Tangent classes for matroid building sets

Abstract: Let (M) be a loopless matroid on a finite ground set (E), and let (\G) be a building set containing the top flat (E). We define a tangent class (T_{M,\G}) in the (K)-ring (K(M,\G)), which extends the tangent bundle class of the de Concini--Procesi wonderful model from realizable matroids to arbitrary matroids with building sets. The class (T_{M,\G}) satisfies a matroidal Hirzebruch--Riemann--Roch package. More precisely, its Hirzebruch class [ \operatorname{ch}(λy T{M,\G}^{\vee})\operatorname{td}(T_{M,\G}) ] specializes to the Todd class and computes the Chow polynomial of ((M,\G)). In the realizable case, these identities agree with the usual tangent-bundle computations on the corresponding wonderful model. As an application, we prove Chern-number inequalities for (T_{M,\G}), including a Miyaoka--Yau type inequality with respect to the hyperplane class.

• The paper constructs an intrinsic K-theory tangent class for every loopless matroid and top-containing building set, extending the maximal-building-set construction through Aluffi’s blow-up recursion and K-theoretic Stanley–Reisner relations.
• The resulting formal Hirzebruch–Riemann–Roch package identifies the Todd and Hirzebruch classes with expressions from the tangent class and recovers the Chow polynomial without requiring the matroid to be realizable.
• The theory produces canonical-class formulas, formal Serre duality, projective-space Chern-number lower bounds, and a Miyaoka–Yau-type inequality whose defect is explicitly determined by the number of rank-2 flats.

Overview

This paper, by Ronnie Cheng, extends the construction of a tangent class for matroids from the maximal building set to arbitrary building sets. Given a loopless matroid $M$ of rank $r$ on a ground set $E$ and a building set $\mathcal G$ containing the top flat $E$, the paper defines a $K$-class $T_{M,\mathcal G}$ in the Feichtner–Yuzvinsky $K$-ring $K(M,\mathcal G)$ that behaves as the tangent bundle of the de Concini–Procesi wonderful model when $M$ is realizable by a linear subspace. The central result is a matroidal Hirzebruch–Riemann–Roch package: the Hirzebruch class $r$0 specializes to the Todd class and computes the Chow polynomial of $r$1. The construction is entirely formal in the combinatorially defined Chow and $r$2-rings, so it applies without any realizability hypothesis.

The motivation is geometric. For a realizable matroid $r$3, realized by $r$4, the building set $r$5 determines the wonderful compactification $r$6 of $r$7, which embeds as the closure of the torus orbit inside the smooth toric variety $r$8 associated to the nested-set fan. The Feichtner–Yuzvinsky presentation identifies the Chow rings of $r$9 and $E$0, and the analogous statement for $E$1-rings follows from [LLPP]. A tangent class for the maximal building set was previously constructed in [ChengTangent]; the present work removes the maximality assumption.

The intrinsic definition of the tangent class

The definition is modeled on Aluffi's complete-intersection blow-up formula. If $E$2 blows up a smooth complete intersection $E$3 of divisors meeting transversely, Aluffi's correction term in $E$4-theory is

$E$5

For each non-top flat $E$6, the paper introduces the cutting class

$E$7

where $E$8 is the common value of the Feichtner–Yuzvinsky linear relations (the hyperplane class in realizable cases) and $E$9 are the boundary divisor classes. In the realizable picture, $\mathcal G$0 is the class of the strict transform of a general hyperplane containing the stratum indexed by $\mathcal G$1. The tangent class is then defined intrinsically as

$\mathcal G$2

with total Chern class

$\mathcal G$3

The class decomposes as a logarithmic part $\mathcal G$4 plus boundary-normal terms $\mathcal G$5; in the realizable case this recovers the exact sequence relating $\mathcal G$6, the logarithmic tangent bundle $\mathcal G$7, and the boundary normals. When $\mathcal G$8 is maximal, the class agrees with the earlier construction via two telescoping arguments built on the $\mathcal G$9-theoretic Stanley–Reisner relation: incomparable flats $E$0 satisfy $E$1. The same argument covers building sets in which every incomparable pair has its join in $E$2, corresponding to polymatroids.

A concrete illustration is given for the braid matroid $E$3 with minimal building set: starting from $E$4, blowing up five points and ten lines, formula yields the familiar Chern class expression involving $E$5, the exceptional classes $E$6, and $E$7.

Compatibility with one-step blow-ups

If $E$8 is a one-step enlargement, the nested-set fan refines by stellar subdivision at the cone over the $E$9-factors $K$0 of $K$1, giving a toric blow-up $K$2 with center $K$3 and normal bundle $K$4. Two lemmas drive the compatibility: cutting classes pull back cleanly ($K$5), and a "factor hyperplane cancellation" identity shows that the hyperplane-correction contributions of the old factors exactly cancel the new $K$6-summand, using again the $K$7-theoretic Stanley–Reisner relations. The result is the relative Aluffi identity:

$K$8

which is precisely the Aluffi $K$9-recursion for the toric blow-up.

The paper also defines a normal class $T_{M,\mathcal G}$0, motivated in the realizable case by the normal bundle of $T_{M,\mathcal G}$1 inside the Boolean ambient toric variety $T_{M,\mathcal G}$2; under one-step enlargements it satisfies $T_{M,\mathcal G}$3. The author poses as an open question whether $T_{M,\mathcal G}$4 arises as the restriction of a Berget–Eur–Spink–Tseng type quotient class on the Boolean ambient variety, which would extend the quotient-bundle picture to arbitrary building sets.

The formal HRR package and its realization

Since $T_{M,\mathcal G}$5 refines $T_{M,\mathcal G}$6 for the maximal building set $T_{M,\mathcal G}$7, there is a proper toric morphism $T_{M,\mathcal G}$8. The Euler characteristic is defined by pullback to the maximal building set, and the formal Todd class by pushforward:

$T_{M,\mathcal G}$9

Formal HRR, $K$0, follows immediately from the projection formula. A recursive Hirzebruch class $K$1 is defined along one-step chains, with correction term $K$2 where $K$3; the resulting degree identity reproduces the one-step Hilbert-series formula for the Chow polynomial $K$4 from [EFMPV]. A priori this recursion depends on the chosen chain; the main theorem eliminates this dependence.

The main theorem states that for every building set,

$K$5

so in particular $K$6 and the Chow polynomial is the specialization at $K$7:

$K$8

The proof combines three inputs. First, a universal one-step identity for blow-ups along transverse complete intersections: if the tangent classes satisfy the Aluffi recursion and the center restriction satisfies $K$9, then $K(M,\mathcal G)$0. This is reduced, via multiplicativity of $K(M,\mathcal G)$1 and the projection formula, to the known case $K(M,\mathcal G)$2 treated in [BSY]. Second, the relative Aluffi identity supplies the first hypothesis. Third, the star-normal identity

$K(M,\mathcal G)$3

supplies the second. Its proof is the most delicate part of the paper: restricting the defining summands to the closed star $K(M,\mathcal G)$4, the leftover rank-shift terms are cancelled using a right-factor telescoping identity derived from the fact that products of $K(M,\mathcal G)$5-theoretic Stanley–Reisner factors vanish off chains. Descending induction along a one-step chain then completes the proof. Since everything is formal in the Feichtner–Yuzvinsky rings, no realization of $K(M,\mathcal G)$6 is required — this is the substantive point distinguishing the argument from a purely geometric verification.

Numerical consequences

The realization theorem yields several concrete consequences. The formal canonical class is

$K(M,\mathcal G)$7

with the one-step transformation rule $K(M,\mathcal G)$8, matching the behavior of canonical classes under blow-ups. A formal Serre duality holds: $K(M,\mathcal G)$9, proved by the symmetry of the Todd product under $M$0.

Hyperplane degrees of the Todd class are computed explicitly: $M$1, obtained by comparing $M$2 with the HRR expansion. A key vanishing lemma gives $M$3 for a rank-$M$4 flat, interpreted geometrically as an exceptional-divisor self-intersection.

The main numerical result is a Chern-number comparison with projective space. Writing $M$5, for every $M$6:

$M$7

The proof uses a truncation mechanism: intersecting with $M$8 general hyperplanes corresponds to passing to the $M$9-fold truncation $r$00 with the truncated building set $r$01, and the degree functional descends through the natural surjection of Chow rings. Combined with Vandermonde's identity and the observation that $r$02 (since the truncated ring has nonzero classes in all degrees up to $r$03), this yields the inequality. Taking $r$04 together with the Todd-class identities produces a Miyaoka–Yau type inequality with respect to the polarization $r$05:

$r$06

of the same form as the higher-dimensional Miyaoka–Yau inequality for polarized varieties [GrebKebekusTaji]. Notably, the paper's final remark sharpens this to an equality: the difference equals $r$07, where $r$08 is the number of rank-2 flats in $r$09 — so the inequality is strict precisely when the building set contains more than one rank-2 flat beyond the atoms forced by the ground set structure.

Limitations and open questions

Several points deserve plain acknowledgment. First, the Euler characteristic $r$10 is defined by pullback to the maximal building set rather than derived from a cohomology theory intrinsic to $r$11; the package is formal, and while it agrees with genuine sheaf cohomology on wonderful models in the realizable case, the paper does not construct an intrinsic categorical interpretation for arbitrary matroids. Second, the recursive Hirzebruch class is defined via a choice of one-step chain, and well-definedness is a consequence of the main theorem rather than established independently. Third, the question of whether the normal class $r$12 restricts from a BEST-type quotient class on the Boolean ambient variety remains open, as does the expected regular-section description of $r$13 as a zero locus extending the quotient-bundle picture to arbitrary building sets. Finally, the Miyaoka–Yau inequality is stated only with respect to the specific polarization $r$14; inequalities for other nef classes are not addressed.

Conclusion

The paper completes the program initiated for the maximal building set by constructing, for every matroid and every top-containing building set, a single $r$15-class $r$16 that simultaneously realizes the Todd class, the Hirzebruch class, and the Chow polynomial of the pair $r$17, and that agrees with the honest tangent bundle on de Concini–Procesi wonderful models in the realizable case. The proof technique — combining Aluffi's blow-up recursion with a star-normal factorization identity verified through $r$18-theoretic Stanley–Reisner relations — is purely combinatorial and transfers geometric blow-up calculus verbatim to the non-realizable setting. The resulting Chern-number inequalities, including the Miyaoka–Yau type bound with its explicit defect $r$19, demonstrate that the formal package carries genuine numerical content beyond the realizable world.