Combinatorial description of Hard Lefschetz for toric varieties 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 $H^*(X)$?
 A: 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 $\sum_{i=1}^m (-\min_{m \in P \cap M} (m,u_{i})) \tau_i$.
