The Lefschetz hyperplane section theorem says that cohomologically a nonsingular complex variety looks like its hyperplane? sections. More precisely,
Let $X$ be an algebraic subvariety of complex projective space and $H$ a generic hyperplane in ${\u2102}^{n}$. Then the $i$-th relative cohomology ${H}_{i}(X,X\cap H)=0$ for $i<n$.
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
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