# Higher degree generalizations of the Hard Lefschetz Theorem

Let $M$ be a $2d$-dimensional manifold. We say that $\omega \in H^2(M)$ has the Hard Lefschetz Property (HLP) if multiplication with $\omega^j$ is an isomorphism $H^{d-j} \to H^{d+j}$. This holds for the first Chern class of an ample line bundle on a smooth projective variety, this is the classical Hard Lefschetz Theorem.

My question is whether there is a generalization to classes in $H^4, H^6,$ ... etc.

My proposed definition is that $\eta \in H^k(M)$ has the k-HLP if multiplication by $\eta^j$ from $H^i$ to $H^{i+jk}$ is invective (surjective) if $|i-d|$ is greater (smaller) than $|i+jk-d|$. Then 2-HLP is HLP and if $\omega$ has the HLP then $\omega^j$ has the $2j$-HLP.

Are there other examples of classes with the $2j$-HLP? E.g. Chern classes of vector bundles satisfying an appropriate ampleness condition?

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I like this question! One little fact which is relevant but doesn't answer your question: if one chooses any $d$ ample classes and uses them to define maps $H^0 \to H^2 \to H^4 \dots \to H^{2d}$ then hard Lefschetz and the Hodge-Riemann relations are still satisfied. Hence any product of $j$ ample classes will give you a $2j$-HLP. Cattani is the expert, e.g. arxiv.org/abs/0707.1352. –  Geordie Williamson Oct 5 '13 at 22:51
Hard Lefschetz can be seen as "coming from $H^*(\mathbb{P}^n)$". Next on the scale of "easy" cohomology rings are Grassmannians. I wonder if the Chern classes of the tautological bundle give $2j$-HLP's. There are explicit formulas (Pieri rule) so it shouldn't be difficult to decide... –  Geordie Williamson Oct 5 '13 at 22:59

As Geordie pointed out in the comments, products of Kähler classes have the $2j$-HLP.

In the paper "The mixed Hodge-Riemann bilinear relations for compact Kähler manifolds" by T-C. Dinh and V-A. Nguyên, the authors introduce the notion of Hodge-Riemann form.

A real form $\Omega$ of type $(k,k)$ is called a Hodge-Riemann form if at each point there is a continuous path $\Omega_t$ with $\Omega_0 = \Omega$ and $\Omega_1 = \omega^k$, where $\omega$ is a fixed Kähler form, in such a way that for each $t$ the map $\alpha \mapsto \Omega_t \wedge \omega^{2r} \wedge \alpha$ is an isomorphism for $0 \leq r \leq \min\{p,q\}$.

They proove that the Hodge-Riemann forms satisfy the Hard Lefschetz Theorem. Using a result by V. Timorin this result apllies for $\Omega = \omega_1 \wedge \cdots \wedge \omega_k$, where $\omega_j$ are Kähler forms.

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I guess I knew that I should've looked in part 2 of Lazarsfeld's book before posting this question, but it was checked out of the library...

Anyway, the speculation in the last paragraph is correct: if $X$ is a smooth projective variety and $E$ is an ample vector bundle on $X$ of rank $e$, then $c_e(E)$ has the $2e$-HLP. This is exactly the statement of Theorem 7.1.10 in Positivity in Algebraic Geometry II. (Lazarsfeld only considers multiplication by a single top Chern class, not powers of it, but this is equivalent by replacing $E$ by $E^{\oplus r}$).

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