Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

The Weak Lefschetz Theorem states that for a compact Kahler manifold, $Pic(X) \rightarrow H^{1,1}(X, \mathbb{Z})$ is surjective.

The Hard Lefschetz Theorem states that for a compact Kahler manifold with Lefschetz operator $L$, $$L^{n-k}: H^k(X,\mathbb{R}) \cong H^{2n-k}(X,\mathbb{R})$$ and for any $k$ $$H^k(X, \mathbb{R}) = \bigoplus_{i \geq 0}L^iH^{k-2i}(X, \mathbb{R})_p$$ The subscript $p$ denotes the primitive cohomology groups.

The decomposition above respects the $p,q$ form decomposition of Dolbeault cohomology.

My question is: what is the relation between these two Lefschetz theorems? Does one imply the other? Any good references?

share|improve this question
To avoid confusion: I think what you call the Weak Lefschetz Theorem is usually called the Lefschetz (1,1) Theorem, and the Weak Lefschetz Theorem is the theorem also called the Lefschetz Hyperplane Theorem. (At least according to Iskovskikh: encyclopediaofmath.org/index.php/Lefschetz_theorem) –  Artie Prendergast-Smith Apr 23 '13 at 15:30

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.