Lefschetz hyperplane theorem

The Lefschetz hyperplane theorem relates a smooth projective variety to a smooth hyperplane section: their homotopy groups agree below the section’s dimension, and inclusion is surjective in that dimension.

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Lefschetz pencil: A one-parameter family of hyperplane sections with finitely many singular members and a common base locus. Pencils turn hyperplane sections into a fibration whose singularities explain the theorem’s topological control.
Projective variety: An algebraic variety that can be embedded as a closed subvariety of projective space. Projective embedding supplies the hyperplanes whose sections the theorem compares.
Solomon Lefschetz: A mathematician who developed foundational methods linking algebraic geometry and topology. His work established the topological principles behind hyperplane-section results.
Lefschetz theorem on (1,1)-classes: A theorem identifying integral cohomology classes of type (1,1) on a smooth projective variety with divisor classes. It is a separate Lefschetz result showing how topology and Hodge type constrain algebraic cycles.
Morse theory: A method for studying a manifold through the critical points of a smooth function. The proof builds the variety from the section using cells constrained by Morse indices.
Hyperplane section: The intersection of a projective variety with a hyperplane in its ambient projective space. This is the subvariety included into the original variety in the theorem.
Lefschetz hyperplane theorem (homology): A theorem comparing the homology of a smooth projective variety with that of a smooth hyperplane section. The homology formulation is historically central and distinct from the homotopy statement given here.
Lefschetz hyperplane theorem for homology: A homological comparison theorem for a smooth projective variety and a smooth hyperplane section. Its homology consequences often provide the first usable form of the section’s connectivity.
Handle decomposition: A presentation of a manifold as a sequence of pieces attached along their boundaries. The relative handle indices show why no low-dimensional topology is added beyond the section.
Smoothness: A property of a space that locally has no singular points and is modeled on a manifold. Smoothness of both variety and section is part of the classical theorem’s hypotheses.