nLab hyperplane section theorem

The Lefschetz hyperplane section theorem says that cohomologically a nonsingular complex variety looks like its hyperplane sections. More precisely,

Hyperplane section theorem

Let XX be an algebraic subvariety of complex projective space and HH a generic hyperplane in n\mathbb{C}^n. Then the ii-th relative cohomology H i(X,XH)=0H_i(X,X\cap H) = 0 for i<ni\lt n.

There is a related deeper theorem, also due to Lefschetz, the hard Lefschetz theorem.

There is also a version of the quantum hyperplane section theorem due to Y.-P. Lee, where the cohomology is replaced by the quantum cohomology.

  • Goresky, Mac Pherson, Stratified Morse theory

  • Kyle Hofmann, The Lefschetz hyperplane section theorem, pdf

  • wikipedia

  • Y-P. Lee, Quantum Lefschetz hyperplane theorem, Inventiones Math. 145, 1, 121–149, 2001 (doi)

  • Mark Andrea A. de Cataldo, Luca Migliorini, The perverse filtration and the Lefschetz hyperplane theorem, accepted to Annals of Math. pdf

Last revised on January 27, 2010 at 22:10:15. See the history of this page for a list of all contributions to it.