Lefschetz theorem on hyperplane cuts

In mathematics , especially in algebraic geometry and algebraic topology , Lefschetz's theorem on hyperplane intersections establishes a connection between the shape of an algebraic variety and the shape of its sub- varieties . It says that for a hyperplane cut in a projective variety the homotopy, homology and cohomology groups are already determined up to a certain dimension by those of . The statement is named after Solomon Lefschetz . ${\displaystyle Y\subset X}$ ${\displaystyle X}$ ${\displaystyle X}$

Sentence (general formulation)

Theorem : Let it be a complex- dimensional projective variety and a hyperplane that contains all singularities of . Then ${\displaystyle V\subset \mathbb {C} P^{n}}$ ${\displaystyle k}$ ${\displaystyle H\subset \mathbb {C} P^{n}}$ ${\displaystyle V}$

${\displaystyle \pi _{r}(V,V\cap H)=0\ \ \forall r<k-1}$.

In particular, the inclusion induces an isomorphism of the homotopy , homology and cohomology groups up to grade and an epimorphism (or a monomorphism in the case of cohomology) in grade . ${\displaystyle V\cap H\to V}$ ${\displaystyle k-2}$ ${\displaystyle k-1}$

The sentence is a consequence of the following stronger sentence by Andreotti-Frankel.

Theorem : Every complex submanifold of the complex dimension is homotopy-equivalent to a -dimensional CW-complex , in particular is for . ${\displaystyle M\subset \mathbb {C} ^{n}}$ ${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle \pi _{i}(M)=H_{i}(M;\mathbb {Z} )=0}$ ${\displaystyle i>k}$

Hypersurfaces in projective space

Probably the most important application are nonsingular hypersurfaces , i.e. sub-varieties given by a single homogeneous polynomial without simultaneous zeros of all partial derivatives ${\displaystyle X\subset \mathbb {C} P^{k}}$ ${\displaystyle P\in \mathbb {C} \left[z_{0},\ldots ,z_{k}\right]}$ ${\displaystyle {\frac {\partial P}{\partial z_{i}}}}$

${\displaystyle X=\left\{\left[z_{0}:\ldots :z_{k}\right]\in \mathbb {C} P^{k}\colon P(z_{0},\ldots ,z_{k})=0\right\}}$.

For this, one embeds using the Veronese embedding ( ) as a sub-variety ${\displaystyle \mathbb {C} P^{k}}$ ${\displaystyle \iota _{d}\colon \mathbb {C} P^{k}\to \mathbb {C} P^{N}}$ ${\displaystyle d=deg(P)}$

${\displaystyle V:=\iota _{d}(\mathbb {C} P^{k})\subset \mathbb {C} P^{N}}$

into a higher-dimensional one with . The image from below the Veronese map is the intersection of with a hyperplane , because the monomials of the degree d polynomial correspond to the components of the Veronese map, so the image is described by a linear equation. One can then apply Lefschetz's theorem to and and get because of that ${\displaystyle \mathbb {C} P^{N}}$ ${\displaystyle N={\tbinom {k+d}{d}}-1}$ ${\displaystyle X}$ ${\displaystyle V}$ ${\displaystyle H\subset \mathbb {C} P^{N}}$ ${\displaystyle P}$ ${\displaystyle V}$ ${\displaystyle H}$ ${\displaystyle X\simeq H\cap V}$

${\displaystyle \pi _{r}X\to \pi _{r}\mathbb {C} P^{k}}$

is an isomorphism for and an epimorphism for . ${\displaystyle r<k-1}$ ${\displaystyle r=k-1}$

In particular, for non-singular hypersurfaces im are simply connected . ${\displaystyle k\geq 3}$ ${\displaystyle \mathbb {C} P^{k}}$

literature

• Lefschetz, S .: L'analysis situs et la géométrie algébrique. Gauthier-Villars, Paris, 1950.
• Andreotti, Aldo ; Frankel, Theodore : The Lefschetz theorem on hyperplane sections. Ann. of Math. (2) 69 1959 713-717.
• Milnor, J .: Morse theory. Based on lecture notes by M. Spivak and R. Wells . Annals of Mathematics Studies, No. 51 Princeton University Press, Princeton, NJ 1963
• Lamotke, K .: The topology of complex projective varieties after S. Lefschetz , Topology 20, 15-51 (1981). On-line

Web links

• Lefschetz Theorem (Encyclopedia of Mathematics)
• Topology of algebraic varieties, Hodge decomposition and applications