The ordinary cohomology ring $H^\*(X)$ of a smooth projective toric variety $X$ has a combinatorial description: the (quotient) Stanley-Reisner ring of its fan. This ring is generated by $T$-invariant divisors of $X$, where $T$ is the torus acting on $X$. (With these assumptions, $H^\*(X)$ is also $A^\*(X)$, the Chow ring.) Can one describe hard Lefschetz in a combinatorial way in this context? Namely, how does $sl\_2$ act on $H^\*(X)$?



Combinatorial description of Hard Lefschetz for toric varieties
Hard Lefschetz depends on the choice of integral ample Cartier class. In combinatorial terms, this is the choice of integral strictly convex support function $\phi : N\_{\mathbb R} \to \mathbb R$. If you have such function then Lefshetz operator is just multiplication by $\sum\_{i=1}^m -\phi(u\_i) \tau\_i$, in terms of your SE question, where $u\_i$ is a ray generators corresponding to toric invariant divisor $D\_i$. In general, no canonical choice is available.


If $X=X\_{\Sigma}$ is a Gorenstein Fano complete toric variety then anticanonical class $-K\_{\Sigma} = \sum\_{\rho \in \Sigma(1)} D\_{\rho}$ is a natural choice. In this case the Lefshetz operator (or the action of $e \in \mathfrak{sl}\_2$, if you prefer) is a multiplication by $\sum\_{i=1}^m \tau\_i$, in terms of your SE question.


If toric variety $X=X\_P$ comes from the full dimensional lattice polytope $P \subset M\_{\mathbb R}$ then $\phi\_P(n) = \min\_{m \in P \cap M} (n,m)$ is a natural choice of integral strictly convex support function. Hence, the Lefshetz operator is a multiplication by $\sum\_{i=1}^m (-\min\_{m \in P \cap M} (m,u\_{i})) \tau\_i$.


