Hard Lefschetz theorem

In mathematics, specifically in algebraic geometry and algebraic topology, the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape of an algebraic variety and the shape of its subvarieties. More precisely, the theorem says that for a variety X embedded in projective space and a hyperplane section Y, the homology, cohomology, and homotopy groups of X determine those of Y. A result of this kind was first stated by Solomon Lefschetz for homology groups of complex algebraic varieties. Similar results have since been found for homotopy groups, in positive characteristic, and in other homology and cohomology theories.

A far-reaching generalization of the hard Lefschetz theorem is given by the decomposition theorem.

Let X be an n-dimensional complex projective algebraic variety in CP^N, and let Y be a hyperplane section of X such that U = X ∖ Y is smooth. The Lefschetz theorem refers to any of the following statements:

Using a long exact sequence, one can show that each of these statements is equivalent to a vanishing theorem for certain relative topological invariants. In order, these are:

Lefschetz used his idea of a Lefschetz pencil to prove the theorem. Rather than considering the hyperplane section Y alone, he put it into a family of hyperplane sections Y_t, where Y = Y₀. Because a generic hyperplane section is smooth, all but a finite number of Y_t are smooth varieties. After removing these points from the t-plane and making an additional finite number of slits, the resulting family of hyperplane sections is topologicaly trivial. That is, it is a product of a generic Y_t with an open subset of the t-plane. X, therefore, can be understood if one understands how hyperplane sections are identified across the slits and at the singular points. Away from the singular points, the identification can be described inductively. At the singular points, the Morse lemma implies that there is a choice of coordinate system for X of a particularly simple form. This coordinate system can be used to prove the theorem directly.

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