The Intersection Cohomology of a Fan and the Hodge Conjecture for Toric Varieties

We formulate a combinatorial version of the Intersection Hodge Conjecture for projective toric varieties. The conjecture asserts that the subspace of rational Hodge classes in the intersection cohomology $I{H}^{\ast }(X_{\mathrm{\Sigma }})$ is generated by the classes of algebraic cycles. We define the space of combinatorial Hodge classes, $Hd{g}_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{b}}^k(\mathrm{\Sigma })\subset I{H}_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{b}}^{2k}(\mathrm{\Sigma },\mathbb{Q})$, using the combinatorial intersection cohomology theory for fans developed by Barthel, Brasselet, Fieseler, and Kaup. We conjecture that this space is spanned by the combinatorial cycle classes corresponding to torus-invariant subvarieties. We verify this conjecture for all projective toric varieties of dimension $n\le 3$ and for the class of simplicial projective toric varieties. Finally, we provide an algorithmic framework to verify the conjecture for arbitrary rational fans.