Heavy Lefschetz sentence

In mathematics is serious Lefschetz theorem (ger .: hard Lefschetz theorem ) is a central tenet of the complex differential geometry .

The term serves to distinguish it from Lefschetz's sentence about hyperplane cuts, which is called weak Lefschetz theorem .

The Lefschetz decomposition of the De Rham cohomology of Kahler manifolds given by the theorem has (partly assumed, partly proven) analogues in completely different areas of mathematics, in particular the standard conjectures of algebraic geometry or the results of matroid theory in combinatorics .

Theorem and Consequences

Let be a -dimensional Kähler manifold with Kähler form . Be ${\displaystyle M}$ ${\displaystyle 2n}$ ${\displaystyle \omega \in \Omega ^{2}(M)}$

${\displaystyle L\colon \Omega ^{i}(M)\to \Omega ^{i+2}(M)}$

the mapping defined by the outer product with the Kähler form on the space of the differential forms . Then is for the mapping induced in De Rham cohomology ${\displaystyle k=0,\ldots ,n}$

${\displaystyle L^{k}\colon H^{n-k}(M)\to H^{n+k}(M)}$

an isomorphism that corresponds to the isomorphism given by Poincaré duality .

Since is a -form, one obtains in particular an isomorphism of the Dolbeault cohomology groups ${\displaystyle \omega }$ ${\displaystyle (1,1)}$

${\displaystyle H^{p,q}(M)\simeq H^{n-p,n-q}(M)}$

and thus the symmetry of the Hodge diamond .

Be

${\displaystyle PH^{r}(M)=\left\{\alpha \in H^{r}(M)\colon L^{r+1}\alpha =0\right\}}$

the so-called primitive cohomology of , then the Lefschetz decomposition follows from the heavy Lefschetz theorem ${\displaystyle M}$

${\displaystyle H^{k}(M)=\bigoplus _{s}L^{s}(PH^{k-2s}(M))}$.

literature

• RO Wells: Differential analysis on complex manifolds , Prentice-Hall, Englewood Cliffs, NJ, 1973.
• C. Voisin: Hodge theory and Complex algebraic geometry , Cambridge Stud. In Adv. Math. 76, 77, 2002/3

Web links

• Notes on Lefschetz Decomposition