In the combinatorial intersection cohomology theory of polyhedral fans, Karu (2019) proved the Relative Hard Lefschetz theorem for all perverse sheaves on a fan, but proved the Relative Hodge–Riemann bilinear relations only for the indecomposable perverse sheaves and their images under projective subdivisions, explicitly deferring the general extension. The problem is to complete the Kähler package (Poincaré duality, Hard Lefschetz, Hodge–Riemann) at the same generality: prove the relative Hodge–Riemann relations for arbitrary perverse sheaves.

Background

A polyhedral fan Σ\Sigma is a finite collection of closed convex polyhedral cones in a real vector space VV, closed under taking faces and with any two cones intersecting in a common face; a fan is complete if its cones cover all of VV. To a rational fan one can associate a toric variety, and the fan then encodes the geometry of that variety, but for non-rational fans no variety exists and the fan itself must carry the structure. Barthel–Brasselet–Fieseler and Kaup (2002) and Bressler–Lunts (2003) made a fan into a ringed space and constructed a purely combinatorial analogue of the intersection cohomology sheaf: one works in a category M(Σ)M(\Sigma) of graded locally free flabby sheaves of finite type (flabby meaning that every restriction map between open sets is surjective), in which every sheaf decomposes as a direct sum ⨁σ∈ΣWσ⊗Lσ\bigoplus_{\sigma\in\Sigma}W_\sigma\otimes L_\sigma of shifted copies of indecomposable sheaves LσL_\sigma. The sheaf LΣL_\Sigma plays the role of the intersection cohomology sheaf, and its global sections, reduced modulo the polynomial functions on VV, give the intersection cohomology IH(Σ)IH(\Sigma). Mimicking the perverse t-structure of geometric intersection theory, Karu (2019) defines a perverse filtration on M(Σ)M(\Sigma); the category M0(Σ)M_0(\Sigma) of perverse sheaves consists of finite direct sums of the sheaves Lτ(codim τ)L_\tau(\mathrm{codim}\,\tau) for cones τ∈Σ\tau\in\Sigma, the shift by codimension being the combinatorial analogue of the standard perverse shift on a smooth variety.

For a projective variety, intersection cohomology satisfies the Kähler package: Poincaré duality, the Hard Lefschetz theorem, and the Hodge–Riemann bilinear relations, which state that multiplication by an ample class ll gives isomorphisms IHn−i→IHn+iIH^{n-i}\to IH^{n+i} and that the quadratic form Ql(w)=⟨liw,w⟩Q_l(w)=\langle l^i w,w\rangle is definite, with sign (−1)(n−i)/2(-1)^{(n-i)/2}, on the primitive part (the kernel of multiplication by li+1l^{i+1}) of IHn−iIH^{n-i}. In the combinatorial setting the role of the ample class is played by a strictly convex piecewise-linear function on the fan, i.e. a convex piecewise-linear function whose maximal domains of linearity are exactly the maximal cones; a complete fan admits such a function exactly when it is polytopal (the normal fan of a polytope). For polytopal fans the combinatorial Kähler package is known: Hard Lefschetz was proved for non-rational polytopes by Karu (2004), and the Hodge–Riemann relations for intersection cohomology of non-rational polytopes by Bressler–Lunts (2005), both by purely combinatorial inductions simulating the Hodge-theoretic arguments.

Karu (2019) extended this to the relative situation of a projective subdivision π:Σ^→Σ\pi:\hat\Sigma\to\Sigma of fans (a refinement of Σ\Sigma induced by a piecewise-linear function), the fan analogue of a resolution, where the sheaf π∗LΣ^\pi_*L_{\hat\Sigma} encodes Stanley's local hh-vector of the subdivision. He proved the Relative Hard Lefschetz theorem for all perverse sheaves G∈M0(Σ^)G\in M_0(\hat\Sigma) (his Theorem 4.6), but the Relative Hodge–Riemann bilinear relations (his Theorem 4.16) only for the indecomposable sheaves G=Lτ(codim τ)G=L_\tau(\mathrm{codim}\,\tau) and their images π∗G\pi_*G. He states explicitly that extending the theorem to general perverse sheaves 'is not as simple as in the case of the RHL theorem, so we will not do it', because the sign of the quadratic form on the primitive part of the multiplicity spaces WσW_\sigma depends on the dimension of τ\tau, and 'one needs to introduce an additional filtration to take this into account'. Likewise, his global Hodge–Riemann statements for the space l⋅IH(Σ^)l\cdot IH(\hat\Sigma) of a complete fan Σ^\hat\Sigma are proved under strict convexity of l^\hat l, i.e. exactly for the polytopal case.

Problem Statement

Within the combinatorial intersection cohomology framework on polyhedral fans (not assumed rational), complete the Hodge–Riemann part of the Kähler package to the same generality as the decomposition theorem and the Relative Hard Lefschetz theorem, specifically:

  1. Relative Hodge–Riemann relations for arbitrary perverse sheaves. Let π:Σ^→Σ\pi:\hat\Sigma\to\Sigma be a projective subdivision of fans and let l^\hat l be a piecewise-linear function on Σ^\hat\Sigma that is relatively strictly convex with respect to π\pi, i.e. strictly convex on the inverse image of every cone σ∈Σ\sigma\in\Sigma (Karu's hypothesis in Theorems 4.6 and 4.16). Karu's Relative Hard Lefschetz theorem holds for every perverse sheaf G∈M0(Σ^)G\in M_0(\hat\Sigma), i.e. every finite direct sum ⨁jWτj⊗Lτj(codim τj)\bigoplus_j W_{\tau_j}\otimes L_{\tau_j}(\mathrm{codim}\,\tau_j), but his Relative Hodge–Riemann theorem covers only the indecomposable summands G=Lτ(codim τ)G=L_\tau(\mathrm{codim}\,\tau). Formulate and prove the relative Hodge–Riemann bilinear relations for arbitrary perverse GG: the definite sign (−1)(c−i)/2(-1)^{(c-i)/2}, c=codim τ−codim σc=\mathrm{codim}\,\tau-\mathrm{codim}\,\sigma, must be well defined on the graded pieces of π∗G\pi_*G (as Karu notes, the sign of Ql^Q_{\hat l} on the primitive part of the multiplicity spaces WσW_\sigma depends on dim⁡τ\dim\tau), and (−1)(c−i)/2 Ql^(-1)^{(c-i)/2}\,Q_{\hat l} must be positive definite on the appropriate primitive parts (kernels of multiplication by l^i+1\hat l^{i+1}) of the associated graded, recovering Karu's Theorem 4.16 as the indecomposable case.

The objective is to remove the special-sheaf restriction under which the Hodge–Riemann bilinear relations for combinatorial intersection cohomology are currently known, so that the relative Kähler package holds at the generality the rest of the theory (decomposition theorem, Poincaré duality, Relative Hard Lefschetz) already reaches.

References

  1. Kalle Karu, Relative Hard Lefschetz theorem for fans, Advances in Mathematics 347 (2019) 859–903. DOI: 10.1016/j.aim.2019.03.006; arXiv:1710.06468, https://arxiv.org/abs/1710.06468
  2. Gottfried Barthel, Jean-Paul Brasselet, Karl-Heinz Fieseler, Ludger Kaup, Combinatorial intersection cohomology for fans, Tohoku Mathematical Journal (2) 54 (2002) 1–41. DOI: 10.2748/tmj/1113247177, https://doi.org/10.2748/tmj/1113247177
  3. Paul Bressler, Valery A. Lunts, Hard Lefschetz theorem and Hodge-Riemann relations for intersection cohomology of nonrational polytopes, Indiana University Mathematics Journal 54 (2005) 263–308. DOI: 10.1512/iumj.2005.54.2528, https://doi.org/10.1512/iumj.2005.54.2528
  4. Kalle Karu, Hard Lefschetz theorem for nonrational polytopes, Inventiones mathematicae 157 (2004) 419–447. DOI: 10.1007/s00222-004-0358-3, https://doi.org/10.1007/s00222-004-0358-3
  5. Richard P. Stanley, Subdivisions and local h-vectors, Journal of the American Mathematical Society 5 (1992) 805–851. DOI: 10.1090/S0894-0347-1992-1157293-9, https://doi.org/10.1090/S0894-0347-1992-1157293-9
  6. Ling Hei Tsang, Filtrations on combinatorial intersection cohomology and invariants of subdivisions, Advances in Mathematics 409 (2022), article 108666. DOI: 10.1016/j.aim.2022.108666; arXiv:2205.03992, https://arxiv.org/abs/2205.03992
  7. Omid Amini, June Huh, Matt Larson, A decomposition theorem for Lefschetz modules, preprint (2025), arXiv:2511.02026, https://arxiv.org/abs/2511.02026
  8. Tom Braden, Nicholas Proudfoot, Intersection cohomology without spaces, Proceedings of the International Congress of Mathematicians 2026, to appear; arXiv:2510.09488, https://arxiv.org/abs/2510.09488
  9. Omid Amini, Matthieu Piquerez, Hodge theory for tropical fans, preprint (2023); arXiv:2310.15367, https://arxiv.org/abs/2310.15367