Hard Lefschetz theorem - Imported Knowlpedia Content

Statement

Let $X$ be a compact Kähler manifold of complex dimension $n$ , and let $L$ denote cup product with its Kähler class $[\omega]$ . The Hard Lefschetz theorem states that for every $0\leq k\leq n$ ,

L^{\,n-k}:H^k(X;\mathbb R)\longrightarrow H^{2n-k}(X;\mathbb R), \qquad [\alpha]\longmapsto[\omega]^{\,n-k}\smile[\alpha],

is an isomorphism. The same holds with complex coefficients. Equivalently, $L^r:H^{n-r}(X)\to H^{n+r}(X)$ is an isomorphism for $0\leq r\leq n$ . Compactness and the Kähler condition are essential hypotheses of this statement Voisin, Theorem 6.25.

Proof mechanism

The Kähler identities show that the Lefschetz operator $L$ , its adjoint $\Lambda$ , and the degree operator act as an $\mathfrak{sl}_2$ -triple on harmonic forms. Finite-dimensional $\mathfrak{sl}_2$ -representation theory then makes the displayed powers of $L$ bijective. Harmonic representatives transfer these linear-algebraic isomorphisms to de Rham cohomology.

Primitive decomposition

For $k\leq n$ , the primitive cohomology is

P^k(X)=\ker\!\left(L^{\,n-k+1}:H^k(X)\to H^{2n-k+2}(X)\right).

Hard Lefschetz yields the direct-sum decomposition

H^k(X)=\bigoplus_{j\geq0}L^jP^{k-2j}(X),

with only terms of nonnegative degree included. This decomposition organizes the cohomology into irreducible strings for the Lefschetz $\mathfrak{sl}_2$ -action.

Scope and consequences

The theorem implies, among other things, that multiplication by $[\omega]$ is injective below the middle degree and surjective at or above it. It is a statement over $\mathbb R$ or $\mathbb C$ ; an integral isomorphism is not asserted, and an arbitrary Kähler class need not be integral. Closed symplectic manifolds need not satisfy Hard Lefschetz, so symplecticity alone is a decisive near-miss.

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 and Theorem 6.25, Lefschetz decomposition and the Hard Lefschetz theorem.