is, however, slightly weaker because of the additional assumptions on Y. Artin and Grothendieck's proof for constructible sheaves Michael Artin and Alexander Grothendieck found a generalization of the Lefschetz hyperplane theorem to the case where the coefficients of the cohomology lie not in a field but instead in a constructible sheaf. They prove that for a constructible sheaf F on an affine variety U, the cohomology groups vanish whenever . The Lefschetz theorem in other cohomology theories The motivation behind Artin and Grothendieck's proof for constructible sheaves was to give a proof that could be adapted to the setting of étale and -adic cohomology. Up to some restrictions on the constructible sheaf, the Lefschetz theorem remains true for constructible sheaves in positive characteristic. The theorem can also be generalized to intersection homology. In this setting, the theorem holds for highly singular spaces. A Lefschetz-type theorem also holds for Picard groups. Hard Lefschetz theorem Let X be a n-dimensional non-singular complex projective variety in . Then in the cohomology ring of X, the k-fold product with the cohomology class of a hyperplane gives an isomorphism between and . This is the hard Lefschetz theorem, christened in French by Grothendieck more colloquially as the Théorème de Lefschetz vache. It immediately implies the injectivity part of the Lefschetz hyperplane theorem. The hard Lefschetz theorem in fact holds for any compact Kähler manifold, with the isomorphism in de Rham cohomology given by multiplication by a power of the class of the Kähler form. It can fail for non-Kähler manifolds: for example, Hopf surfaces have vanishing second cohomology groups, so there is no analogue of the second cohomology class of a hyperplane section. The hard Lefschetz theorem was proven for -adic cohomology of smooth projective varieties over algebraically closed fields of positive characteristic by . References Bibliography Reprinted in Topological methods of algebraic geometry Morse theory Theorems in algebraic geometry Theorems in algebraic topology