Lefschetz hyperplane 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.
- 1 The Lefschetz hyperplane theorem for complex projective varieties
- 2 The Lefschetz theorem in other cohomology theories
- 3 Hard Lefschetz theorem
- 4 References
- 5 Bibliography
The Lefschetz hyperplane theorem for complex projective varieties
Let X be an n-dimensional complex projective algebraic variety in CPN, 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:
- The natural map Hk(Y, Z) → Hk(X, Z) in singular homology is an isomorphism for k < n − 1 and is surjective for k = n − 1.
- The natural map Hk(X, Z) → Hk(Y, Z) in singular cohomology is an isomorphism for k < n − 1 and is injective for k = n − 1.
- The natural map πk(Y, Z) → πk(X, Z) is an isomorphism for k < n − 1 and is surjective for k = n − 1.
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:
- The relative singular homology groups Hk(X, Y, Z) are zero for .
- The relative singular cohomology groups Hk(X, Y, Z) are zero for .
- The relative homotopy groups πk(X, Y) are zero for .
Solomon 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 Yt, where Y = Y0. Because a generic hyperplane section is smooth, all but a finite number of Yt 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 Yt 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.
Andreotti and Frankel's proof
Aldo Andreotti and Theodore Frankel recognized that Lefschetz's theorem could be recast using Morse theory. Here the parameter t plays the role of a Morse function. The basic tool in this approach is the Andreotti–Frankel theorem, which states that a complex affine variety of complex dimension n (and thus real dimension 2n) has the homotopy type of a CW-complex of (real) dimension n. This implies that the relative homology groups of Y in X are trivial in degree less than n. The long exact sequence of relative homology then gives the theorem.
Thom's and Bott's proofs
Neither Lefschetz's proof nor Andreotti and Frankel's proof directly imply the Lefschetz hyperplane theorem for homotopy groups. An approach that does was found by René Thom no later than 1957 and was simplified and published by Raoul Bott in 1959. Thom and Bott interpret Y as the vanishing locus in X of a section of a line bundle. An application of Morse theory to this section implies that X can be constructed from Y by adjoining cells of dimension n or more. From this, it follows that the relative homology and homotopy groups of Y in X are concentrated in degrees n and higher, which yields the theorem.
Kodaira and Spencer's proof for Hodge groups
Kunihiko Kodaira and Donald C. Spencer found that under certain restrictions, it is possible to prove a Lefschetz-type theorem for the Hodge groups Hp,q. Specifically, assume that Y is smooth and that the line bundle is ample. Then the restriction map Hp,q(X) → Hp,q(Y) is an isomorphism if p + q < n − 1 and is injective if p + q = n − 1. By Hodge theory, these cohomology groups are equal to the sheaf cohomology groups and . Therefore, the theorem follows from applying the Akizuki–Nakano vanishing theorem to and using a long exact sequence.
Combining this proof with the universal coefficient theorem nearly yields the usual Lefschetz theorem for cohomology with coefficients in any field of characteristic zero. It 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 Hk(U, F) vanish whenever k > n.
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.
Hard Lefschetz theorem
Let X be a n-dimensional non-singular complex projective variety in CPN. Then in the cohomology ring of X, the k-fold product with the cohomology class of a hyperplane gives an isomorphism between Hn − k and Hn + k.
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.
- Milnor 1969, Theorem 7.3 and Corollary 7.4
- Voisin 2003, Theorem 1.23
- Lefschetz 1924
- Griffiths, Spencer & Whitehead 1992
- Andreotti & Frankel 1959
- Milnor 1969, p. 39
- Bott 1959
- Lazarsfeld 2004, Example 3.1.24
- Voisin 2003, Theorem 1.29
- Lazarsfeld 2004, Theorem 3.1.13
- Lazarsfeld 2004, Example 3.1.25
- Sabbah 2001
- Andreotti, Aldo; Frankel, Theodore (1959), "The Lefschetz theorem on hyperplane sections", Annals of Mathematics, Second Series, 69: 713–717, doi:10.2307/1970034, ISSN 0003-486X, MR 0177422
- Beauville, Arnaud, The Hodge Conjecture, CiteSeerX
- Bott, Raoul (1959), "On a theorem of Lefschetz", Michigan Mathematical Journal, 6 (3): 211–216, doi:10.1307/mmj/1028998225, MR 0215323, retrieved 2010-01-30
- Deligne, Pierre (1980), "La conjecture de Weil. II", Publications Mathématiques de l'IHÉS (52): 137–252, ISSN 1618-1913, MR 0601520
- Griffiths, Phillip; Spencer, Donald C.; Whitehead, George W. (1992), "Solomon Lefschetz", in National Academy of Sciences, Office of the Home Secretary, Biographical Memoirs, 61, The National Academies Press, ISBN 978-0-309-04746-3
- Lazarsfeld, Robert (2004), Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 48, Berlin, New York: Springer-Verlag, ISBN 978-3-540-22533-1, MR 2095471
- Lefschetz, Solomon (1924), L'Analysis situs et la géométrie algébrique, Collection de Monographies publiée sous la Direction de M. Émile Borel (in French), Paris: Gauthier-Villars Reprinted in Lefschetz, Solomon (1971), Selected papers, New York: Chelsea Publishing Co., ISBN 978-0-8284-0234-7, MR 0299447
- Milnor, John Willard (1963), Morse theory, Annals of Mathematics Studies, No. 51, Princeton University Press, MR 0163331
- Sabbah, Claude (2001), Theorie de Hodge et theoreme de Lefschetz "difficile" (PDF), archived from the original (PDF) on 2004-07-07
- Voisin, Claire (2003), Hodge theory and complex algebraic geometry. II, Cambridge Studies in Advanced Mathematics, 77, Cambridge University Press, ISBN 978-0-521-80283-3, MR 1997577