Last modified date: <%+ tp.file.last_modified_date() %>

• Tags
  • #AG
• Refs:
  • #todo/add-references
• Links:
  • right derived pushforward

Lefschetz theorems

Hard Lefschetz

Application: Signatures

For a Kahler surface

See Hodge index theorem

Weak Lefschetz

Hard Lefschetz

For symplectic manifolds \((X, \omega)\) of real dimension \(2n\): define a map \(L: H_\mathrm{dR}(X) \to \Sigma^{2}H_{\mathrm{dR}}(X)\) by \([\alpha] \mapsto [\omega \wedge \alpha]\). Then the iterates \(L^i\) restrict to \(L^i: H^{n-i}_\mathrm{dR}(X) \to H_\mathrm{dR}^{n+i}(X)\), and Hard Lefschetz states that this is an isomorphism for compact Kahlers.

For smooth complex projective varieties of complex dimension \(n\), replace \(\omega\) with the class of a hyperplane.