1819 Alan Huckleberry May 1, 2010 1 Introduction Recall that by using the short exact sequence → O * → M * → Div → of sheaves and the first piece of the long exact cohomology sequence, we have identified the group Pic( X ) of divisors modulo linear equivalence on a projective algebraic manifold with the cohomology group H 1 ( X, O * ). Here we begin by isolating this cohomology group in another exact cohomology sequence which involves the topological invariant called the Chern class. This enables us to give an exact computation of H 1 ( X, O * ) in the case where X is projective space. After introducing the basic notions for the study of fiber bundles, in particu lar vector bundles, we show that in general H 1 ( X, O * ) can be interpreted as as the space of equivalence classes of holomorphic line bundles on X . Since Pic( X ) ∼ = H 1 ( X, O * ) , given a divisor D we look for a geometric interpretation of its line bundle L ( D ). In the case where the support  D  is smooth this is given by the normal bundle of  D  . This interpretation arises naturally during our discussion of the tangent bundle sequence defined by an arbitrary submanifold of a complex manifold. 1 2 Expsequence For any complex space X we may consider the short exact sequence→ Z→ O exp→ O *→ of sheaves where exp( f ) := e 2 πif . In the previous lecture we introduced the boundary map δ which in this case goes from O * ( X ) to H 1 ( X, Z ). Now one can in fact show that H 1 ( X, Z ) is a topological invariant of X . For example, if X is smooth it is the integral deRham cohomology. Thus given a nowhere vanishing holomorphic fuction f ∈ O * ( X ) the image δ ( f ) is an obstruction to f having a global logarithm. Exercise. Suppose that on a covering U = { U α } the restrictions f α of a globally defined nowhere vanishing holomorphic function f have logarithms and that the cohomology class δ ( f ) ∈ H 1 ( U , Z ) is zero. Show in concrete terms how to build a global logarithm of f from the logarithms of the locally defined functions f α . Extending the long exact sequence In general if→ S→ S → S 00→ is a short exact sequence of sheaves, the long exact sequence continues by formal considerations to ...→ S 00 ( X ) δ→ H 1 ( X, S )→ H 1 ( X, S )→ H 1 ( X, S 00 ) . Thus in our concrete example of the expsequence we have the continuation→ H 1 ( X, Z )→ H 1 ( X, O )→ H 1 ( X, O * ) . Since we are in particular interested in understanding H 1 ( X, O * ), the next boundary map δ is of particular importance. As would be expected this is first defined on a cover. For this let ξ = [ { f αβ } ] be a realization of a class ξ ∈ H 1 ( X, O * ) on a cover U . Recall that we are looking for the obstruction to ξ being in the image of a class from H 1 ( X, O ). We may refine U if necessary so that at least locally this is the case, i.e., there exit g αβ ∈ O...
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This note was uploaded on 05/18/2010 for the course MATHEMATIC 120102 taught by Professor Xxxxxxxxxyyyyyy during the Spring '10 term at Jacobs University Bremen.
 Spring '10
 xxxxxxxxxyyyyyy
 Algebra, Geometry

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