Lefschetz hyperplane theorem
- Lefschetz hyperplane theorem
In mathematics, the Lefschetz hyperplane theorem states that a hyperplane section "W" of a non-singular complex algebraic variety "V", in complex projective space, inherits most of its algebraic topology from "V". This allows certain geometrical questions to be investigated by induction on dimension. Results in this direction were first given by Solomon Lefschetz in the early 1920s. They were subsequently revisited, in the light first of the foundational work that had been done in topology; and then in order to extend them to areas of algebraic geometry that were outside the classical setting.
A modern approach deduces the Lefschetz results from the Andreotti-Frankel theorem, which states that the "affine part" "V""W" is, viewed by homotopy theory, of low dimension. This means that relative homology and its long exact sequence, together with Lefschetz duality, can be applied to recover the basic theorem on homology groups: if "V" has complex dimension "d", then the inclusion map
:"W" → "V"
induces an isomorphism on "H""i" when "i" < "d" − 1, and a surjective homomorphism on "H""d" − 1.
Hard Lefschetz theorem
A related result states that in the cohomology ring of "W", the "k"-fold product with the cohomology class of a hyperplane gives an isomorphism between
:"H""d" − "k"
and
:"H""d" + "k",
if "W" is non-singular. This is the hard Lefschetz theorem (in French, "vache", which here may be best translated into colloquial English as 'pig' or 'bastard'). It immediately implies the injectivity part of the Lefschetz hyperplane theorem.
References
*John Milnor (1963) "Morse theory" Sect. 7
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