Geometric Manifestations of Positivity

  • Robert Lazarsfeld
Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics book series (MATHE3, volume 48)


This chapter focuses on a number of results that in one way or another express geometric consequences of positivity. In the first section we prove the Lefschetz hyperplane theorem following the Morse-theoretic approach of Andreotti-Frankel [7]. Section 3.2 deals with subvarieties of small codimension in projective space: we prove Barth’s theorem and give an introduction to the conjectures of Hartshorne. The connectedness theorems of Bertini and Fulton-Hansen are established in Section 3.3, while applications of the Fulton-Hansen theorem occupy Section 3.4. Like the results from 3.2, these reflect the positivity of projective space itself. Finally, some extensions and variants are presented in Section 3.5.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Robert Lazarsfeld
    • 1
  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA

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