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

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