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 $H^*(X)$?
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$\begingroup$ Crosspost from math.stackexchange.com/questions/4459555/… $\endgroup$– Alvaro MartinezJun 1, 2022 at 20:00
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$\begingroup$ The action of $sl_2$ is not by ring automorphisms, so describing its action on the $sl_2$-invariant divisors is not the same as describing its action on the full ring. Which one do you want? $\endgroup$– Will SawinJun 1, 2022 at 20:11
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$\begingroup$ @Will Sawin Didn’t realize that! Fair enough, I suppose I’d like to describe the action on the generators as a module (monomials) $\endgroup$– Alvaro MartinezJun 1, 2022 at 20:32
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1$\begingroup$ Are you aware of the paper by Fleming and Karu, "Hard Lefschetz theorem for simple polytopes" (personal.math.ubc.ca/~karu/papers/simple.pdf)? $\endgroup$– Sam HopkinsJun 1, 2022 at 21:11
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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 $\sum_{i=1}^m (-\min_{m \in P \cap M} (m,u_{i})) \tau_i$.
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1$\begingroup$ The action of $h \in \mathfrak{sl}_2$ is multiplication by $d-n$ for $d$ the degree and $n$ the dimension of $X$, which acts by multiplication on monomials. What about $f\in \mathfrak{sl}_2$? Is there a description of it better than "the unique operator which satisfies the relations"? $\endgroup$ Jun 1, 2022 at 22:04
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$\begingroup$ @WillSawin In complex-analytical setting it is $\Lambda = \star^{-1} L \star$ conjugation of Lefshetz operator with Hodge star. I don't know any specific combinatorial/toric description of $f$. $\endgroup$ Jun 1, 2022 at 22:51