Hard Lefschetz theorem

Remarks

The classical proof uses Hodge theory on the Kähler manifold $X(\C)$. The operator $L$, its adjoint $\Lambda$ and the degree operator form a representation of $\mathfrak{sl}_2$ on harmonic forms, and $L^k$ is an isomorphism by the representation theory of $\mathfrak{sl}_2$. For $X = \mathbb{P}^n$, $H^{2i}(X, \Q) = \Q \, \omega^i$ for $0 \le i \le n$ and the odd groups vanish, so $L^k$ sends $\omega^{(n-k)/2}$ to $\omega^{(n+k)/2}$, which is a basis. The theorem shows that $L \colon H^j \to H^{j+2}$ is injective for $j < n$, so the Betti numbers satisfy $b_j \le b_{j+2}$ for $j < n$, and also $b_j = b_{2n-j}$. Over a field of positive characteristic, Deligne proved the analogue for $\ell$-adic cohomology. The hypothesis that $\omega$ comes from an ample class matters: for a class $\omega$ with $\omega^n = 0$, such as $0$, the map $L^n \colon H^0 \to H^{2n}$ is zero.

Going deeper

For a student

The theorem gives the Lefschetz decomposition of $H^*(X, \Q)$ into primitive parts, and together with the Hodge–Riemann bilinear relations it shows that the cohomology carries a polarized Hodge structure. It is one of the standard tools that restrict which manifolds can be projective varieties: for example, the sequences $b_0, b_2, b_4, \ldots$ and $b_1, b_3, b_5, \ldots$ cannot decrease before the middle degree. It complements the Lefschetz hyperplane theorem, which compares the cohomology of $X$ with that of a hyperplane section below the middle degree. Grothendieck's Lefschetz-type standard conjecture asks that the inverse of $L^k$ come from an algebraic cycle.