Codimension 2
From Mathematics
(→Schedule) |
(→Cech cohomology and the Hodge conjecture in codim 1) |
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= Cech cohomology and the Hodge conjecture in codim 1 = | = Cech cohomology and the Hodge conjecture in codim 1 = | ||
| − | * Classification of vector bundles by Cech-cocycles, Theorem I.5.10<ref name="Wei13"/>. Mention also that for a ''projective'' complex algebraic variety <math>X</math>, complex analytic vector bundles, classified by <math>H^1(X^{an}, \mathcal{O}_X^{an})</math> (analytic Cech cohomology), are the same as algebraic vector bundles on <math>X</math>, classified by <math>H^1(X, \mathcal{O}_X)</math> (Zariski Cech cohomology). | + | * Review of Cech cohomology and the long exact sequence, explicit construction for the <math>\delta</math> homomorphism. Treat the analytic and Zariski topology parallel. |
| + | * Classification of vector bundles by Cech-cocycles, Theorem I.5.10<ref name="Wei13"/>. | ||
| + | * Mention also that for a ''projective'' complex algebraic variety <math>X</math>, complex analytic vector bundles, classified by <math>H^1(X^{an}, \mathcal{O}_X^{an})</math> (analytic Cech cohomology), are the same as algebraic vector bundles on <math>X</math>, classified by <math>H^1(X, \mathcal{O}_X)</math> (Zariski Cech cohomology). | ||
| + | * The exponential sequence | ||
| + | <math> 1 \rightarrow 2\pi i \mathbb{Z}_X \rightarrow \mathcal{O}_X \rightarrow \mathcal{O}_X^* \rightarrow 0</math> | ||
| + | * Proof of the Hodge conjecture for divisors using the exponential sequence | ||
* Need also Lemma 2.1<ref name="Blo74"/> on a non-Abelian version of the long exact sequence. | * Need also Lemma 2.1<ref name="Blo74"/> on a non-Abelian version of the long exact sequence. | ||
