Lefschetz operator - Imported Knowlpedia Content

Definition

Let $(X,\omega)$ be a $2n$ -dimensional symplectic manifold. The Lefschetz operator on differential forms is the degree-two linear map

L:\Omega^k(X)\longrightarrow\Omega^{k+2}(X),\qquad L(\alpha)=\omega\wedge\alpha.

Because $d\omega=0$ , it commutes with the exterior derivative and induces

L:H^k_{\mathrm{dR}}(X)\longrightarrow H^{k+2}_{\mathrm{dR}}(X),\qquad L[\alpha]=[\omega]\smile[\alpha].

When $X$ is Kähler, $\omega$ has type $(1,1)$ , so $L$ also maps $(p,q)$ -forms to $(p+1,q+1)$ -forms. The same symbol $L$ is conventionally used for all these compatible operators.

Linear-algebraic structure

On a symplectic vector space, multiplication by $\omega$ has an adjoint lowering operator $\Lambda$ , defined using contraction with the inverse bivector after conventions are fixed. Together with their grading commutator, $L$ and $\Lambda$ form an $\mathfrak{sl}_2$ -triple. This representation-theoretic structure yields the decomposition of forms into powers of $L$ applied to primitive forms Voisin, §6.2.

Cohomological consequences

On a compact Kähler manifold, the Hard Lefschetz theorem says that appropriate powers of the cohomological operator are isomorphisms between complementary degrees. It follows that cohomology also has a primitive Lefschetz decomposition. For a general symplectic manifold, $L$ is still defined, but these cohomological isomorphisms can fail.

Conventions and examples

On complex projective space with its Fubini–Study form, $L$ is multiplication by the degree-two generator of real cohomology. Some authors normalize $\omega$ by a scalar or define the lowering operator first; these choices change numerical formulas but not the definition $L(\alpha)=\omega\wedge\alpha$ . This operator is unrelated to the Lefschetz number of a self-map.

References

Claire Voisin, Hodge Theory and Complex Algebraic Geometry I, Cambridge Studies in Advanced Mathematics 76, Cambridge University Press, 2002. DOI record. Relevant: §6.2, Lefschetz operators, primitive forms, and the associated $\mathfrak{sl}_2$ -action.