Lefschetz hyperplane theorem

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, specifically in <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a> and <a href="/facts/Algebraic_topology/2wA8yrjT">algebraic topology</a>, the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape of an <a href="/facts/Algebraic_variety/Vk7f97vn">algebraic variety</a> and the shape of its subvarieties. More precisely, the theorem says that for a variety X embedded in <a href="/facts/Projective_space/7MSpA9cM">projective space</a> and a <a href="/facts/Hyperplane_section/o5gl0vDk">hyperplane section</a> Y, the <a href="/facts/Homology_(mathematics)/VFJmryEW">homology</a>, <a href="/facts/Cohomology/J7FJzTNh">cohomology</a>, and <a href="/facts/Homotopy_group/xjogL95Y">homotopy groups</a> of X determine those of Y. A result of this kind was first stated by <a href="/facts/Solomon_Lefschetz/Ut2VFwGr">Solomon Lefschetz</a> 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 <a href="/facts/Decomposition_theorem/jUUyrrp1">decomposition theorem</a>.

Lefschetz hyperplane theorem open-in-new

Lefschetz hyperplane theorem