Hodge theory for combinatorial projective bundles

Abstract: We prove the Hard Lefschetz theorem and Hodge-Riemann relations for certain rings which resemble the cohomology rings of projectivizations of globally generated vector bundles over toric varieties. This proves new cases of the standard conjecture of Hodge type and gives Bloch-Gieseker-type results for tautological classes of matroids.

• The paper proves the Kähler package—Poincaré duality, Hard Lefschetz, and Hodge–Riemann relations—for projective bundle rings built from matroid Chow rings and Lefschetz fans.
• The authors establish the projective bundle relation through a combinatorial canonical-expansion and cancellation argument on biflags, avoiding algebraic-geometric methods and extending to iterated bundles.
• The results imply Bloch–Gieseker-type positivity for matroid tautological classes and verify cases of Grothendieck’s standard conjecture of Hodge type for projectivized globally generated toric vector bundles in positive characteristic.

Overview

The paper proves the Kähler package—Poincaré duality, Hard Lefschetz, and Hodge–Riemann relations—for a family of graded algebras modeled on the Chow rings of projectivizations of globally generated toric vector bundles. The main result, Theorem 1.1, states that for a projective simplicial fan $\Sigma$ refining the normal fan of $\varphi^*(P(\mathrm{M}))$ for a matroid $\mathrm{M}$ of rank $r$, the "projective bundle ring"

$$B = A(\Sigma)[\zeta]/(\zeta^r + c_1\zeta^{r-1} + \dotsb + c_r),$$

where $c_1, \dotsc, c_r$ are the Chern classes of the tautological bundle associated to $\mathrm{M}$, satisfies the Kähler package with respect to the degree map and the interior of the cone generated by the ample cone of $\Sigma$ and $\zeta$. The proof is entirely combinatorial and does not invoke algebraic geometry. The result was conjectured by Hunter Spink in 2021 and posed as an open problem by Christopher Eur at a 2023 BIRS workshop; prior work had established only the degree $0$ and $\varphi^*(P(\mathrm{M}))$0 Hodge–Riemann relations, using algebro-geometric methods.

Combinatorial setup

The starting point is the observation that a globally generated $\varphi^*(P(\mathrm{M}))$1-equivariant vector bundle $\varphi^*(P(\mathrm{M}))$2 on a toric variety $\varphi^*(P(\mathrm{M}))$3 arises as the pullback of the dual universal subbundle from a Grassmannian along a $\varphi^*(P(\mathrm{M}))$4-equivariant map whose image is the toric variety of the normal fan of a matroid polytope $\varphi^*(P(\mathrm{M}))$5. The Chern classes of $\varphi^*(P(\mathrm{M}))$6 are therefore encoded by piecewise polynomial functions $\varphi^*(P(\mathrm{M}))$7 on $\varphi^*(P(\mathrm{M}))$8, where on a maximal cone indexed by a basis $\varphi^*(P(\mathrm{M}))$9 the function $\mathrm{M}$0 is the $\mathrm{M}$1th elementary symmetric function of the coordinates of $\mathrm{M}$2. Consequently, classifying Chern classes of globally generated toric vector bundles reduces to the combinatorics of realizable matroids.

The paper works with fans permitted to have nontrivial lineality spaces, since the distinguished piecewise polynomial functions do not descend to the quotient by the lineality space. A central technical object is the bipermutohedral fan $\mathrm{M}$3 in $\mathrm{M}$4, defined via biflags of proper bisubsets, and its subfan $\mathrm{M}$5 (the "projective bundle fan") whose cones correspond to biflags of $\mathrm{M}$6. This fan serves as the tropical model for the projectivization of the tautological quotient bundle. The framework also recovers known classes: the tautological classes of $\mathrm{M}$7 on the permutohedral fan, augmented tautological classes of $\mathrm{M}$8 on the stellahedral variety, and delta-matroid classes of $\mathrm{M}$9 on the type $r$0 permutohedral variety.

The projective bundle relation

The key computation, Theorem 2.x, establishes that in $r$1 one has

$r$2

i.e., $r$3 satisfies the expected projective bundle relation. This is the combinatorial analogue of the relation defining the Chow ring of a projective bundle, and it is what allows $r$4 to be realized as a subring of the Chow ring of an auxiliary Lefschetz fan.

The proof is a delicate induction on lexicographically decreasing biflags. The central difficulty is that the class $r$5 has many representatives $r$6, and the computation requires choosing the correct representative at each step via a "canonical expansion" (Definition 2.x), which expresses each product as a signed sum of squarefree monomials with coefficients in $r$7. A cancellation mechanism (Lemma 2.x and the accompanying Proposition 2.x) shows that the negative terms of the canonical expansion inject into the positive terms, leaving a nonnegative remainder that vanishes by induction on the pair $r$8 where $r$9. The final step uses Poincaré duality on the bipermutohedral fan to convert the vanishing into an intersection-theoretic statement, then proceeds by induction on the rank of $$B = A(\Sigma)[\zeta]/(\zeta^r + c_1\zeta^{r-1} + \dotsb + c_r),$$0 using truncation: multiplying the Minkowski weight $$B = A(\Sigma)[\zeta]/(\zeta^r + c_1\zeta^{r-1} + \dotsb + c_r),$$1 by $$B = A(\Sigma)[\zeta]/(\zeta^r + c_1\zeta^{r-1} + \dotsb + c_r),$$2 yields the Minkowski weight of the truncation $$B = A(\Sigma)[\zeta]/(\zeta^r + c_1\zeta^{r-1} + \dotsb + c_r),$$3.

Proof of the main theorem

The proof of Theorem 1.1 proceeds by embedding $$B = A(\Sigma)[\zeta]/(\zeta^r + c_1\zeta^{r-1} + \dotsb + c_r),$$4 into the Chow ring of a carefully constructed Lefschetz fan $$B = A(\Sigma)[\zeta]/(\zeta^r + c_1\zeta^{r-1} + \dotsb + c_r),$$5 refining $$B = A(\Sigma)[\zeta]/(\zeta^r + c_1\zeta^{r-1} + \dotsb + c_r),$$6, whose support is $$B = A(\Sigma)[\zeta]/(\zeta^r + c_1\zeta^{r-1} + \dotsb + c_r),$$7. The projective bundle relation ensures that the subring of $$B = A(\Sigma)[\zeta]/(\zeta^r + c_1\zeta^{r-1} + \dotsb + c_r),$$8 generated by $$B = A(\Sigma)[\zeta]/(\zeta^r + c_1\zeta^{r-1} + \dotsb + c_r),$$9 and the pullback $c_1, \dotsc, c_r$0 of $c_1, \dotsc, c_r$1 is isomorphic to $c_1, \dotsc, c_r$2; injectivity follows from Poincaré duality by verifying that $c_1, \dotsc, c_r$3 is nonzero, which reduces to the known fact that $c_1, \dotsc, c_r$4 in the Chow ring of a matroid. Since $c_1, \dotsc, c_r$5 is a Lefschetz fan and $c_1, \dotsc, c_r$6 is generated by classes lying in the closure of the Lefschetz cone, the authors invoke the Hodge–Riemann descent criterion (a form of the Lefschetz module machinery of Cattani–Kaplan–Schnell and Kashiwara–Kawai) to conclude that $c_1, \dotsc, c_r$7 has the Kähler package.

Two structural features of the argument deserve emphasis. First, the proof crucially relies on the Hodge–Riemann relations for matroid Chow rings $c_1, \dotsc, c_r$8, so it does not give a new proof of that result—although, conversely, the special case of the permutohedral fan recovers $c_1, \dotsc, c_r$9 via the quotient construction of Proposition 2.x, indicating the strength of the statement. Second, and importantly, the standard inductive strategy for proving Kähler packages—which simultaneously proves ampleness of the Lefschetz elements—cannot be used here: as the paper shows in Example 2.x, in the permutohedral case $\mathrm{M}$0 is the Chow ring of a smooth projective toric variety (even for non-realizable $\mathrm{M}$1), but the cone in question is strictly larger than the nef cone, since the splitting line bundles $\mathrm{M}$2 are essentially never nef. The theorem is therefore genuinely outside the reach of ampleness-based arguments.

Theorem 1.1 extends in two directions (Theorem 2.x): to iterated projective bundles over multiple matroids, and to arbitrary Lefschetz fans as the base, yielding in particular the Kähler package for rings over the Chow ring of an arbitrary matroid $\mathrm{M}$3 built from tautological classes of other matroids $\mathrm{M}$4.

Applications

Two classes of consequences follow. First, adapting the Bloch–Gieseker argument, the paper proves (Theorem 3.x) that if multiplication by $\mathrm{M}$5 on $\mathrm{M}$6 has full rank, then multiplication by $\mathrm{M}$7 on $\mathrm{M}$8 is injective below the middle dimension and surjective above it, where $\mathrm{M}$9. Combined with a "twisting" trick—replacing $\Sigma$0 by modified classes $\Sigma$1 via the substitution $\Sigma$2—this yields Bloch–Gieseker-type positivity statements for tautological classes of matroids, including the nonvanishing sign statement $\Sigma$3 when $\Sigma$4.

Second, the theorem gives new cases of Grothendieck's standard conjecture of Hodge type in positive characteristic. Specifically, for a smooth projective toric variety $\Sigma$5 and a globally generated toric vector bundle $\Sigma$6, the conjecture $\Sigma$7 holds for the ample class $\Sigma$8 on $\Sigma$9 (Corollary 1.x). This is notable because the étale cohomology of such a projectivization is generated by algebraic cycles, yet the conjecture was previously open over fields of positive characteristic. The paper argues that a lifting-to-characteristic-$\zeta$0 approach is unlikely to work in general: since the moduli of toric vector bundles satisfies Murphy's law, one can readily construct toric vector bundles in positive characteristic that do not lift, although verifying that the projectivization itself does not lift remains open.

Limitations and open questions

The paper is candid about several boundaries of the result. The Kähler package is established only for the specific cone generated by $\zeta$1 and $\zeta$2; Remark 3.x shows that some Lefschetz cone always exists for such a projective bundle ring, but the argument gives no control over it, rendering that existence unusable for applications. The Bloch–Gieseker statement requires the full-rank hypothesis on multiplication by $\zeta$3, which the main theorem does not itself supply and which must be arranged via twisting. The standard-conjecture application covers only ample classes of the form $\zeta$4, not arbitrary ample classes on $\zeta$5. Finally, the authors raise the question of how the global-generation notion for the combinatorial abstraction of toric vector bundles in $\zeta$6 compares with the classes appearing in the main theorem, and they note that whether $\zeta$7 can fail to lift to characteristic $\zeta$8 is unresolved.

Conclusion

This paper extends the Kähler package to projective bundle rings over Lefschetz fans built from matroid tautological classes, with a proof that is combinatorial throughout and that circumvents—rather than follows—the ampleness-based induction paradigm. The projective bundle relation in the Chow ring of the projective bundle fan is the technical heart, and its cancellation argument via lexicographically decreasing biflags is the paper's main methodological contribution. The results yield Bloch–Gieseker-type positivity for matroid tautological classes and new verifications of the Hodge type standard conjecture over fields of positive characteristic, while leaving open the behavior of the Kähler package for larger cones and the lifting question for projectivizations of non-liftable toric vector bundles.