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Cohomology of Coherent Sheaves and Kodaira’s Embedding Theorem

In: Analytic Function Theory of Several Variables

Author

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  • Junjiro Noguchi

    (The University of Tokyo
    Tokyo Institute of Technology)

Abstract

Up to the present we have been dealt with open domains and open complex manifolds. In this chapter we also deal with compact ones. We will introduce a topology in the space of sections of a coherent sheaf. As a consequence we will see that all cohomologies of a coherent sheaf over a compact complex space are finite dimensional (Cartan–Serre Theorem). Furthermore, we will extend Grauert’s Theorem 7.5.26 for a general coherent sheaf. Then, as an application, we prove Kodaira’s Embedding Theorem to embed a Hodge manifold into a complex projective space. Kodaira’s Embedding Theorem provides a bridge between the theory of compact Kähler manifolds and that of complex projective algebraic varieties; it is nice to see such a theorem being naturally proved on the extended line of the theory of coherent sheaves.

Suggested Citation

  • Junjiro Noguchi, 2016. "Cohomology of Coherent Sheaves and Kodaira’s Embedding Theorem," Springer Books, in: Analytic Function Theory of Several Variables, chapter 0, pages 343-366, Springer.
    • Handle: RePEc:spr:sprchp:978-981-10-0291-5_8
    DOI: 10.1007/978-981-10-0291-5_8
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