LLAGRANGIAN GEOMETRY OF MATROIDS
FEDERICO ARDILA, GRAHAM DENHAM, AND JUNE HUHA
BSTRACT . We introduce the conormal fan of a matroid M , which is a Lagrangian analog of theBergman fan of M . We use the conormal fan to give a Lagrangian interpretation of the Chern–Schwartz–MacPherson cycle of M . This allows us to express the h -vector of the broken circuitcomplex of M in terms of the intersection theory of the conormal fan of M . We also develop gen-eral tools for tropical Hodge theory to prove that the conormal fan satisfies Poincaré duality, thehard Lefschetz theorem, and the Hodge–Riemann relations. The Lagrangian interpretation of theChern–Schwartz–MacPherson cycle of M , when combined with the Hodge–Riemann relations forthe conormal fan of M , implies Brylawski’s and Dawson’s conjectures that the h -vectors of thebroken circuit complex and the independence complex of M are log-concave sequences. C ONTENTS
1. Introduction 21.1. Geometry of matroids 21.2. Conormal fans and their geometry 31.3. Inequalities for matroid invariants 71.4. Tropical Hodge theory 102. The bipermutohedral fan 122.1. The normal fan of the simplex 132.2. The normal fan of the permutohedron 132.3. The bipermutohedral fan as a subdivision 152.4. The bipermutohedral fan as a configuration space 162.5. The bipermutohedral fan as a common refinement 182.6. The bipermutohedral fan in terms of its rays and cones 192.7. The bipermutohedral fan as the normal fan of the bipermutohedron 223. The conormal intersection theory of a matroid 253.1. Homology and cohomology 253.2. The Bergman fan of a matroid 273.3. The Chow ring of the Bergman fan 28 a r X i v : . [ m a t h . C O ] M a y FEDERICO ARDILA, GRAHAM DENHAM, AND JUNE HUH
NTRODUCTION
Geometry of matroids. A matroid M on a finite set E is a nonempty collection of subsets of E , called flats of M , that satisfies the following properties:(1) The intersection of any two flats is a flat.(2) For any flat F , any element in E ´ F is contained in exactly one flat that is minimal amongthe flats strictly containing F .The set L p M q of all flats of M is a geometric lattice, and all geometric lattices arise in this wayfrom a matroid [Wel76, Chapter 3]. The theory of matroids captures the combinatorial essenceshared by natural notions of independence in linear algebra, graph theory, matching theory, thetheory of field extensions, and the theory of routings, among others.Gian-Carlo Rota, who helped lay down the foundations of the field, was one of its mostenergetic ambassadors. He rejected the “ineffably cacophonous" name of matroids, preferringto call them combinatorial geometries instead. This alternative name never really caught on, butthe geometric roots of the field have since grown much deeper, bearing many new fruits. The AGRANGIAN GEOMETRY OF MATROIDS 3 geometric approach to matroid theory has recently led to solutions of long-standing conjectures,and to the development of fascinating mathematics at the intersection of combinatorics, algebra,and geometry.There are at least three useful polyhedral models of a matroid M . For a short survey, see[Ard18]. The first one is the basis polytope of M introduced by Edmonds in optimization andGelfand–Goresky-MacPherson-Serganova in algebraic geometry. It reveals the intricate rela-tionship of matroids with the Grassmannian variety and the special linear group. The secondmodel is the Bergman fan of M , introduced by Sturmfels and Ardila–Klivans in tropical geom-etry. It was used by Adiprasito–Huh–Katz to prove the log-concavity of the f -vectors of theindependence complex and the broken circuit complex of M . The third model, which we callthe conormal fan of M , is the main character of this paper. We use its intersection-theoretic andHodge-theoretic properties to prove conjectures of Brylawski [Bry82], Dawson [Daw84], andSwartz [Swa03] on the h -vectors of the independence complex and the broken circuit complexof M .1.2. Conormal fans and their geometry.
Throughout the paper, we write r ` for the rank of M , write n ` for the cardinality of E , and suppose that n is positive. Following [MS15], wedefine the tropical projective torus of E to be the n -dimensional vector space N E “ R E { span p e E q , e E “ (cid:88) i P E e i . The tropical projective torus is equipped with the functions α j p z q “ max i P E p z j ´ z i q , one for each element j of E .These functions are equal to each other modulo global linear functions on N E , and we write α for the common equivalence class of α j . The Bergman fan of M , denoted Σ M , is an r -dimensionalfan in the n -dimensional vector space N E whose underlying set is the tropical linear space trop p M q “ (cid:110) z | min i P C p z i q is achieved at least twice for every circuit C of M (cid:111) Ď N E . It is a subfan of the permutohedral fan Σ E cut out by the hyperplanes x i “ x j for each pair ofdistinct elements i and j in E . This is the normal fan of the permutohedron Π E . The functions α j are piecewise linear on the permutohedral fan, and hence piecewise linear on the Bergman fanof M . Tropical linear spaces are central objects in tropical geometry: For any linear subspace V of C E , the tropicalization of the intersection of P p V q with the torus of P p C E q is the tropical linearspace of the linear matroid on E represented by V [Stu02]. Furthermore, tropical linear spacesare precisely the tropical fans of degree one with respect to α , that is, the tropical analogs of There are exactly two matroids on a single element ground set, the loop and the coloop , which are dual to each other.These matroids will play a special role in our inductive arguments. A continuous function f is said to be piecewise linear on a fan Σ if the restriction of f to any cone in Σ is linear. In thiscase, we say that the fan Σ supports the piecewise linear function f . FEDERICO ARDILA, GRAHAM DENHAM, AND JUNE HUH linear spaces [Fin13]. Tropical manifolds are thus defined to be spaces that locally look likeBergman fans of matroids [IKMZ19].Adiprasito, Huh, and Katz showed that the Chow ring of the Bergman fan of M satisfiesPoincaré duality, the hard Lefschetz theorem, and the Hodge–Riemann relations [AHK18]. Fur-thermore, they interpreted the entries of the f -vector of the reduced broken circuit complex of M – an invariant of the matroid generalizing the chromatic polynomial for graphs – as intersectionnumbers in the Chow ring of Σ M . The geometric interpretation then implied the log-concavityof the coefficients of the characteristic polynomial and the reduced characteristic polynomial χ M p q q – (cid:88) F P L p M q µ p ∅ , F q q corank p F q , χ M p q q – χ M p q q{p q ´ q , where µ is the Möbius function on the geometric lattice L p M q for a loopless matroid M . The conormal fan Σ M , M K is an alternative polyhedral model for M . Its construction uses thedual matroid M K , the matroid on E whose bases are the complements of bases of M . We refer to[Oxl11] for background on matroid duality and other general facts on matroids. A central roleis played by the addition map N E,E – N E ‘ N E ÝÑ N E , p z, w q ÞÝÑ z ` w. The function α j on N E pulls back to a function δ j on N E,E under the addition map. Explicitly, δ j p z, w q “ max i P E p z j ` w j ´ z i ´ w i q . The function δ j is piecewise linear on a fan that we construct, called the bipermutohedral fan Σ E,E .This is the normal fan of a convex polytope Π E,E that we call the bipermutohedron . The functions δ j for j in E are equal to each other modulo global linear functions on N E,E , and we write δ fortheir common equivalence class.The cotangent fan Ω E is the subfan of the bipermutohedral fan Σ E,E whose underlying set isthe tropical hypersurface trop p δ q “ (cid:110) p z, w q | min i P E { z i ` w i } is achieved at least twice (cid:111) Ď N E,E . We show the containment trop p M q ˆ trop p M K q Ď trop p δ q , and define the conormal fan Σ M , M K to be the subfan of the cotangent fan Ω E that subdivides theproduct trop p M q ˆ trop p M K q . For our purposes, it is necessary to work with the conormal fanof M instead of the product of the Bergman fans of M and M K , because the function δ j need notbe piecewise linear on the product of the Bergman fans. If M has a loop, by definition, the characteristic polynomial and the reduced characteristic polynomial of M are zero. AGRANGIAN GEOMETRY OF MATROIDS 5
The projections to the summands of N E,E define morphisms of fans π : Σ M , M K ÝÑ Σ M and π : Σ M , M K ÝÑ Σ M K . Thus, in addition to the functions δ j , the conormal fan of M supports the pullbacks of α j in M and α j in M K , which are the piecewise linear functions γ j p z, w q “ max i P E p z j ´ z i q and γ j p z, w q “ max i P E p w j ´ w i q . These define the equivalence classes γ and γ of functions on N E,E .The conormal fan is a tropical analog of the incidence variety appearing in the classical theoryof projective duality. For a subvariety X of a projective space P , the incidence variety I X is asubvariety of the product of P with the dual projective space P _ that projects onto X and its dual X _ . Over the smooth locus of X , the incidence variety I X is the total space of the projectivizedconormal bundle of X and, over the smooth locus of X _ , it is the total space of the projectivizedconormal bundle of X _ . We refer to [GKZ94] for a modern exposition of the theory of projectiveduality.We use the conormal fan of M to give a geometric interpretation of the polynomial χ M p q ` q ,whose coefficients form the h -vector of the broken circuit complex of M with alternating signs.In particular, we give a geometric formula for Crapo’s beta invariant β p M q – p´ q r χ M p q . This new tropical geometry is inspired by the Lagrangian geometry of conormal varieties inclassical algebraic geometry, as we now explain.Consider the category of complex algebraic varieties with proper morphisms. According toa conjecture of Deligne and Grothendieck, there is a unique natural transformation “ csm ” fromthe functor of constructible functions on complex algebraic varieties to the homology of complexalgebraic varieties such that, for any smooth variety X , csm p X q “ c p T X q X r X s “ p the total homology Chern class of the tangent bundle of X q . The conjecture was proved by MacPherson [Mac74], and it was recognized later in [BS81] thatthe class csm p X q , for possibly singular X , coincides with a class constructed earlier by Schwartz[Sch65]. For any constructible subset X of Y , the k -th Chern–Schwartz–MacPherson class of X in Y is the homology class csm k p X q P H k p Y q . Aiming to introduce a tropical analog of this theory, López de Medrano, Rincón, and Shawintroduced the Chern–Schwartz–MacPherson cycle of the Bergman fan of M in [LdMRS20]: The A morphism from a fan Σ in N “ R b N , Z to a fan Σ in N “ R b N , Z is an integral linear map from N to N such that the image of any cone in Σ is a subset of a cone in Σ . In the context of toric geometry, a morphism from Σ to Σ can be identified with a toric morphism from the toric variety of Σ to the toric variety of Σ [CLS11, Chapter 3]. Thus, to be precise, the conormal fan is a tropical analog of the projectivized conormal variety and the cotangent fan isa tropical analog of the projectivized cotangent space. We trust that the omission of the term “projectivized” will causeno confusion.
FEDERICO ARDILA, GRAHAM DENHAM, AND JUNE HUH k -th Chern–Schwartz–MacPherson cycle of M is the weighted fan csm k p M q supported on the k -dimensional skeleton of Σ M with the weights w p σ F q “ p´ q r ´ k β p M r F sq where σ F is the k -dimensional cone corresponding to a flag of flats F of M , M p i q is the minorof M corresponding to the i -th interval in F , and β p M r F sq – β p M p qq ¨ ¨ ¨ β p M p k ` qq is the betainvariant of the flag F in M . This weighted fan behaves well combinatorially and geometrically.First, the weights satisfy the balancing condition in tropical geometry [LdMRS20, Theorem 1.1],so that we may view the Chern–Schwartz–MacPherson cycle as a Minkowski weight csm k p M q P MW k p Σ M q . Second, when trop p M q is the tropicalization of the intersection P p V qXp C ˚ q E { C ˚ , the Minkowskiweight can be identified with the k -th Chern–Schwartz–MacPherson class of P p V q X p C ˚ q E { C ˚ in the toric variety of the permutohedron Π E [LdMRS20, Theorem 1.2]. Third, the Chern-Schwartz-MacPherson cycles of M satisfy a deletion-contraction formula, a matroid version ofthe inclusion-exclusion principle [LdMRS20, Proposition 5.2]. It follows that the degrees of theseMinkowski weights determine the reduced characteristic polynomial of M by the formula χ M p q ` q “ r (cid:88) k “ deg p csm k p M qq q k , where the degrees are taken with respect to the class α [LdMRS20, Theorem 1.4]. Fourth, theChern-Schwartz-MacPherson cycles of matroids can be used to define Chern classes of smoothtropical varieties. In codimension , the class agrees with the anticanonical divisor of a tropi-cal variety defined by Mikhalkin in [Mik06]. For smooth tropical surfaces, these classes agreewith the Chern classes of tropical surfaces introduced in [Car] and [Sha] to formulate Noether’sformula for tropical surfaces.Schwartz’s and MacPherson’s constructions of csm are rather subtle. Sabbah later observedthat the Chern-Schwartz-MacPherson classes can be interpreted more simply as “shadows” ofthe characteristic cycles in the cotangent bundle of X . Sabbah summarizes the situation in thefollowing quote from [Sab85]: la théorie des classes de Chern de [Mac74] se ramène à une théorie de Chow sur T ˚ X ,qui ne fait intervenir que des classes fondamentales. The functor of constructible functions is replaced with a functor of Lagrangian cycles of T ˚ X ,which are exactly the linear combinations of the conormal varieties of the subvarieties of X . Inthe Lagrangian framework, key operations on constructible functions become more geometric.Similarly, López de Medrano, Rincón, and Shaw’s original definition of the Chern–Schwartz–MacPherson cycles of a matroid M is somewhat intricate combinatorially. We prove that theyare “shadows” of much simpler cycles under the pushforward map π ˚ : MW k p Σ M , M K q ÝÑ MW k p Σ M q . AGRANGIAN GEOMETRY OF MATROIDS 7
See Section 3.1 for a review of basic tropical intersection theory.
Theorem 1.1.
When M has no loops and no coloops, for every nonnegative integer k ď r , csm k p M q “ p´ q r ´ k π ˚ p δ n ´ k ´ X M , M K q , where M , M K is the top-dimensional constant Minkowski weight on the conormal fan of M .It follows from Theorem 1.1 and the projection formula that the reduced characteristic poly-nomial of M can be expressed in terms of the intersection theory of the conormal fan as follows: Theorem 1.2.
When M has no loops and no coloops, we have χ M p q ` q “ r (cid:88) k “ p´ q r ´ k deg p γ k δ n ´ k ´ q q k , where the degrees are taken with respect to the top-dimensional constant Minkowski weight M , M K on the conormal fan.When M is representable over C , the third author gave an algebro-geometric version of The-orem 1.1 in [Huh13]. The complex geometric version of the identity boils down to the fact thatthe Chern–Schwartz–MacPherson class of a smooth variety X in its normal crossings compact-ification Y is the total Chern class of the logarithmic tangent bundle: csm p X q “ c p Ω Y p log Y ´ X q _ q X r Y s . In fact, the logarithmic formula can be used to construct the natural transformation csm [Alu06].For precursors of the logarithmic viewpoint, see [Alu99] and [GP02]. The current paper demon-strates that a similar geometry exists for arbitrary M in the tropical setting.1.3. Inequalities for matroid invariants.
Let a , a , . . . , a n be a sequence of nonnegative inte-gers, and let d be the largest index with nonzero a d . ‚ The sequence is said to be unimodal if a ď a ď ¨ ¨ ¨ ď a k ´ ď a k ě a k ` ě ¨ ¨ ¨ ě a n for some ď k ď n . ‚ The sequence is said to be log-concave if a k ´ a k ` ď a k for all ă k ă n . ‚ The sequence is said to be flawless if a k ď a d ´ k for all ď k ď d { .Many enumerative sequences are conjectured to have these properties, but proving them oftenturns out to be difficult. Combinatorialists have been interested in these conjectures because We say that M is representable over a field F if there exists a linear subspace V Ď F E such that S Ď E is independentin M if and only if the projection from V to F S is surjective. Almost all matroids are not representable over any field[Nel18]. FEDERICO ARDILA, GRAHAM DENHAM, AND JUNE HUH their solution typically requires a fundamentally new construction or connection with a distantfield, thus revealing hidden structural information about the objects in question. For surveys ofknown results and open problems, see [Bre94] and [Sta89, Sta00].A simplicial complex ∆ is a collection of subsets of a finite set, called faces of ∆ , that is down-ward closed. The face enumerator of ∆ and the shelling polynomial of ∆ are the polynomials f ∆ p q q “ (cid:88) S P ∆ q | S | “ (cid:88) k ě f k p ∆ q q k and h ∆ p q q “ f ∆ p q ´ q “ (cid:88) k ě h k p ∆ q q k . The f -vector of a simplicial complex is the sequence of coefficients of its face enumerator, and the h -vector of a simplicial complex is the sequence of coefficients of its shelling polynomial. When ∆ is shellable , the shelling polynomial of ∆ enumerates the facets used in shelling ∆ , and hencethe h -vector of ∆ is nonnegative.We study the f -vectors and h -vectors of the following shellable simplicial complexes associ-ated to M . For a gentle introduction, and for the proof of their shellability, see [Bjö92]. ‚ The independence complex IN p M q , the collection of subsets of E that are independent in M . ‚ The broken circuit complex BC p M q , the collection of subsets of E which do not contain anybroken circuit of M .Here a broken circuit is a subset obtained from a circuit of M by deleting the least element relativeto a fixed ordering of E . The notion was developed by Whitney [Whi32], Rota [Rot64], Wilf[Wil76], and Brylawski [Bry77], for the “chromatic” study of matroids. The f -vector and the h -vector of the broken circuit complex of M are determined by the characteristic polynomial of M , and in particular they do not depend on the chosen ordering of E : χ M p q q “ r ` (cid:88) k “ p´ q k f k p BC p M qq q r ´ k ` , χ M p q ` q “ r ` (cid:88) k “ p´ q k h k p BC p M qq q r ´ k ` . Conjecture 1.3.
The following holds for any matroid M .(1) The f -vector of IN p M q is unimodal, log-concave, and flawless.(2) The h -vector of IN p M q is unimodal, log-concave, and flawless.(3) The f -vector of BC p M q is unimodal, log-concave, and flawless.(4) The h -vector of BC p M q is unimodal, log-concave, and flawless.Welsh [Wel71] and Mason [Mas72] conjectured the log-concavity of the f -vector of the inde-pendence complex. Dawson conjectured the log-concavity of the h -vector of the independence An r -dimensional pure simplicial complex is said to be shellable if there is an ordering of its facets such that each facetintersects the simplicial complex generated by its predecessors in a pure p r ´ q -dimensional complex. In [Mas72], Mason proposed a stronger conjecture that the f -vector of the independence complex of M satisfies f k (cid:0) n ` k (cid:1) ě f k ´ (cid:0) n ` k ´ (cid:1) f k ` (cid:0) n ` k ` (cid:1) for all k . AGRANGIAN GEOMETRY OF MATROIDS 9 complex in [Daw84], and independently, Colbourn conjectured the same in [Col87] in the con-text of network reliability. Hibi conjectured that the h -vector of the independence complex mustbe flawless [Hib92]. The unimodality and the log-concavity conjectures for the f -vector of thebroken circuit complex are due to Heron [Her72], Rota [Rot71], and Welsh [Wel76]. The sameconjectures for the chromatic polynomials of graphs were given earlier by Read [Rea68] andHoggar [Hog74]. We refer to [Whi87, Chapter 8] and [Oxl11, Chapter 15] for overviews andhistorical accounts. Brylawski [Bry82] conjectured the log-concavity of the h -vector of the bro-ken circuit complex. That the h -vector of the broken circuit complex is flawless stated as anopen problem in [Swa03] and reproduced in [JKL18] as a conjecture. We deduce all the abovestatements using the geometry of conormal fans. Theorem 1.4.
Conjecture 1.3 holds.We prove the log-concavity of the h -vector of the broken circuit complex using Theorem 1.1.This log-concavity implies all other statements in Conjecture 1.3, thanks to the following knownobservations: ‚ For any simplicial complex ∆ , the log-concavity of the h -vector implies the log-concavity ofthe f -vector [Bre94, Corollary 8.4]. ‚ For any pure simplicial complex ∆ , the f -vector of ∆ is flawless. More generally, any pureO-sequence is flawless [Hib89, Theorem 1.1]. ‚ For any shellable simplicial complex ∆ , the h -vector of ∆ has no internal zeros, being anO-sequence [Sta77, Theorem 6]. Therefore, if the h -vector of ∆ is log-concave, then it is uni-modal. ‚ The broken circuit complex of M is the cone over the reduced broken circuit complex of M , andthe two simplicial complexes share the same h -vector. The independence complex of M isthe reduced broken circuit complex of another matroid, the free dual extension of M [Bry77,Theorem 4.2]. ‚ If the h -vector of the broken circuit complex of M is unimodal for all M , then the h -vector ofthe broken circuit complex of M is flawless for all M [JKL18, Theorem 1.2]. Previous work.
The log-concavity of the f -vector of the broken circuit complex was proved in[Huh12] for matroids representable over a field of characteristic . The result was extended to In [Bry82], Brylawski conjectures the same set of inequalities for the f -vector of the broken circuit complex of M . Ma-son’s stronger conjecture was recently proved in [ALOGV] and [BHa, BHb]. An extension of the same result to matroidquotients was obtained in [EH]. In [Bry82], Brylawski proposed a stronger conjecture that the h -vector of the broken circuit complex of M satisfies h k (cid:0) n ´ kn ´ r ´ (cid:1) ě h k ´ (cid:0) n ´ k ` n ´ r ´ (cid:1) h k ` (cid:0) n ´ k ´ n ´ r ´ (cid:1) for all k . A sequence of nonnegative integers h , h , . . . is an O-sequence if there is an order ideal of monomials O such that h k is the number of degree k monomials in O . The sequence is a pure O-sequence if the order ideal O can be chosen so that allthe maximal monomials in O have the same degree. See [BMMR `
12] for a comprehensive survey of pure O-sequences. matroids representable over some field in [HK12] and to all matroids in [AHK18]. An alternativeproof of the same fact using the volume polynomial of a matroid was obtained in [BES]. It wasobserved in [Len13] that the log-concavity of the f -vector of the broken circuit complex impliesthat of the independence complex.For matroids representable over a field of characteristic , the log-concavity of the h -vectorof the broken circuit complex was proved in [Huh15]. The algebraic geometry behind the log-concavity of the h -vector, which became a model for the Lagrangian geometry of conormal fansin the present paper, was explored in [DGS12] and [Huh13]. In [JKL18], Juhnke-Kubitzke and Leused the result of [Huh15] to deduce that the h -vector of the broken circuit complex is flawlessfor matroids representable over a field of characteristic . The flawlessness of the h -vector ofthe independence complex was first proved by Chari using a combinatorial decomposition ofthe independence complex [Cha97]. The result was recovered by Swartz [Swa03] and Hausel[Hau05], who obtained stronger algebraic results. The other cases of Conjecture 1.3 remainedopen.Our solution of Conjecture 1.3 was announced in [Ard18]. Very recently, Berget, Spink, andTseng [BST] have announced an alternative proof of the log-concavity of the h -vector of theindependence complex (Dawson’s Conjecture 1.3.2). The relationship between our approachand theirs is still to be understood. The h -vector of the broken circuit complex (Brylawski’sConjecture 1.3.4) is not currently accessible through their methods.1.4. Tropical Hodge theory.
Let us discuss in more detail the strategy of [AHK18] that led tothe log-concavity of the f -vector of the broken circuit complex of M . For the moment, supposethat there is a linear subspace V Ď C E representing M over C , and consider the variety Y V “ the closure of P p V q X p C ˚ q E { C ˚ in the toric variety of the permutohedron X p Σ E q .If nonempty, Y V is an r -dimensional smooth projective complex variety which is, in fact, con-tained in the torus invariant open subset of X p Σ E q corresponding to the Bergman fan of M : Y V Ď X p Σ M q Ď X p Σ E q . The work of Feichtner and Yuzvinsky [FY04], which builds up on the work of De Concini andProcesi [DCP95], reveals that the inclusion maps induce isomorphisms between integral coho-mology and Chow rings: H ‚ p Y V , Z q » A ‚ p Y V , Z q » A ‚ p X p Σ M q , Z q . As a result, the Chow ring of the n -dimensional variety X p Σ M q has the structure of the evenpart of the cohomology ring of an r -dimensional smooth projective variety. Remarkably, thisstructure on the Chow ring of X p Σ M q persists for any matroid M , even if M does not admitany representation over any field. In particular, the Chow ring of X p Σ M q satisfies the Poincaré Throughout the paper, the toric variety of a fan in N E refers to the one constructed with respect to the lattice Z E { Z .Similarly, the toric variety of a fan in N E,E refers to the one constructed with respect to the lattice Z E { Z ‘ Z E { Z . AGRANGIAN GEOMETRY OF MATROIDS 11 duality, the hard Lefschetz theorem, and the Hodge–Riemann relations [AHK18]. For a sim-pler proof of the three properties of the Chow ring, based on its semi-small decomposition, see[BHM ` ].For a simplicial fan Σ , let A p Σ q be the ring of real-valued piecewise polynomial functionson Σ modulo the ideal of the linear functions on Σ , and let K p Σ q be the cone of strictly convex piecewise linear functions on Σ . We formalize the above properties of the Bergman fan of M asfollows. Definition 1.5. A d -dimensional simplicial fan Σ is Lefschetz if it satisfies the following.(1) (Fundamental weight) The group of d -dimensional Minkowski weights on Σ is generatedby a positive Minkowski weight w . We write deg for the corresponding linear isomorphism deg : A d p Σ q ÝÑ R , η ÞÝÑ η X w. (2) (Poincaré duality) For any ď k ď d , the bilinear map of the multiplication A k p Σ q ˆ A d ´ k p Σ q A d p Σ q R deg is nondegenerate.(3) (Hard Lefschetz property) For any ď k ď d and any (cid:96) P K p Σ q , the multiplication map A k p Σ q Ñ A d ´ k p Σ q , η ÞÝÑ (cid:96) d ´ k η is a linear isomorphism.(4) (Hodge–Riemann relations) For any ď k ď d and any (cid:96) P K p Σ q , the bilinear form A k p Σ q ˆ A k p Σ q ÞÝÑ R , p η , η q ÞÝÑ p´ q k deg p (cid:96) d ´ k η η q is positive definite when restricted to the kernel of the multiplication map (cid:96) d ´ k ` .(5) (Hereditary property) For any ă k ď d and any k -dimensional cone σ in Σ , the star of σ in Σ is a Lefschetz fan of dimension d ´ k .The Hodge–Riemann relations give analogs of the Alexandrov–Fenchel inequality amongstdegrees of products of convex piecewise linear functions (cid:96) , (cid:96) , . . . , (cid:96) d on Σ : deg p (cid:96) (cid:96) (cid:96) ¨ ¨ ¨ (cid:96) d q ě deg p (cid:96) (cid:96) (cid:96) ¨ ¨ ¨ (cid:96) d q deg p (cid:96) (cid:96) (cid:96) ¨ ¨ ¨ (cid:96) d q . The Bergman fan of a matroid M is Lefschetz, and the log-concavity of the f -vector of the bro-ken circuit complex of M follows from the Hodge–Riemann relations for the Bergman fan of M [AHK18].We establish the log-concavity of the h -vector of the broken circuit complex of M in the sameway, using the conormal fan of M in place of the Bergman fan of M . Theorem 1.2 relates theintersection theory of the conormal fan of M to the h -vector of the broken circuit complex of M via the Chern-Schwartz-MacPherson cycles of M . In order to proceed, we need to show that theconormal fan of M is Lefschetz. We obtain this from the following general result. Theorem 1.6.
Let Σ and Σ be simplicial fans that have the same support | Σ | “ | Σ | . If K p Σ q and K p Σ q are nonempty, then Σ is Lefschetz if and only if Σ is Lefschetz.Theorem 1.6 implies, for example, that the reduced normal fan of any simple polytope isLefschetz, because the reduced normal fan of a simplex is Lefschetz. In the context of matroidtheory, Theorem 1.6 implies that the conormal fan of M is Lefschetz, because the Bergman fansof M and M K are Lefschetz and the product of Lefschetz fans is Lefschetz. Acknowledgments.
The first author thanks the Mathematical Sciences Research Institute, theSimons Institute for the Theory of Computing, the Sorbonne Université, the Università di Bolo-gna, and the Universidad de Los Andes for providing wonderful settings to work on this project,and Laura Escobar, Felipe Rincón, and Kristin Shaw for valuable conversations; his researchis supported by NSF grant DMS-1855610 and Simons Fellowship 613384. The second authorthanks the University of Sydney School of Mathematics and Statistics for hospitality during anearly part of this project; his research is supported by NSERC of Canada. The third authorthanks Karim Adiprasito for helpful conversations; his research is supported by NSF GrantDMS-1638352 and the Ellentuck Fund.2. T
HE BIPERMUTOHEDRAL FAN
Let E be a finite set of cardinality n ` . For notational convenience, we often identify E withthe set of nonnegative integers at most n . As before, we let N E be the n -dimensional space N E “ R E { span p e E q , e E “ (cid:88) i P E e i . Let N E,E be the n -dimensional space N E ‘ N E , and let µ be the addition map µ : N E,E ÝÑ N E , p z, w q ÞÝÑ z ` w. Throughout the paper, all fans in N E will be rational with respect to the lattice Z E { Z e E , and allfans in N E,E will be rational with respect to the lattice Z E { Z e E ‘ Z E { Z e E . We follow [CLS11]when using the terms fan and generalized fan : A generalized fan is a fan if and only if each of itscone is strongly convex. The notion of morphism of fans is extended to morphism of generalizedfans in the obvious way. For any subset S of E , we write e S and f S for the vectors e S “ (cid:88) i P S e i , f S “ (cid:88) i P S f i , where e i are the standard basis vectors of R E defining the first summand of N E,E and f i are thestandard basis vectors of R E defining the second summand of N E,E .In this section, we construct a complete simplicial fan Σ E,E in N E,E that will play a centralrole in this paper. We offer five equivalent descriptions; each one of them will play an important McMullen gave an elementary proof of this fact in [McM93]. See [Tim99] and [FK10] for alternative presentations. Ourproof of Theorem 1.6 is modeled on these arguments. Theorem 1.6 gives another proof of the necessity of McMullen’sbounds [McM93] on the face numbers of simplicial polytopes.
AGRANGIAN GEOMETRY OF MATROIDS 13 role for us. We call it the bipermutohedral fan because it is the normal fan of a polytope which wecall the bipermutohedron . Before we begin defining the bipermutohedral fan Σ E,E in N E,E , werecall some basic facts on the permutohedral fan Σ E in N E .2.1. The normal fan of the simplex.
Consider the standard n -dimensional simplex conv { e i } i P E Ď R E . Its normal fan in R E has the lineality space spanned by e E . For any convex polytope, we callthe quotient of the normal fan by its lineality space the reduced normal fan of the polytope. Forexample, the reduced normal fan of the standard simplex, denoted Γ E , is the complete fan in N E with the cones σ S – cone { e i } i P S Ď N E , for every proper subset S of E .The cone σ S consists of the points z P N E such that min i P E z i “ z s for all s not in S . For each element j of E , the function α j “ max i P E { z j ´ z i } is piecewise linear on the fan Γ E . These piecewise linearfunctions are equal to each other modulo global linear functions on N E , and we write α for thecommon equivalence class of α j .2.2. The normal fan of the permutohedron.
Let Π E be the n -dimensional permutohedron conv (cid:110) p x , x , . . . , x n q | x , x , . . . , x n is a permutation of , , . . . , n (cid:111) Ď R E . The permutohedral fan Σ E , also known as the braid fan or the type A Coxeter complex , is the reducednormal fan of the permutohedron Π E . It is the complete simplicial fan in N E whose chambersare separated by the n -dimensional braid arrangement , the real hyperplane arrangement in N E consisting of the (cid:0) n ` (cid:1) hyperplanes z i “ z j , for distinct elements i and j of E .The face of the permutohedral fan containing a given point z in its relative interior is determinedby the relative order of its homogeneous coordinates p z , . . . , z n q . Therefore, the faces of thepermutohedral fan correspond to the ordered set partitions P “ p E “ P \ ¨ ¨ ¨ \ P k ` q , which are in bijection with the strictly increasing sequences of nonempty proper subsets S “ p ∅ Ĺ S Ĺ ¨ ¨ ¨ Ĺ S k Ĺ E q , S m “ m (cid:91) (cid:96) “ P (cid:96) . The collection of ordered set partitions of E form a poset under adjacent refinement , where P ď P if P can be obtained from P by merging adjacent parts. The normal fan of a convex polytope P in a vector space is a generalized fan in the dual space whose face poset isanti-isomorphic to the face poset of P . Unlike the reduced normal fan, the normal fan of a polytope is a generalized fan,and need not be a fan. We trust that the use of the term “normal fan” will cause no confusion. Proposition 2.1.
The face poset of the permutohedral fan Σ E is isomorphic to the poset of or-dered set partitions of E .Thus the permutohedral fan has p n ´ q rays corresponding to the nonempty proper subsetsof E and p n ` q ! chambers corresponding to the permutations of E .We now describe the permutohedral fan in terms of its rays. Two subsets S and S of E aresaid to be comparable if S Ď S or S Ě S . A flag in E is a set of pairwise comparable subsets of E . For any flag S of subsets of E , we define σ S “ cone { e S } S P S Ď N E . We identify a flag in E with the strictly increasing sequence obtained by ordering the subsets inthe flag. Proposition 2.2.
The permutohedral fan Σ E is the complete fan in N E with the cones σ S “ cone { e S } S P S , where S is a flag of nonempty proper subsets of E .For example, the cone corresponding to the ordered set partition | | is cone p e , e q “ { z P N E | z “ z ě z “ z “ z ě z } . Proposition 2.2 shows that the permutohedral fan is a unimodular fan : The set of primitive raygenerators in any cone in Σ E is a subset of a basis of the free abelian group Z E { Z . It also showsthat the permutohedral fan is a refinement of the fan Γ E in Section 2.1.It will be useful to view the permutohedral fan as a configuration space as follows. Regard N E as the space of E -tuples of points p p , . . . , p n q moving in the real line, modulo simultaneoustranslation: p “ p p , . . . , p n q “ p p ` λ, . . . , p n ` λ q for any λ P R .The ordered set partition of p , denoted π p p q , is obtained by reading the labels of the points in thereal line from right to left, as shown in Figure 1. This model gives the permutohedral fan Σ E the following geometric interpretation.
569 7 1 04 28 3
ÞÝÑ | | | | | F IGURE
1. An E -tuple of points p and its ordered set partition π p p q “ | | | | | . Proposition 2.3.
The permutohedral fan Σ E is the configuration space of E -tuples of points inthe real line modulo simultaneous translation, stratified according to their ordered set partition.In Section 2.4, we give an analogous description of the bipermutohedral fan Σ E,E as a con-figuration space of E -tuples of points in the real plane. AGRANGIAN GEOMETRY OF MATROIDS 15
The bipermutohedral fan as a subdivision.
Denote a point in N E,E by p z, w q . We constructthe bipermutohedral fan Σ E,E in N E,E as follows.First, we subdivide N E,E into the charts C , C , . . . , C n , where C k is the cone C k “ (cid:110) p z, w q | min i P E p z i ` w i q “ z k ` w k (cid:111) . These form the chambers of a complete generalized fan in N E,E , denoted ∆ E . The chamber C k is the inverse image of the cone σ E ´ k under the addition map, and hence ∆ E is the coarsestcomplete generalized fan in N E,E for which the addition map is a morphism to the fan Γ E inSection 2.1. To each chart C k , we associate the linear functions Z i “ z i ´ z k , W i “ ´ w i ` w k , for every i in E .Omitting the zero function Z k “ W k , we obtain a coordinate system p Z, W q for N E,E such that C k “ (cid:110) p Z, W q | Z i ě W i for every i in E (cid:111) . This coordinate system depends on k , but we will drop k from the notation for better readability.Second, we consider the subdivision Σ k of the cone C k obtained from the braid arrangementof (cid:0) n ` (cid:1) hyperplanes Z a “ Z b , W a “ W b , Z a “ W b , for all a and b in E .Note that the arrangement contains the n hyperplanes that cut out C k in N E,E . One may viewthe subdivision Σ k of C k as a copy of { n -th of the n -dimensional permutohedral fan. Proposition 2.4.
The union of the fans Σ i for i P E is a fan in N E,E . We call it the bipermutohedralfan Σ E,E . Proof.
To check that Σ E,E is indeed a fan, we need to check that the fans Σ i glue compatiblyalong the boundaries of C i . For this, we verify that Σ i and Σ j induce the same subdivision on C i X C j for all i ‰ j .Consider the system of linear functions p Z, W q for C i and the system of linear functions p Z , W q for C j . It is straightforward to check that, for any point in N E,E , we have Z a ´ Z b “ Z a ´ Z b and W a ´ W b “ W a ´ W b for all a and b in E .Furthermore, on the intersection of C i and C j , where z i ` w i “ z j ` w j , we have Z a ´ W b “ p z a ´ z i q ´ p w i ´ w b q “ p z a ´ z j q ´ p w j ´ w b q “ Z a ´ W b . Thus the hyperplanes separating the chambers of Σ i and Σ j have the same intersections with C i X C j . (cid:3) The following subfan of the bipermutohedral fan will serve as a guide toward Theorem 1.1.
Definition 2.5.
The cotangent fan Ω E is the union of the fans Σ i X Σ j for i ‰ j P E . In other words, Ω E is the subfan of Σ E,E whose support is the tropical hypersurface trop p δ q “ (cid:110) p z, w q | min i P E p z i ` w i q is achieved at least twice (cid:111) Ď N E,E . In Section 3.4, we show that the cotangent fan contains the conormal fan of any matroid on E .2.4. The bipermutohedral fan as a configuration space.
It will be useful to view the bipermu-tohedral fan Σ E,E as a configuration space as follows. Regard N E,E as the space of E -tuples ofpoints p p , . . . , p n q moving in the real plane, modulo simultaneous translation: p p , . . . , p n q “ p p ` λ, . . . , p n ` λ q for any λ P R .The point p z, w q in N E,E corresponds to the points p i “ p z i , w i q in R for i in E . Definition 2.6. A bisequence on E is a sequence B of nonempty subsets of E , called the parts of B , such that(1) every element of E appears in at least one part of B ,(2) every element of E appears in at most two parts of B , and(3) some element of E appears in exactly one part of B .The trivial bisequence on E is the bisequence with exactly one part E . A bisubset of E is a nontriv-ial bisequence on E of minimal length . A bipermutation of E is a bisequence on E of maximallength n ` .We will write bisequences by listing the elements of its parts, separated by vertical bars. Forexample, the bisequence { } , { , } , { } , { } on { , , } will be written | | | . Definition 2.7.
Let p “ p p , . . . , p n q be an E -tuple of points in R .(1) The supporting line of p , denoted (cid:96) p p q , is the lowest line of slope ´ containing a point in p .(2) For each point p i , the vertical and horizontal projections of p i onto (cid:96) p p q will be labelled i .(3) The bisequence of p , denoted B p p q , is obtained by reading the labels on (cid:96) p p q from right to left.See Figure 2 for an illustration of Definition 2.7. Remark . One can recover any configuration p from their projections onto the supporting line (cid:96) p p q and their labels. Therefore, modulo translations, we may also consider p as a configurationof n ` points on the real line labeled , , , , . . . , n, n such that at least one pair of points withthe same label coincide. This is illustrated at the bottom of Figure 2.This model gives the bipermutohedral fan Σ E,E the following geometric interpretation.
Proposition 2.9.
The bipermutohedral fan Σ E,E is the configuration space of E -tuples of pointsin the real plane modulo simultaneous translation, stratified according to their bisequence. AGRANGIAN GEOMETRY OF MATROIDS 17 p p (cid:96) p p q p p p p ÞÝÑ | | | | |
00 24 1 035 2 34 F IGURE
2. An E -tuple of points p “ p p , . . . , p q in the plane, their vertical andhorizontal projections onto the supporting line (cid:96) p p q , and the bisequence B p p q . Proof.
Consider a point p z, w q in N E,E and the associated configuration of points p i in the plane.The chart C k consists of configurations p where k appears exactly once in the bisequence B p p q .In other words, p is in C k if and only if p k is on the supporting line (cid:96) p p q . We consider the systemof linear functions p Z, W q for C k discussed in Section 2.3. The cones in the subdivision Σ k of C k encode the relative order of Z , . . . , Z n , W , . . . , W n , where Z k “ W k “ and Z i ě W i for every i in E .On the other hand, the bisequence B p p q keeps track of the relative order of the vertical andhorizontal projections of p i onto (cid:96) p p q . As shown in Figure 3, after the translation by p´ z k , ´ w k q ,the vertical and horizontal projections of p i onto (cid:96) p p q are p z i , z k ` w k ´ z i q ´ p z k , w k q “ p Z i , ´ Z i q and p z k ` w k ´ w i , w i q ´ p z k , w k q “ p W i , ´ W i q . Their relative order along (cid:96) p p q is given by the relative order of Z , . . . , Z n , W , . . . , W n . (cid:3) p W i , ´ W i q “ p w k ´ w i , w i ´ w k q p k p Z i , ´ Z i q “ p z i ´ z k , z k ´ z i q (cid:96) p p q p i F IGURE
3. The vertical and horizontal projections of p i onto the supporting line (cid:96) p p q , after the translation by p´ z k , ´ w k q . The collection of bisequences on E form a poset under adjacent refinement , where B ď B if B can be obtained from B by merging adjacent parts. The poset of bisequences on E is a gradedposet. Its k -th level consists of the bisequences of k ` nonempty subsets of E , and the top levelconsists of the bipermutations of E . Proposition 2.10.
The face poset of the bipermutohedral fan Σ E,E is isomorphic to the poset ofbisequences on E . Proof.
Remark 2.8 shows that, given any bisequence B on E , there is a configuration p with B p p q “ B . Thus, by Proposition 2.9, the cones in Σ E,E are in bijection with the bisequenceson E . If a configuration p moves into more special position, then some adjacent parts of B p p q merge. (cid:3) For a bisequence B on E , we write σ B for the corresponding cone defined by σ B “ closure (cid:110) configurations p satisfying B p p q “ B (cid:111) Ď N E,E . In terms of the cones σ B , the fan Σ i subdividing the chart C i can be described as the subfan Σ i “ { σ B | i appears exactly once in the bisequence B } Ď Σ E,E . See Figure 4 for an illustration of Proposition 2.10 when n “ . | | | | | | | | | | | | | | | |
01 1 | | µ Σ Σ | | F IGURE
4. The map µ : Σ { , } , { , } Ñ Σ { , } from the bipermutohedral fan tothe permutohedral fan, and the labelling of their cones with bisequences on { , } and ordered set partitions on { , } , respectively.2.5. The bipermutohedral fan as a common refinement.
The importance of the bipermutohe-dral fan Σ E,E stems from its relationship with the normal fan Γ E of the standard simplex andthe permutohedral fan Σ E described in Sections in Section 2.1 and 2.2. Recall that a morphism from a fan Σ in N to a fan Σ in N is an integral linear map from N to N that maps any conein Σ into a cone in Σ . Proposition 2.11.
The bipermutohedral fan Σ E,E has the following properties.
AGRANGIAN GEOMETRY OF MATROIDS 19 (1) The projections π p z, w q “ z and π p z, w q “ w are morphisms of fans from Σ E,E to Σ E .(2) The addition map µ p z, w q “ z ` w is a morphism of fans from Σ E,E to Γ E . Proof.
That Σ E,E has the stated properties follows from the interpretation of Σ E and Σ E,E asconfiguration spaces, as we now explain. Suppose p z, w q is a point in N E,E and p is the corre-sponding E -tuple of points in R modulo simultaneous translation, with corresponding bise-quence B p p q . Then the smallest cone of Γ E containing z ` w is given by the entries that appeartwice in B p p q . The ordered set partition of z in N E is given by the first occurrence of each i in B p p q . Similarly, the ordered set partition of w in N E is given by the order of the last occurrenceof each i in B p p q . For example, if a point p z, w q has the bisequence | | | | | , as in Figure2, then the sum z ` w is in the cone of in Γ E , the first projection z is in the cone of | | | in Σ E , and the second projection w is in the cone of | | | in Σ E . (cid:3) The bipermutohedral fan in terms of its rays and cones.
The rays of the bipermutohedralfan Σ E,E correspond to the bisubsets of E . In other words, the rays of Σ E,E correspond to theordered pairs of nonempty subsets S | T of E such that S Y T “ E and S X T ‰ E. Proposition 2.12.
The p n ´ q rays of the bipermutohedral fan Σ E,E are generated by e S | T – e S ` f T , where S | T is a bisubset of E . Proof.
The configuration p corresponding to e S | T has points with labels in S X T located at p , q ,the points with labels in S ´ T located at p , q , and the points with labels in T ´ S located at p , q . The bisequence of p is indeed S | T , and hence the conclusion follows from Proposition2.9. (cid:3) Proposition 2.13.
The bipermutohedral fan Σ E,E has p n ` q ! { n ` chambers. Proof.
By Proposition 2.10, the chambers correspond to the bipermutations. These are obtainedbijectively from the p n ` q ! { n ` permutations of the multiset { , , . . . , n, n } by dropping thelast letter in the one-line notation for permutations. For example, the bipermutation | | | | | | correspond to the permutation of { , , , , , , , } . (cid:3) It is worth understanding Proposition 2.13 in a different way. Recall that the bipermutohedralfan is obtained by gluing copies of { n -th of the n -dimensional permutohedral fan. Thereare p n ` q such copies, and each copy contains p n ` q ! { n chambers, producing the totalof p n ` q ! { n ` chambers. This viewpoint explains why Figure 4 deceivingly looks like apermutohedral fan: For n “ , the bipermutohedral fan consists of two glued copies of half ofthe permutohedral fan.We now describe the cones in the bipermutohedral fan in terms of their generating rays. Let B “ B | B | ¨ ¨ ¨ | B k be a bisequence on E . Propositions 2.10 and 2.12 show that the rays of the k -dimensional cone σ B are generated by the vectors e S | T , . . . , e S k | T k , where S i “ i ´ (cid:91) j “ B j and T i “ k (cid:91) j “ i B j . See Figure 5 for an illustration. We use the following table to record the rays of σ B : ∅ Ĺ S Ď S Ď ¨ ¨ ¨ Ď S k Ď EE Ě T Ě T Ě ¨ ¨ ¨ Ě T k Ľ ∅ For each index j such that S j Ĺ S j ` and T j Ľ T j ` , we mark those two strict inclusions in blue.We write S p B q| T p B q for the collection of bisubsets S i | T i constructed from B as above by mergingadjacent parts. For convenience, we also refer to the pairs S | T “ ∅ | E and S k ` | T k ` “ E | ∅ . B “ | | | |
00 102012 | |
01 01 0202 |
01 01 22 | ∅ Ĺ Ď Ď Ď Ď EE Ě Ě Ě Ě Ľ ∅ F IGURE
5. The cone of | | | | has the rays generated by e | , e | , e | , e | .Conversely, we may ask which subsets of k rays in Σ E,E generate a k -dimensional cone in Σ E,E . To answer this question, we introduce the notion of a flag of bisubsets.
Definition 2.14.
We say that two bisubsets S | T and S | T of E are comparable if( S Ď S and T Ě T ) or ( S Ě S and T Ď T ) . A flag of bisubsets in E , or a biflag in E , is a set S | T of pairwise comparable bisubsets of E satisfy-ing (cid:91) S | T P S | T S X T ‰ E. The length of a biflag is the number of bisubsets in it.We have the following useful alternative characterization of biflags in E . Proposition 2.15.
Let S be an increasing sequence of k nonempty subsets of E , say S “ p ∅ Ĺ S Ď ¨ ¨ ¨ Ď S k Ď E q , AGRANGIAN GEOMETRY OF MATROIDS 21 and let T be a decreasing sequence of k nonempty subsets of E , say T “ p E Ě T Ě ¨ ¨ ¨ Ě T k Ľ ∅ q . Then the set S | T consisting of the pairs S | T , . . . , S k | T k is a flag of bisubsets if and only if S j Y T j “ E for every ď j ď k and S j Y T j ` ‰ E for some ď j ď k. Proof. If S | T is a biflag in E , then each S j | T j is a bisubset of E , and hence S j Y T j “ E for all j .Now let e be an element not in the union of all S j X T j , and consider the largest index i for which e R S i . Then e P S i ` , which implies e R T i ` by the definition of e . Therefore, S i Y T i ` ‰ E .Conversely, if S and T satisfy the stated conditions, then the pairs S j | T j form a set of pairwisecomparable bisubsets of E . If e is an element not in S j Y T j ` for some index j , then e is not in S k for all indices k ď j and e is not in T k for all indices k ą j . Therefore, e is not in the union ofall S k X T k , as desired. (cid:3) Note that S j Y T j ` ‰ E implies that S j Ĺ S j ` and T j Ľ T j ` , so the table of any biflag hasat least one pair of strict inclusions marked in blue.For a biflag S | T of length k , we write S for the increasing sequence of k nonempty subsets S “ p ∅ Ĺ S Ď ¨ ¨ ¨ Ď S k Ď E q , where S j are the first parts of the bisubsets in S | T , and write T for the decreasing sequence of k nonempty subsets T “ p E Ě T Ě ¨ ¨ ¨ Ě T k Ľ ∅ q , where T j are the second parts of the bisubsets in S | T . We use S and T to define B p S | T q as the sequence of k ` nonempty sets B | B | ¨ ¨ ¨ | B k , where B j “ p S j ` ´ S j q Y p T j ´ T j ` q . The above construction is an isomorphism between the poset of bisequences under adjacentrefinement and the poset of biflags under inclusion.
Proposition 2.16.
The bisequences on E are in bijection with the biflags in E . More precisely,(1) if B is a bisequence on E , then S p B q| T p B q is a biflag in E ,(2) if S | T is a biflag in E , then B p S | T q is a bisequence on E , and(3) the constructions S p B q| T p B q and B p S | T q are inverses to each other.Note that a bisubset S | T corresponds to the biflag { S | T } under the above bijection. Forsimplicity, we use the two symbols interchangeably. Proof. (1) Since every element of E appears at least once in the bisequence B , the increasing flag S p B q and the decreasing flag T p B q satisfy S j Y T j “ E for all j . In addition, since some elementof E appears exactly once in B , say in B j , we have S j Y T j ` ‰ E for some j . Therefore, byProposition 2.15, the pair S p B q| T p B q is a biflag in E . (2) Conversely, suppose that S | T is a biflag in E . Since S | T , . . . , S k | T k are pairwise distinct, B j must be nonempty for all j . Clearly, every element in E must appear in B j for some j . Inaddition, each element e in E can occur at most twice in B p S | T q , namely, in the parts B a and B b whose indices satisfy e P S a ` ´ S a and e P T b ´ T b ` . Furthermore, by Proposition 2.15, thereis an element e not in S c Y T c ` for some index c , and in this case we must have a “ b “ c . Thatelement e can occur only in the part B a of B p S | T q , and hence B p S | T q is indeed a bisequence.(3) It is straightforward to check that the constructions S p B q| T p B q and B p S | T q are inverses toeach other. (cid:3) We identify a biflag S | T in E with the sequence of bisubsets of E obtained by ordering thebisubsets in S | T as above. For any sequence S | T of bisubsets of E , we define σ S | T “ cone { e S | T } S | T P S | T Ď N E,E . Thus, for any bisequence B on E , we have σ B “ σ S p B q| T p B q . Corollary 2.17.
The bipermutohedral fan Σ E,E is the complete fan in N E,E with the cones σ S | T “ cone { e S | T } S | T P S | T , where S | T is a flag of bisubsets of E . Proof.
The statement is straightforward, given Propositions 2.10 and 2.16. (cid:3)
Corollary 2.17 can be used to show that the bipermutohedral fan is a unimodular fan. Proposition 2.18.
The set of primitive ray generators of any chamber of Σ E,E is a basis of thefree abelian group Z E { Z e E ‘ Z E { Z f E . Proof.
Let S “ S p B q and T “ T p B q for a bipermutation B of E . If is the unique element of E that appears exactly once in B , then (cid:110) e S j ` | T j ` ´ e S j | T j | is contained in S j Y T j ` (cid:111) “ (cid:110) e , . . . , e n , f , . . . , f n (cid:111) . Therefore, the set of n primitive ray generators of σ B generates Z E { Z e E ‘ Z E { Z f E . (cid:3) The bipermutohedral fan as the normal fan of the bipermutohedron.
In this section weconstruct a polytope Π E,E , called the bipermutohedron , whose reduced normal fan is Σ E,E . Webegin by identifying each permutation of the multiset E Y E : “ { , , , , . . . , n.n } , written as aword, with a bijection π : E Y E ÝÑ { ´p n ` q , ´p n ´ q , . . . , ´ , ´ , , , . . . , p n ´ q , p n ` q } Alternatively, one may appeal to the unimodularity of the n -dimensional braid arrangement fan in p Z, W q -coordinates discussed in Section 2.3. AGRANGIAN GEOMETRY OF MATROIDS 23 that sends the letters of the word to ´p n ` q , . . . , ´ , , . . . , p n ` q in increasing order. Forexample, the permutation gives rise to the following bijection π : ÞÝÑ π “ (cid:32) ´ ´ ´ ´ (cid:33) . To the bijection π we associate a vector u π “ p x, y q P R E ˆ R E with coordinates x i “ π p i q and y i “ ´ π p i q for i P E . Notice that u π is on the hyperplane (cid:80) i P E x i ´ (cid:80) i P E y i “ , so we maydefine s π “ (cid:80) i P E x i “ (cid:80) i P E y i . Writing vectors p x, y q P R E ˆ R E in a ˆ E table whose top andbottom rows are x and y respectively, we have, for example, u “ ´ ´ ´ ´ ´ ´ , s π “ ´ . Now, for each bipermutation B on E with only one occurrence of k P E , let π p B q be the permu-tation of E Y E obtained by replacing the first and second occurrences of each i ‰ k with i and i respectively, and replacing k with kk . Then define v B “ u π p B q ´ s π p B q p e k ` f k q , where e k and f k are the k th unit vectors in the first and second copies of R E . For example, v | | | | | | “ u ´ s p e ` f q“ ´ ´ ´ ´ ´ ´ ` “ ´ ´ ´ ´ ´ The row sums of v B equal , so v B P M E ‘ M E where M E is the dual vector space to N E . Definition 2.19.
The bipermutohedron of E is Π E,E : “ conv { v B : B is a bipermutation on E } Ă M E ‘ M E . Theorem 2.20.
The bipermutohedral fan Σ E,E is the normal fan of the bipermutohedron Π E,E . Proof.
Let B be a bipermutation on E . We claim that the cone of the normal fan of Π E,E corre-sponding to v B is precisely the maximal cone σ B of the bipermutohedral fan Σ E,E : N Π E,E p v B q “ σ B . (2.7.1)This will prove the desired result. It will also show that each v B is indeed a vertex of the biper-mutohedron Π E,E . Ď : Consider a linear functional p z, w q P N E,E in N Π E,E p v B q , so the p z, w q -minimal face of Π E,E contains the vertex v B . We need to show that p z, w q P σ B .For any adjacent letters i, j of B , let B be the bipermutation obtained by swapping them: B “ . . . | i | j | . . . , B “ . . . | j | i | . . . , Notice that the tables of v B and v B can only differ in columns i, j , and k , where k is the letterthat is not repeated in B . We use this fact to simplify the inequality w p B q ď w p B q , rewriting itin terms of the coordinate system of chart C k of N E ˆ N E : Z i “ z i ´ z k , W i “ ´p w i ´ w k q for i P E. There are eight cases:
Case 1: i, j P E ´ k . The permutations π “ π p B q and π “ π p B q satisfy π p i q “ a ´ , π p j q “ a ` and π p i q “ a ` , π p j q “ a ´ for some a . Also s p π q “ s p π q . Therefore p z, w q (cid:32) i k j ¨ a ´ ¨ ´ s ¨ a ` ¨¨ ¨ ¨ ´ s ¨ ¨ ¨ (cid:33) ď p z, w q (cid:32) i k j ¨ a ` ¨ ´ s ¨ a ´ ¨¨ ¨ ¨ ´ s ¨ ¨ ¨ (cid:33) p a ´ q z i ` p a ` q z j ď p a ` q z i ` p a ´ q z j z j ď z i Z j ď Z i Case 2: i, j P E ´ k . This is the reverse of Case 1. Similarly, we have: p z, w q (cid:32) i k j ¨ ¨ ¨ ´ s ¨ ¨ ¨¨ ´p a ´ q ¨ ´ s ¨ ´p a ` q ¨ (cid:33) ď p z, w q (cid:32) i k j ¨ ¨ ¨ ´ s ¨ ¨ ¨¨ ´p a ` q ¨ ´ s ¨ ´p a ´ q ¨ (cid:33) ´p a ´ q w i ´ p a ` q w j ď ´p a ` q w i ´ p a ´ q w j ´ w j ď ´ w i W j ď W i Case 3: i P E ´ k and j P E ´ k . Again, π p i q “ a ´ , π p j q “ a ` and π p i q “ a ` , π p j q “ a ´ for some a . Now we have that s p π q “ s p π q ` , so p z, w q (cid:32) i k j ¨ a ´ ¨ ´ s ¨ ¨ ¨¨ ¨ ¨ ´ s ¨ ´p a ` q ¨ (cid:33) ď p z, w q (cid:32) i k j ¨ a ` ¨ ´p s ` q ¨ ¨ ¨¨ ¨ ¨ ´p s ` q ¨ ´p a ´ q ¨ (cid:33) p a ´ q z i ´ p a ` q w j ď p a ` q z i ´ p a ´ q w j ´ z k ´ w k ´p w j ´ w k q ď z i ´ z k W j ď Z i Case 4: i P E ´ k and j P E ´ k . This is the reverse of Case 3. We obtain Z j ď W i . AGRANGIAN GEOMETRY OF MATROIDS 25
Case 5: i “ k and j P E ´ k . Now π and π satisfy π p i q “ a ´ , π p i q “ a, π p j q “ a ` and π p i q “ a, π p i q “ a ` , π p j q “ a ´ for some a . In this case s p π q “ s p π q ´ , so p z, w q (cid:32) i “ k j ¨ p a ´ q ´ s ¨ a ` ¨¨ ´ a ´ s ¨ ¨ ¨ (cid:33) ď p z, w q (cid:32) i j “ k ¨ a ´ p s ´ q ¨ a ´ ¨¨ ´p a ` q ´ p s ´ q ¨ ¨ ¨ (cid:33) p a ´ ´ s q z i ´ p a ` s q w i ` p a ` q z j ď p a ` ´ s q z i ´ p a ` s q w i ` p a ´ q z j z j ď z i Z j ď Z i Case 6: i P E ´ k and j “ k . This is the reverse of Case 5. We obtain Z j ď Z i . Case 7: i “ k and j P E ´ k . An argument analogous to Case 5 gives W j ď W i . Case 8: i P E ´ k, j “ k . This is the reverse of Case 7. We obtain W j ď W i .Applying the above analysis to each pair of adjacent letters of B , we conclude that the relativeorder of Z , . . . , Z n , W , . . . , W n is (weakly) the same as the opposite order of , . . . , n, , . . . , n in π p B q . In particular, since i precedes i for all i P E , we have that Z i ě W i for all i ; that is, min i P E p z i ` w i q “ z k ` w k . We conclude that p z, w q is in the cone C k and it satisfies the defininginequalities of σ B Ă C k . Therefore p z, w q P σ B , as desired. Ě : Consider a point p z, w q in the interior of σ B . If p z, w q were not in the normal cone N Π E,E p v B q ,then it would have to be in the normal cone N Π E,E p v σ q for some other vertex v B correspondingto a bipermutation B ‰ B . But then p z, w q P σ B by the first part of this proof, and this wouldmean that one maximal cone in the fan Σ E,E intersects the interior of another, a contradiction.We conclude that N Π E,E p v B q contains int σ B and, being closed, it must contain all of σ B asdesired. (cid:3)
3. T
HE CONORMAL INTERSECTION THEORY OF A MATROID
In this section, we construct the conormal fan of a matroid M on E , and describe its Chow ring.Our running example will be the graphic matroid M p G q of the graph G of the square pyramid,whose dual is the graphic matroid of the dual graph G K shown in Figure 6.3.1. Homology and cohomology.
Throughout this section we fix a simplicial rational fan Σ in N “ R b N Z . For each ray ρ in Σ , we write e ρ for the primitive generator of ρ in N Z , and introducea variable x ρ . ‚ Let S p Σ q be the polynomial ring with real coefficients that has x ρ as its variables, one for eachray ρ of Σ . ‚ Let I p Σ q be the Stanley-Reisner ideal of S p Σ q , generated by the square-free monomials index-ing the subsets of rays of Σ which do not generate a cone in Σ . IGURE
6. The graph G of the square pyramid and its dual graph G K . ‚ Let J p Σ q be the ideal of S p Σ q generated by the linear forms (cid:80) ρ (cid:96) p e ρ q x ρ , where (cid:96) is any linearfunction on N and the sum is over all the rays in Σ . Definition 3.1.
The
Chow ring of Σ , denoted A p Σ q , is the graded algebra S p Σ q{p I p Σ q ` J p Σ qq .Billera [Bil89] constructed an isomorphism from the monomial quotient S p Σ q{ I p Σ q to thealgebra of continuous piecewise polynomial functions on Σ by identifying the variable x ρ withthe piecewise linear tent or Courant function on Σ determined by the condition x ρ p e ρ q “ , if ρ is equal to ρ , , if ρ is not equal to ρ .Thus, under this isomorphism, a piecewise linear function (cid:96) on Σ is identified with the linearform (cid:96) “ (cid:88) ρ (cid:96) p e ρ q x ρ . We regard the elements of A p Σ q as equivalence classes of piecewise polynomial functions on Σ ,modulo the restrictions of global linear functions to Σ .Brion [Bri96] showed that the Chow ring of the toric variety X p Σ q of Σ with real coefficientsis isomorphic to A p Σ q . Under this isomorphism, the class of the torus orbit closure of a cone σ in Σ is identified with mult p σ q x σ , where x σ is the monomial (cid:81) ρ Ď σ x ρ and mult p σ q is the indexof the subgroup (cid:16) (cid:88) ρ Ď σ Z e ρ (cid:17) Ď N Z X (cid:16) (cid:88) ρ Ď σ R e ρ (cid:17) . All the fans appearing in this section will be unimodular, so mult p σ q “ for every σ in Σ .We write Σ p k q for the set of k -dimensional cones in Σ . A k -dimensional Minkowski weight on Σ is a real-valued function ω on Σ p k q that satisfies the balancing condition : For every p k ´ q -dimensional cone τ in Σ , (cid:88) τ Ă σ ω p σ q e σ { τ “ in the quotient space N { span p τ q , When Σ is complete, this description of the Chow ring can be deduced from a classical result of Danilov and Ju-rkiewicz [CLS11, Theorem 12.5.3]. AGRANGIAN GEOMETRY OF MATROIDS 27 where e σ { τ is the primitive generator of the ray p σ ` span p τ qq{ span p τ q . We say that w is pos-itive if w p σ q is positive for every σ in Σ p k q . We write MW k p Σ q for the space of k -dimensionalMinkowski weights on Σ , and set MW p Σ q “ (cid:77) k ě MW k p Σ q . We will make use of the basic fact that the Chow group of a toric variety is generated by theclasses of torus orbit closures [CLS11, Lemma 12.5.1]. Thus, there is an injective linear mapfrom the dual of A k p Σ q to the space of k -dimensional weights on Σ , whose image turns out tobe MW k p Σ q , as noted in [FS97]. Explicitly, the inverse isomorphism from the image is MW k p Σ q ÝÑ Hom p A k p Σ q , R q , w ÞÝÑ p mult p σ q x σ ÞÝÑ w p σ qq . Following [AHK18, Section 5], we define the cap product , denoted η X w , using the composition A (cid:96) p Σ q ÝÑ Hom p A k ´ (cid:96) p Σ q , A k p Σ qq ÝÑ Hom p MW k p Σ q , MW k ´ (cid:96) p Σ qq , η ÞÝÑ p w ÞÝÑ η X w q , where the first map is given by the multiplication in the Chow ring of Σ . In short, MW p Σ q hasthe structure of a graded A p Σ q -module given by the isomorphism MW p Σ q » Hom p A p Σ q , R q .Let f : Σ Ñ Σ be a morphism of simplicial fans. The pullback of functions define the pullbackhomomorphism between the Chow rings f ˚ : A p Σ q ÝÑ A p Σ q , whose dual is the pushforward homomorphism between the space of Minkowski weights f ˚ : MW p Σ q ÝÑ MW p Σ q . Since f ˚ is a homomorphism of graded rings, f ˚ is a homomorphism of graded modules. Inother words, the pullback and the pushforward homomorphisms satisfy the projection formula η X f ˚ w “ f ˚ p f ˚ η X w q . The Bergman fan of a matroid.
The
Bergman fan of a matroid M on E , denoted Σ M , is the r -dimensional subfan of the n -dimensional permutohedral fan Σ E whose underlying set is the tropical linear space trop p M q “ (cid:110) z | min i P C p z i q is achieved at least twice for every circuit C of M (cid:111) Ď N E . The Bergman fan of M is equipped with the piecewise linear functions α j “ max i P E p z j ´ z i q , and the space of linear functions on the Bergman fan is spanned by the differences α i ´ α j “ z i ´ z j . Note that trop p M q is nonempty if and only if M is loopless . In the remainder of this section, wesuppose that M has no loops. In this case, the Bergman fan of M is the induced subfan of Σ E generated by the rays corresponding to the nonempty proper flats of M [AK06]. Proposition 3.2.
The Bergman fan of M is the unimodular fan in N E with the cones σ F “ cone { e F } F P F , where F is a flag of flats of M .The most important geometric property of Σ M is the following description of its top-dimensionalMinkowski weights. For a proof, see, for example, [AHK18, Proposition 5.2]. Proposition 3.3. An r -dimensional weight on Σ M is balanced if and only if it is constant.We write M for the fundamental weight on Σ M , the r -dimensional Minkowski weight on theBergman fan that has the constant value .3.3. The Chow ring of the Bergman fan.
In the context of matroids, for simplicity, we set S M “ S p Σ M q , I M “ I p Σ M q , J M “ J p Σ M q , A M “ A p Σ M q . We identify the elements of S M { I M with the piecewise linear functions on Σ M as before.Let x F be the variable of the polynomial ring corresponding to the ray generated by e F in theBergman fan. For any set F of nonempty proper flats of M , we write x F for the monomial x F “ (cid:89) F P F x F . The variable x F , viewed as a piecewise linear function on the Bergman fan, is given by x F p e F q “ , if F is equal to F , , if F is not equal to F ,and hence the piecewise linear function α j on the Bergman fan satisfies the identity α j “ (cid:88) F α j p e F q x F “ (cid:88) j P F x F . Thus, in the above notation, ‚ S M is the ring of polynomials in the variables x F , where F is a nonempty proper flat of M , ‚ I M is the ideal generated by the monomials x F , where F is not a flag, and ‚ J M is the ideal generated by the linear forms α i ´ α j , for any i and j in E .We write α for the common equivalence class of α j in the Chow ring of the Bergman fan. Definition 3.4.
The fundamental weight M defines the degree map deg : A r M ÝÑ R , x F ÞÝÑ x F X M “ if F is a flag, if F is not a flag. AGRANGIAN GEOMETRY OF MATROIDS 29
By Proposition 3.3, the degree map is an isomorphism. In other words, for any maximal flag F of nonempty proper flats of M , the class of the monomial x F in the Chow ring of the Bergmanfan of M is nonzero and does not depend on F .3.4. The conormal fan of a matroid.
The conormal fan of a matroid M on E , denoted Σ M , M K , isthe p n ´ q -dimensional subfan of the n -dimensional bipermutohedral fan Σ E,E whose supportis the product of tropical linear spaces | Σ M , M K | “ trop p M q ˆ trop p M K q . Equivalently, the conormal fan is the largest subfan of the bipermutohedral fan for which theprojections to the factors are morphisms of fans π : Σ M , M K ÝÑ Σ M and π : Σ M , M K ÝÑ Σ M K . The addition map p z, w q ÞÑ z ` w is also a morphism of fans Σ M , M K Ñ Γ E .The conormal fan of M is equipped with the piecewise linear functions γ j “ max i P E p z j ´ z i q , γ j “ max i P E p w j ´ w i q , δ j “ max i P E p z j ` w j ´ z i ´ w i q , which are the pullbacks of α j under the projections π and π and the addition map, respectively.The space of linear functions on the conormal fan is spanned by the differences γ i ´ γ j “ z i ´ z j and γ i ´ γ j “ w i ´ w j . Note that the support of the conormal fan of M is nonempty if and only if M is loopless and coloopless . In the remainder of this section, we suppose that M has no loops and no coloops. Definition 3.5. A biflat F | G of M consists of a flat F of M and a flat G of M K that form a bisubset;that is, they are nonempty, they are not both equal to E , and their union is E . A biflag of M is aflag of biflats.We give an analog of Proposition 3.2 for conormal fans in terms of biflats. Proposition 3.6.
The conormal fan of M is the unimodular fan in N E,E with the cones σ F | G “ cone { e F | G } F | G P F | G , for F | G a flag of biflats of M . Proof.
The proof is straightforward, given Corollary 2.17 and Proposition 3.2: If F | G is a flag ofbiflats of M , then F is an increasing sequence of flats of M and G is a decreasing sequence of flatsof M K , and hence σ F | G Ď σ F ˆ σ G P Σ M ˆ Σ M K . Therefore, the conormal fan of M contains the induced subfan of Σ E,E generated by the rayscorresponding to the biflats of M . The other inclusion follows from the easy implication e F | G is in the support of the conormal fan of M ùñ F | G is a biflat of M . (cid:3) We also have the following analog of Proposition 3.3 for conormal fans.
Proposition 3.7. An p n ´ q -dimensional weight on Σ M , M K is balanced if and only if it is constant.We write M , M K for the fundamental weight on Σ M , M K , the top-dimensional Minkowski weighton the conormal fan that has the constant value . Proof.
Proposition 3.3 applied to M and M K shows that a top-dimensional weight on Σ M ˆ Σ M K satisfies the balancing condition if and only if it is constant. This property of the fan remainsinvariant under any subdivision of its support, as shown in [GKM09, Section 2]. (cid:3) For our purposes, the product of the Bergman fans of M and M K has a shortcoming: The ad-dition map need not be a morphism from the product to the fan Γ E . Thus, in general, we cannotdefine the class of δ j in the Chow ring of the product. This is our motivation for subdividing itfurther, to obtain the conormal fan Σ M , M K . Example . Let M and M K be the graphic matroids of the graphs in Figure 6. Consider the cone σ F ˆ σ G in the product of Bergman fans of M and M K , where F “ p ∅ Ĺ Ĺ Ĺ Ĺ E q and G “ p ∅ Ĺ Ĺ Ĺ Ĺ E q . This cone is subdivided into the chambers of Σ M , M K corresponding to the biflags ∅ Ĺ Ď Ď Ď Ď Ď E Ď EE Ě E Ě E Ě E Ě Ě Ě Ľ ∅ , ∅ Ĺ Ď Ď Ď Ď E Ď E Ď EE Ě E Ě E Ě Ě Ě Ě Ľ ∅ , ∅ Ĺ Ď Ď Ď E Ď E Ď E Ď EE Ě E Ě E Ě Ě Ě Ě Ľ ∅ . If p z, w q is inside the first chamber, then the minimum of z i ` w i is attained by z ` w “ z ` w ,and hence z ` w is in the cone σ . If p z, w q is inside the second or the third chamber, thenthe minimum of z i ` w i is attained by z ` w “ z ` w , and hence z ` w is in the cone σ .Thus, the product cone does not map into a cone in Γ E under the addition map.Recall from Definition 2.5 that the cotangent fan Ω E is the subfan of Σ E,E with support trop p δ q “ (cid:110) p z, w q | min i P E p z i ` w i q is achieved at least twice (cid:111) Ď N E,E . In other words, the cotangent fan is the collection of cones σ B for bisequences B on E , where atleast two elements of E appear exactly once in B . We show that the cotangent fan contains allthe conormal fans of matroids on E . Proposition 3.9.
For any matroid M on E , we have trop p M q ˆ trop p M K q Ď trop p δ q .In other words, if the minimum of p z i q i P C is achieved at least twice for every circuit C of M and the minimum of p w i q i P C K is achieved at least twice for every circuit C K of M K , then the AGRANGIAN GEOMETRY OF MATROIDS 31 minimum of p z i ` w i q i P E is achieved at least twice. We deduce Proposition 3.9 from Proposition3.14 below, a stronger statement on the flags of biflats of M . The notion of gaps introduced herefor Proposition 3.14 will be useful in Section 4.Let F | G be a flag of biflats of M . As before, we write F and G for the sequences F “ p ∅ Ĺ F Ď ¨ ¨ ¨ Ď F k Ď E q , where F j are the first parts of the biflats in F | G , G “ p E Ě G Ě ¨ ¨ ¨ Ě G k Ľ ∅ q , where G j are the second parts of the biflats in F | G , where k is the length of F | G . Thus, the bisequence B p F | G q from Proposition 2.16 can be written B | B | ¨ ¨ ¨ | B k , where B j “ p F j ` ´ F j q Y p G j ´ G j ` q . Definition 3.10.
The gap sequence of F | G , denoted D p F | G q , is the sequence of gaps D | D | ¨ ¨ ¨ | D k , where D j “ p F j ` ´ F j q X p G j ´ G j ` q . Note that D j consists of the elements of B j that appear exactly once in the bisequence B p F | G q . Example . The three maximal flags of biflats shown in Example 3.8 have the gap sequences ∅ | ∅ | ∅ | ∅ | ∅ | | ∅ , ∅ | ∅ | | ∅ | ∅ | ∅ | ∅ , ∅ | ∅ | | ∅ | ∅ | ∅ | ∅ . We show in Proposition 3.17 that any maximal flag of biflats has a unique nonempty gap.
Lemma 3.12.
The complement of the gap D j in E is the union of F j and G j ` .Therefore, by Proposition 2.15, at least one of the gaps of F | G must be nonempty. Proof.
Since F j | G j and F j ` | G j ` are bisubsets, we have G cj Ď F j and F cj ` Ď G j ` . Thus, D cj “ p F j ` X F cj X G j X G cj ` q c “ F cj ` Y F j Y G cj Y G j ` “ F j Y G j ` . (cid:3) Lemma 3.13.
Let e P E . There exists an index i for which e P F i X G i if and only if e is not inany gap. In symbols, the union of the gaps of F | G is k (cid:71) j “ D j “ E ´ k (cid:91) i “ ( F i X G i ) . Proof.
First suppose e P F i X G i . Then e P F j for all j ě i , which means e R D j for i ď j ď k .Dually, e P G j for all j ď i , so e R D j for all ď i ď j ´ .Now suppose e is not in any gap, and consider the index ď i ď k ` for which e P F i ´ F i ´ .Since e P F i ´ Y G i , we must have e P G i and hence e P F i X G i . (cid:3) Proposition 3.14.
Every nonempty gap of a biflag F | G of M has at least two distinct elements. Proof.
Recall that, for any matroid, the complement of any hyperplane is a cocircuit [Oxl11,Proposition 2.1.6] and that any flat is an intersection of hyperplanes [Oxl11, Proposition 1.7.8].
Since the complement of a gap of F | G is the union of a flat and a coflat by Lemma 3.12, we maywrite the gap as the intersection (cid:16) (cid:91) C P C C (cid:17) X (cid:16) (cid:91) C K P C K C K (cid:17) , where C is a collection of circuits and C K is a collection of cocircuits. Thus, if the gap is nonempty,there are C P C and C K P C K that intersect nontrivially. Now the first statement follows fromthe classical fact that the intersection of a circuit and a cocircuit is either empty or contains atleast two elements [Oxl11, Proposition 2.11]. (cid:3) For any biflag F | G , there are at least two elements of E that appear exactly once in the bise-quence B p F | G q ; therefore trop p M q ˆ trop p M K q Ď trop p δ q , proving Proposition 3.9.We will often use the following restatement of Proposition 3.14. Recall that | E | “ n ` . Lemma 3.15.
The union of a flat and a coflat cannot have exactly n elements.For later use, we record here another elementary property of the flags of biflats of a matroid. Definition 3.16.
The jump sets of F and G are the sets of indices J p F q “ { j | ď j ď k and F j ‰ F j ` } and J p G q “ { j | ď j ď k and G j ‰ G j ` } . The elements of J p F q X J p G q are called the double jumps of F | G .The double jumps are colored blue in the table of F | G , as shown in Example 3.8. Clearly, j is a double jump whenever the corresponding gap D j is nonempty. We show that the converseholds when F | G is maximal. Proposition 3.17.
Every maximal flag of biflats F | G of M has a unique double jump. Ignoringrepetitions, F and G are complete flags of non-zero flats in M and M K , respectively.In particular, every maximal flag of biflats F | G of M has a unique nonempty gap. Proof.
Recall that at least one of the gaps of F | G is nonempty. In addition, since tropical linearspaces are pure-dimensional, the length of any maximal flag of biflats must be n ´ . Thus, | J p F q X J p G q| ě and | J p F q Y J p G q| “ n. On the other hand, writing r ` for the rank of M as before, we have | J p F q| ď r ` and | J p G q| ď n ´ r. Therefore, n ` ď | J p F q Y J p G q| ` | J p F q X J p G q| “ | J p F q| ` | J p G q| ď n ` , and hence | J p F q| “ r ` , | J p G q| “ n ´ r and | J p F q X J p G q| “ which imply the desired results. (cid:3) AGRANGIAN GEOMETRY OF MATROIDS 33
The Chow ring of the conormal fan.
For notational simplicity, we set S M , M K “ S p Σ M , M K q , I M , M K “ I p Σ M , M K q , J M , M K “ J p Σ M , M K q , A M , M K “ A p Σ M , M K q . We identify the elements of S M , M K { I M , M K with the piecewise linear functions on the conormalfan.Let x F | G be the variable of the polynomial ring corresponding to the ray generated by e F | G in the conormal fan. For any set F | G of biflats of M , we write x F | G for the monomial x F | G “ (cid:89) F | G P F | G x F | G . We note that the piecewise linear function δ j on the conormal fan satisfies the identity δ j “ (cid:88) F | G δ j p e F | G q x F | G “ (cid:88) j P F X G x F | G . Similarly, the piecewise linear functions γ j and γ j satisfy the identities γ j “ (cid:88) j P F ‰ E x F | G and γ j “ (cid:88) j P G ‰ E x F | G . Thus, in the above notation, ‚ S M , M K is the ring of polynomials in the variables x F | G , where F | G is a biflat of M , ‚ I M , M K is the ideal generated by the monomials x F | G , where F | G is not a biflag, and ‚ J M , M K is the ideal generated by the linear forms γ i ´ γ j and γ i ´ γ j , for any i and j in E .We write γ , γ , and δ , respectively, for the equivalence classes of γ j , γ j , and δ j in the Chow ringof the conormal fan. Definition 3.18.
The fundamental weight M , M K of the conormal fan defines the degree map deg : A n ´ , M K ÝÑ R , x F | G ÞÝÑ x F | G X M , M K “ if F | G is a biflag, if F | G is not a biflag.By Proposition 3.7, the degree map is a linear isomorphism. In other words, for maximal flagof biflats F | G of M , the class of the monomial x F | G in the Chow ring of the conormal fan of M isnonzero and does not depend on F | G .Recall that the projection π is a morphism from the conormal fan of M to the Bergman fan of M . The projection has the special property that the image of a cone in the conormal fan is a conein the Bergman fan (and not just contained in one). This property leads to the following simpledescription of the pullback π ˚ : A M Ñ A M , M K . Proposition 3.19.
For any flag of nonempty proper flats F of M , π ˚ p x F q “ (cid:88) G x F | G , where the sum is over all decreasing sequences G such that F | G is a flag of biflats of M .Dually, the pushforward of any Minkowski weight w on the conormal fan is given by π ˚ p w qp σ F q “ (cid:88) G w p σ F | G q , where the sum is over all decreasing sequences G such that F | G is a flag of biflats of M . Proof.
Since π p e F | G q “ e F , the pullback of the piecewise linear function x F satisfies π ˚ p x F q “ (cid:88) G x F | G , where the sum is over all G such that F | G is a biflat of M . Thus, for any given F , π ˚ p x F q “ (cid:89) F P F π ˚ p x F q “ (cid:88) G x F | G , where the sum is over all decreasing sequences G such that F | G is a flag of biflats of M . (cid:3)
4. D
EGREE COMPUTATIONS IN THE C HOW RING OF THE CONORMAL FAN
Recall that the beta invariant of a matroid M of rank r ` is β p M q – p´ q r χ M p q . Given a strictly increasing flag of flats F “ { H Ĺ F Ĺ ¨ ¨ ¨ Ĺ F k Ĺ E } , the beta invariant of F in M is β p M r F sq – k ` (cid:89) i “ β p M r F i ´ , F i sq , (4.0.1)where β p M r F i ´ , F i sq is the beta invariant of the matroid minor M p i q “ M r F i ´ , F i s “ M | F i { F i ´ for ď i ď k ` .The goal of this section is to prove Propositions 4.8 and 4.18, which state that deg p δ n ´ q “ β p M q and, more generally, that for any strictly increasing flag of flats F in M of length k , deg p π ˚ p x F q δ n ´ k ´ q “ (cid:88) F | G biflag deg p x F | G δ n ´ k ´ q “ β p M r F sq , where π ˚ : A M Ñ A M , M K is the pullback of the projection map π : Σ M , M K Ñ Σ M . Thus we seekto compute x F | G δ n ´ k ´ in the Chow ring of Σ M , M K . This will require us to study more closelythe combinatorial structure of conormal fans, and develop algebraic combinatorial techniquesfor computing in their Chow rings. We do so in this section.4.1. Canonical expansions in the Chow ring of the conormal fan.
In order to compute thedegree of δ n ´ in the Chow ring A M , M K – or more generally the degree of x F | G δ n ´ k ´ for a k -biflag F | G — we seek to express it as a sum of square-free monomials, each of which have degree AGRANGIAN GEOMETRY OF MATROIDS 35 one by Definition 3.4. One fundamental feature of this computation, which is simultaneouslyan advantage and a difficulty, is that there are many different ways to carry it out, since we maychoose from n ` different expressions for δ ; namely δ “ δ i for all i P E . To have control overthe computation, we require some structure amidst that freedom. Thus we prescribe a canonicalway of expressing δ m (and more generally, x F | G δ m ) for each m . Definition 4.1. ( Canonical expansion of x F | G δ m . ) For a nonzero monomial x F | G in A M , M K , let e “ e p F | G q : “ max (cid:0) E ´ k (cid:91) i “ p F i X G i q (cid:1) “ max (cid:0) k (cid:71) j “ D j (cid:1) be the largest gap element of F | G , which exists thanks to Lemma 3.13. Define the canonicalexpansion of x F | G δ to be x F | G δ “ x F | G δ e “ (cid:88) F | G P R M , M K e P F X G x F | G x F | G . (4.1.1)This is a sum of monomials in A M , M K . Thus we may recursively obtain the canonical expansionof x F | G δ m for m ě by multiplying each monomial in the canonical expansion of x F | G δ m ´ by δ , again using the canonical expansion.Note that some or all of the summands in (4.1.1) may equal in the Chow ring A M , M K . Thefollowing lemma describes the non-zero terms. Lemma 4.2.
The canonical expansion of x F | G δ is the sum of the monomials x F Y F | G Y G corre-sponding to the cones of the form σ F Y F | G Y G Ľ σ F | G such that e “ e p F | G q P F X G . If e is in gap D j , we must have F j Ď F Ď F j ` , G j Ě G Ě G j ` . Proof.
The first statement follows directly from the definitions. If σ F Y F | G Y G is a cone with e P F X G , then e R F j and e R G j ` imply that the pair F | G must be added in between indices j and j ` of σ F | G . Conversely, any such pair arises in this expansion. (cid:3) We may think of the canonical expansion of δ m as a recursive procedure to produce a list of m -dimensional cones in the conormal fan Σ M , M K , where each cone is built up one ray at a timeaccording to the rules prescribed in Lemma 4.2. Example . For the graph G of the square pyramid in Figure 6, the canonical expansion of thehighest non-zero power of δ in A M , M K , namely δ n ´ “ δ , is δ “ x | E x | E x | E x E | x E | x E | ` x | E x | E x | E x E | x E | x E | ` x | E x | E x | E x E | x E | x E | . This expression is deceivingly short. Carrying out this seemingly simple computation by handis very tedious; if one were to do it by brute force, one would find that the number of terms of the canonical expansions of δ , . . . , δ are the following: δ δ δ δ δ δ δ number of monomials counted with multiplicities number of distinct monomials . This example shows typical behavior: for small k the number of cones in the expansion of δ k increases with k , but as k approaches n ´ , increasingly many products x F | G δ are zero, and thecanonical expansions become shorter.We summarize the properties of the canonical expansion in the following proposition, whichfollows readily from the previous discussion. Proposition 4.4.
For each m ě , the canonical expansion of δ m of Definition 4.1 is the sum ofthe monomials indexed by the collection T m M , M K of all the tables p F | G , e q of M for which(1) F | G is a biflag of length m , and(2) e “ p e , . . . , e m q is a sequence of distinct elements of E such that e i P F i X G i , and e i “ max (cid:0) E ´ (cid:91) j : e j ą e i p F j X G j q (cid:1) for all ď i ď m. In symbols, the following identity holds in the Chow ring A M , M K : δ m “ (cid:88) p F | G , e qP T m M , M K x F | G x F | G ¨ ¨ ¨ x F m | G m . We encode such a pair p F | G , e q in the following table. p F | G , e q : H Ĺ F Ď ¨ ¨ ¨ Ď F d Ď F d ` Ď ¨ ¨ ¨ Ď F m Ď EE Ě G Ě ¨ ¨ ¨ Ě G d Ě G d ` Ě ¨ ¨ ¨ Ě G m Ľ H e ¨ ¨ ¨ e d e d ` ¨ ¨ ¨ e m We adopt the convention that F “ G m ` “ H and G “ F m ` “ E .As Example 4.3 illustrates, the canonical expansion of δ m may contain repeated terms x F | G coming from tables that have the same biflag F | G but different sequences e . Example . Let us revisit the canonical expansion of δ , the highest non-zero power of δ , inExample 4.3. The first monomial arises from the following table p F | G , e q : H Ă Ĺ Ĺ Ĺ E “ E “ E “ EE “ E “ E “ E Ľ Ľ Ľ Ą H e “ e “ e “ e “ e “ e “ The terms x F i | G i arrive to the monomial in the order x E | x | E x | E x | E x E | x E | , indecreasing order of the e i s. The two other monomials are x | E x E | x | E x | E x E | x E | and x | E x | E x E | x | E x E | x E | , where the terms are again listed in order of arrival. AGRANGIAN GEOMETRY OF MATROIDS 37
The beta invariant of a matroid in its conormal intersection theory.
Our next goal is toprove Proposition 4.8, which describes the canonical expansion of δ n ´ in the Chow ring A M , M K ,and uses it to conclude that its degree is Crapo’s beta invariant β p M q .For each basis B Ď E of M , denote the corresponding dual basis of M K by B K – E ´ B. We also let cl K denote the closure function of M K .A broken circuit of M is a set of the form C ´ min C where C is a circuit. An nbc -basis of M is a basis that contains no broken circuits. A β nbc -basis of M is an nbc basis B of M suchthat B K Y { } ´ { } is an nbc basis of M K . The number of nbc basis is the Möbius number | µ p M q| “ | µ p ∅ , E q| , whereas the number of β nbc bases is the beta invariant β p M q [Zie92].It is well known that the independence complex IN p M q and the reduced broken circuit com-plex BC p M q of a matroid M are shellable, and hence homotopy equivalent to wedges of spheres.The nbc bases and β nbc bases of M naturally index the spheres in the lexicographic shellingsof IN p M q and BC p M q , respectively [Bjö92, Zie92]. Definition 4.6.
Let B be a β nbc basis of M and write B ´ “ { e ą ¨ ¨ ¨ ą e r } , B K ´ “ { e r ` ă ¨ ¨ ¨ ă e n ´ } . The maximal biflag F p B q| G p B q and the β cone p B q – σ F p B q| G p B q of B are H Ĺ cl M p e q Ĺ ¨ ¨ ¨ Ĺ cl p e , . . . , e r q Ĺ E “ ¨ ¨ ¨ “ E “ EE “ E “ ¨ ¨ ¨ “ E Ľ cl K p e r ` , . . . , e n ´ q Ľ ¨ ¨ ¨ Ľ cl K p e n ´ q Ľ H To see that F p B q| G p B q is indeed a biflag, we verify that cl p B ´ q Y cl K p B K ´ q ‰ E . Noticethat R cl K p B K ´ q since B K is a basis of M K ; and if we had P cl p B ´ q , then B ´ Y wouldcontain a circuit C whose minimum element is , and hence B would contain the broken circuit C ´ , contradicting that B is nbc . Example . The matroid of Figure 6 has three β nbc basis, namely B “ , B “ , B “ . The corresponding β cones are precisely the ones arising in the expansion of Example 4.3. Thefollowing theorem shows this is a general phenomenon. Proposition 4.8.
Let M be a loopless and coloopless matroid on the ground set E “ { , . . . , n } .Then, in the Chow ring of the conormal fan of M , we have the canonical expansion δ n ´ “ (cid:88) B P β nbc p M q x β cone p B q . It follows that the degree of δ n ´ is the β -invariant of M . This definition is different from the standard one, but they are readily proved to be equivalent.
Proof.
We proceed in a series of lemmas. Proposition 4.4 describes the canonical expansion of δ n ´ in terms of tables p F | G , e q of the form H Ĺ F Ď ¨ ¨ ¨ Ď F d Ĺ F d ` Ď ¨ ¨ ¨ Ď F n ´ Ď EE Ě G Ě ¨ ¨ ¨ Ě G d Ľ G d ` Ě ¨ ¨ ¨ Ě G n ´ Ľ H e ¨ ¨ ¨ e d e d ` ¨ ¨ ¨ e n ´ , which have a unique double jump d “ j p F q X j p G q thanks to Proposition 3.17. A priori, thisdouble jump could occur at d “ or d “ n ´ . We let { e n , e n ` } – E ´ { e , . . . , e n ´ } be the two elements missing from the sequence e . Let us record two simple observations aboutsuch tables, which we will return to often. Lemma 4.9. If i P J p F q ´ J p G q , then e i ą e i ` . If i P J p G q ´ J p F q , then e i ă e i ` . Proof.
By symmetry, it suffices to prove the first assertion. Assume contrariwise that i P J p F q ´ J p G q and e i ă e i ` , so the table p F | G , e q contains F i Ĺ F i ` G i “ G i ` e i ă e i ` . Then the pair F i | G i arrives to the monomial x F | G after F i ` | G i ` , so e i R F i ` X G i ` . Thiscontradicts that e i P F i X G i Ď F i ` X G i “ F i ` X G i ` . (cid:3) Lemma 4.10. If i ă j and e i ă e j , then e i R G j . If i ă j and e i ą e j , then e j R F i . Proof.
It suffices to prove the first assertion. The table p F | G , e q contains F i Ď ¨ ¨ ¨ Ď F j G i Ě ¨ ¨ ¨ Ě G j e i ă e j , which shows that F i | G i appears in the term x F | G after F j | G j , so e i R F j X G j . Since e i P F i Ď F j ,we must have e i R G j . (cid:3) Lemma 4.11.
If the table p F | G , e q arises in the canonical expansion of δ n ´ , then its uniquedouble jump is at d “ r , and its table is of the form H Ĺ F Ĺ ¨ ¨ ¨ Ĺ F r Ĺ E “ ¨ ¨ ¨ “ E “ EE “ E “ ¨ ¨ ¨ “ E Ľ G r ` Ľ ¨ ¨ ¨ Ľ G n ´ Ľ H e ą ¨ ¨ ¨ ą e r { e n , e n ` } e r ` ă ¨ ¨ ¨ ă e n ´ . The unique nonempty gap is D r “ { e n , e n ` } ; we write it under the double jump at r . AGRANGIAN GEOMETRY OF MATROIDS 39
Proof. If d is the unique double jump, D d is the unique nonempty gap. Since { e , . . . , e d } Ď F d and { e d ` , . . . , e n ´ } Ď G d ` , we must have F d Y G d ` “ { e , . . . , e n ´ } by Lemma 3.15.Therefore the gap D d “ E ´ p F d Y G d ` q indeed equals { e n , e n ` } .Now we prove that e ą e ą ¨ ¨ ¨ ą e d and e d ` ă ¨ ¨ ¨ ă e n ´ ă e n ´ . By symmetry, it suffices to show the first claim. For contradiction, suppose that e j ă e j ` fora minimal choice of j ă d . If j ą then e j ´ ą e j implies e j R F j ´ by Lemma 4.9; if j “ this holds trivially. On the other hand e j ă e j ` implies e j R G j ` by Lemma 4.10. Howeverwe have { e , . . . , e j ´ } Ď F j ´ and { e j ` , . . . , e n ´ } Ď G j ` , and also { e n , e n ` } Ď G d Ď G j ` ;therefore F j ´ Y G j ` “ E ´ e j . This contradicts Lemma 3.15, proving the first claim.Now, for j “ , . . . , d ´ , the inequality e j ą e j ` implies that e j ` P F j ` ´ F j and hence j P J p F q . It follows that { , , . . . , d } “ J p F q and similarly { d, . . . , n ´ , n ´ } “ J p G q . Therefore d “ r . Additionally, since J p G q does not contain , , . . . , d ´ , we must have E “ G “ ¨ ¨ ¨ “ G d ,and similarly F d ` “ ¨ ¨ ¨ “ F n ´ “ E . (cid:3) Lemma 4.12.
If a table p F | G , e q arises in the canonical expansion of δ n ´ , then { e , . . . , e n ´ } “ { , , . . . , n } and { e n , e n ` } “ { , } . Moreover, e i “ min F i for ď i ď r and e i “ min G i for r ` ď i ď n ´ , and F i “ cl p e , . . . , e i q for ď i ď r and G i “ cl K p e i , . . . , e n ´ q for r ` ď i ď n ´ . In particular, the sequence e and the biflag F | G determine each other. Proof.
Let us assume without loss of generality that e r ă e r ` , so x F r ,G r is the last term to arrivein the monomial corresponding to p F | G , e q . By definition, e r “ max (cid:0) E ´ (cid:91) ď j ď n ´ j ‰ r p F j X G j q (cid:1) “ max (cid:0) E ´ p F r ´ Y G r ` q (cid:1) . If we had e r ď , then | F r ´ Y G r ` | ě n ´ which would imply | F r Y G r ` | “ n , a contradic-tion by Lemma 3.15. Thus e r “ and the first claim follows. Also F r Y G r ` “ E ´ { , } .Now let us show e i “ min F i for ď i ď r . If that were not the case, then since , R F i ,we would have min F i “ e j ă e i for some j ‰ i . Since e ą ¨ ¨ ¨ ą e i , this would imply i ă j ,and Lemma 4.10 would then tell us that e j R F i , a contradiction. Similarly e i “ min G i for r ` ď i ď n ´ .Finally, since e P F , e P F ´ F , . . . , e i P F i ´ F i ´ and F i has rank i , the elements e , . . . , e i must be independent and span F i . The analogous result holds for G i as well. (cid:3) Lemma 4.13.
If a table p F | G , e q arises in the canonical expansion of δ n ´ , then { , e , . . . , e r } isa β nbc basis. Proof.
Since e r “ min F r , we have R F r “ cl p e , . . . , e r q . Therefore B “ { , e , . . . , e r } isindeed a basis. We prove that B is nbc by contradiction; assume that it contains a broken circuit C ´ min C . Since min C R B , there are two cases:(i) min C “ . Let C “ { , e a , . . . , e a k } where ď a ă . . . ă a k ď r . Then P cl p e a , . . . , e a k q Ď F a k Ď F r . This contradicts that { , } “ E ´ p F r Y G r ` q .(ii) min C “ e s for some s ě r ` . Let C “ { e s , e a , . . . , e a k } where ď a ă . . . ă a k ď r .Then e s P cl p e a , . . . , e a k q Ď F a k . This contradicts Lemma 4.10 since a k ď r ă s and e a k ą e s .An analogous argument shows that B K ´ { } Y { } “ { , e r ` , . . . , e n ´ } is an nbc basis of M K .We conclude that B is β nbc , as desired. (cid:3) We now have all the ingredients to complete the proof of Proposition 4.8.Lemma 4.12 tells us that each monomial x F | G that appears in the canonical expansion of δ n ´ has coefficient ` . Combined with Lemma 4.13, it also tells us that every term that appears is ofthe form x β cone p B q for a β nbc basis B .Conversely, if F | G “ F p B q| G p B q is the biflag of a β nbc basis B , and if we define e by setting B “ { e ą ¨ ¨ ¨ ą e r ą } and E ´ B “ { e n ´ ą ¨ ¨ ¨ ą e r ` ą } , then it is straightforward tocheck that the table p F | G , e q satisfies the conditions of Proposition 4.4, so it does in fact arise inthe canonical expansion of δ n ´ .This proves the formula for δ n ´ and for its degree, in light of Definition 3.4. (cid:3) A vanishing lemma.
Throughout the remainder of this section, we fix a strictly increasingflag of nontrivial flats F “ { F Ĺ ¨ ¨ ¨ Ĺ F k } , following the convention that F “ H and F k ` “ E . We define the orthogonal flag F K of flats of M K by F K “ { F K Ě ¨ ¨ ¨ Ě F K k } , where F K i “ cl K p E ´ F i q for ď i ď k. Note that the flag F K may contain repeated coflats, and it may contain the trivial coflat E . Wecall the interval r F i ´ , F i s short if | F i ´ F i ´ | “ and long otherwise. Recall that we denote thecorresponding minor by M p i q – M r F i ´ , F i s .The following lemma shows that many monomials in the Chow ring A M , M K vanish whenmultiplied by the highest possible power of δ . Lemma 4.14. (Vanishing Lemma) Let F | G be a biflag of M of length k such that F is strictlyincreasing, and suppose that x F | G δ n ´ k ´ ‰ in the Chow ring of the conormal fan of M . Then(1) G “ F K , and AGRANGIAN GEOMETRY OF MATROIDS 41 (2) every long interval M p i q “ M r F i ´ , F i s for ď i ď k ` is loopless and coloopless. Proof.
Let us assume x F | G δ n ´ k ´ ‰ and consider a non-zero term x F ` | G ` arising in the canon-ical expansion of x F | G δ n ´ k ´ . Let F | G “ F k | G k , F k ` | G k ` , . . . , F n ´ | G n ´ “ F ` | G ` be some sequence of biflags obtained by recursively applying Lemma 4.2 to this expansion. For k ď i ď n ´ , the biflag σ F i | G i has i rays. Let D i, | ¨ ¨ ¨ | D i,i be its sequence of gaps as describedin Definition 3.10. With Lemma 3.13 in mind, let Y i “ i (cid:71) j “ D i,j “ E ´ (cid:91) F | G P F i | G i p F X G q (4.3.1)be the union of the gaps in the biflag F i | G i . In particular, D k, | ¨ ¨ ¨ | D k,k “ D | ¨ ¨ ¨ | D k and Y k “ Y are the gap sequence and the union of the gaps of the initial flag F | G . To prove the VanishingLemma 4.14 we need a preliminary lemma. Lemma 4.15.
Suppose the conditions of Lemma 4.14 hold. Then(1) If F | G has z empty gaps, then the union Y of its gaps has size | Y | “ n ` ´ z .(2) For each empty gap D l “ H we have F l ` ´ F l “ { e l } for some e l P E . Furthermore, theunion of the gaps is Y “ E ´ { e l : D l “ H } .(3) For all ď i ď k we have | F i ` ´ F i | “ p r i ` ´ r i q ` p r K i ´ r K i ` q (4.3.2)where we denote r j “ r M p F j q and r K j “ r M K p G j q . Proof of Lemma 4.15.
1. First let us prove that | Y | ď n ` ´ z. (4.3.3)For each empty gap D l “ H , choose an element e l P F l ` ´ F l . Since e l R D l “ E ´ p F l Y G l ` q ,we must have e l P G l ` . This implies that e l P F l ` X G l ` , so (4.3.1) gives e l R Y . There are z such elements e l , which are all distinct by construction; this implies (4.3.3).Now let us prove the opposite inequality | Y | ě n ` ´ z. (4.3.4)We obtain F i ` | G i ` from F i | G i by choosing the largest gap element e “ max Y i , finding theunique gap D i,j of F i | G i containing e , and inserting a new pair F | G with e P F X G between the j th and p j ` q th rays of F i | G i , as follows: F i ` | G i ` : ¨ ¨ ¨ Ď F i,j Ď F Ď F i,j ` Ď ¨ ¨ ¨¨ ¨ ¨ Ě G i,j Ě G Ě G i,j ` Ě ¨ ¨ ¨
Thus the only change between the gaps of F i | G i and F i ` | G i ` is that we are replacing the gap D i,j with two smaller disjoint gaps D i ` ,j and D i ` ,j ` that do not contain e : D i,j “ E ´ p F i,j Y p G i,j ` q ÞÝÑ D i ` ,j “ E ´ pp F i,j Y G q D i ` ,j ` “ E ´ p F Y p G i ` ,j ` q . We have D i,j Ě D i ` ,j \ D i ` ,j ` \ e. (4.3.5)In the end, the final biflag F n ´ | G n ´ has n gaps, of which n ´ are empty and one of them, say D , has size at least .It is helpful to visualize this data as a graded forest of levels k, k ` , . . . , n ´ . The verticesof the top level k are the gaps D , . . . , D k of the original biflag F i | G i ; they are the roots of the k ` trees in the forest. The vertices of the i th level are the gaps D i, , . . . , D i,i of F i | G i . To gofrom level i to level i ` , we connect the split gap D i,j with the gaps D i ` ,j and D i ` ,j ` thatreplace it. Every other gap D i,k is connected to the gap in the next level that is equal to it; this is D i ` ,k if k ă j and D i ` ,k ` if k ą j .Each gap of F ` | G ` “ F n ´ | G n ´ , at the bottom level of the tree, descends from one of theoriginal gaps of F | G “ F k | G k through successive gap replacements. Let d l “ number of gaps of F ` | G ` that descend from the initial gap D l of F | G , for ď l ď k . We consider three cases:Case 1. D l “ H :In this case the gap D l eventually becomes a single empty gap in F ` | G ` , so d l “ .Case 2. D l ‰ H is the progenitor of the unique non-empty gap D of F ` | G ` :Consider the gaps that descend from D l throughout this process. By (4.3.5), every time one suchgap gets replaced by two smaller ones, the size of the union of the gaps strictly decreases. Inthe end this union has size | D | ě . Therefore these gaps were split at most | D l | ´ times, so d l ď | D l | ´ .Case 3. D l ‰ H is not the progenitor of the non-empty gap D :Again, every time a descendant of D l gets replaced by two smaller ones, the size of their uniondecreases. Furthermore, their union can never have size by Proposition 3.14. Thus d l ď | D l | .Since the final number of gaps is n , we conclude that n “ k (cid:88) l “ d l ď z ` (cid:88) l : D l ‰H | D l | ´ “ z ` | Y | ´ , where z is the number of empty gaps D l . This proves (4.3.4). AGRANGIAN GEOMETRY OF MATROIDS 43
Since the two opposite inequalities (4.3.3) and (4.3.4) and hold, we must have | Y | “ n ` ´ z, proving part 1. of the lemma. Furthermore, every inequality we applied along the way must infact have been an equality. Let us record these:a) For (4.3.3) to be an equality, we must have F l ` ´ F l “ { e l } for each empty gap D l “ H , and Y “ E ´ { e l : D l “ H } .b) For (4.3.4) to be an equality, we must have d l “ in case 1, d l “ | D l | ´ in case 2, d l “ | D l | in case 3. (4.3.6)We use this to prove (4.3.2), in two steps. First we prove that d l “ | F l ` ´ F l | if D l is in case 1 or 3 above, | F l ` ´ F l | ´ if D l is in case 2. (4.3.7)Case 1. D l “ H :In this case we have d l “ , and | F l ` ´ F l | “ by a).Cases 2 and 3. D l ‰ H .We claim that D l “ F l ` ´ F l (4.3.8)which will imply the claim by b). The forward inclusion holds by definition. For the backwardinclusion, consider e P F l ` ´ F l . By a) we must have e P Y and since D l is the only gapintersecting F l ` ´ F l , we must have e P D l .Next we prove that d l “ p r l ` ´ r l q ` p r K l ´ r K l ` q if D l is in case 1 or 3 p r l ` ´ r l q ` p r K l ´ r K l ` q ´ if D l is in case 2. (4.3.9)Case 1 and 3. D l is not the progenitor of the double gap D :In these cases, the part of F ` | G ` between F l | G l and F l ` | G l ` contains no double jumps. Ineach of the d l single jumps, either the rank increases by 1 or the corank decreases by 1, but notboth. Therefore d l must equal the sum of the rank increase r l ` ´ r l and the corank decrease r K l ´ r K l ` .Case 2. D l is the progenitor of the double gap D :In these cases, the part of F ` | G ` between F l | G l and F l ` | G l ` contains one double jump. Ineach of the d l ´ single jumps, either the rank increases by 1 or the corank decreases by 1, butnot both. In the double jump, both changes occur. Therefore d l ` must equal the sum of therank increase r l ` ´ r l and the corank decrease r K l ´ r K l ` . The desired result now follows from (4.3.7) and (4.3.9). (cid:3)
With Lemma 4.15 at hand, we are finally ready to prove the Vanishing Lemma 4.14.First, we prove that G “ F K . One readily verifies, using the rank function of the dual matroid,that p r i ` ´ r i q ` p r M K p E ´ F i q ´ r M K p E ´ F i ` qq “ | F i ` ´ F i | for ď i ď k .By Lemma 4.15(3), the sequences p r M K p E ´ F i q : 0 ď i ď k q and p r K i : 0 ď i ď k q satisfy thesame recurrence; they also have the same initial value r M K p E ´ F q “ r K “ r K since F “ H and G “ E . We conclude that r M K p E ´ F i q “ r M K p G i q for ď i ď k .But F i Y G i “ E implies G i Ě E ´ F i , and since G i is a coflat, G i Ě cl K p E ´ F i q “ F K i . It followsthat G i Ě F K i are flats of the same rank in M K , so G i “ F K i for all i as desired.Next, we prove that every long interval M p i q “ M r F i ´ , F i s is loopless and coloopless. Weproceed by contradiction.First assume that M p i q “ p M { F i ´ q|p F i ´ F i ´ q has a loop l . Since restriction cannot createnew loops, l must also be a loop of M { F i ´ . This contradicts the fact that F i ´ is a flat.Now assume that M p i q “ p M | F i q{ F i ´ has a coloop c . Since contraction cannot create newcoloops, c must also be a coloop of M | F i . Thus r M p F i ´ c q “ r M p F i q ´ , which implies that r M K pp E ´ F i q Y c q “ r M K p E ´ F i q . This means that c P cl K p E ´ F i q “ F K i .Now, since M p i q is long, Lemma 4.15(2) implies that D i ‰ H and that c P Y . But then wemust have c P D i “ p F i ´ F i ´ q X p F K i ´ ´ F K i q , contradicting that c P F K i . The desired resultfollows. (cid:3) The beta invariant of a flag in its conormal intersection theory.
In this section we com-plete the proof that deg p π ˚ p x F q δ n ´ k ´ q “ β p M r F sq for any strictly increasing flag of flats F in M of length k . We will first need a lemma relating the conormal fan of M with that of the deletion M { i .Let i be an arbitrary element of E ; recall that M has no coloops, so i K “ E and i | E is a biflatof M . The ambient space of st i | E Σ M , M K is p N E ‘ N E q{p e i ` f E q “ p N E { e i q ‘ N E . We let e S bethe image of e S in N E { e i for S Ď E . We also let x F | G be the variable in the Chow ring of thestar corresponding to a ray F | G ; we set it equal to if F | G is not a ray in this star. Lemma 4.16.
Consider the natural projection ψ : p N E { e i q ‘ N E ÝÑ N E ´ i ‘ N E ´ i .(1) The projection ψ induces a morphism of fans ψ : st i | E Σ M , M K ÝÑ Σ M { i, p M { i q K . AGRANGIAN GEOMETRY OF MATROIDS 45 (2) The corresponding pullback of Chow rings ψ ˚ : A M { i, p M { i q ˚ Ñ A p st i | E Σ M , M K q is given by ψ ˚ p x A | B q “ x p A Y i q| B ` x p A Y i q|p B Y i q , A | B biflat of M { i, where at least one of the terms in the right hand side is nonzero.(3) The pullback ψ ˚ maps the class δ of A M { i, p M { i q K to the following class of A p st i | E Σ M , M K q : δ – ψ ˚ p δ q “ (cid:88) F | G biflat of M i P F, j P F,G x F | G for any j P E. (4) The pullback ψ ˚ commutes with the degree maps of A M { i, p M { i q K and A p st i | E Σ M , M K q ; thatis, deg M { i η “ deg st p ψ ˚ η q for all η P A n ´ { i, p M { i q K . Proof.
1. The image of a ray F | G in the star is ψ p e F ` f G q “ e F ´ i ` f G ´ i , F | G biflat , i P F, which is a ray of the conormal fan Σ M { i, p M { i q K because p F ´ i q|p G ´ i q is a biflat of M { i : cl M { i p F ´ i q “ cl M p F q ´ i “ F ´ i, and cl M K ´ i p G ´ i q “ cl M K p G ´ i q ´ i Ď cl M K p G q ´ i “ G ´ i. Furthermore, if i | E Y F | G is a biflag of M , its gaps occur to the right of i | E , and there will alsobe gaps in the corresponding positions of p F ´ i q|p G ´ i q – { p F ´ i q|p G ´ i q : F | G P F | G} ; sothis will be a biflag of M { i . Therefore ψ maps cones to cones.2. The value of the piecewise linear function ψ ˚ x A | B on a ray e F ` f G of the star is ψ ˚ x A | B p e F ` f G q “ x A | B p e F ´ i ` f G ´ i qq “ if F “ A Y i and G P { B, B Y i } , or otherwise , taking into account that we must have i P F . The fact that B is a flat of M { i implies that cl p B q P { B, B Y i } , so at least one of the summands is nonzero.3. We have ψ ˚ p δ j q “ (cid:88) A | B biflat of M { ij P A,B p x p A Y i q| B ` x p A Y i q|p B Y i q q “ (cid:88) F | G biflat of M i P F, j P F,G x F | G “ δ j .
4. We need to verify that deg M { i x A | B “ deg st ψ ˚ p x A | B q – deg M p x i | E ψ ˚ p x A | B qq for any maximal biflag A | B of M { i . Writing ψ ˚ p x A | B q “ x p A Y i q| B ` x p A Y i q|p B Y i q for each A | B in A | B , we express x i | E ψ ˚ p x A | B q as a sum of squarefree monomials. One of the terms in thisexpression is x i | E x p A Y i q| cl K p B q , where p A Y i q| cl K B – { p A Y i q| cl K p B qq : A | B P A | B} . We needto prove this is the only nonzero term. Consider any term x i | E x A | B that arises in this expression. We automatically have A j “ A j Y i for all j , so it remains to prove B j “ cl K p B j q for all j as well.Let k be the largest index such that i P cl K p B k q . Then cl K p B j q “ B j Y i for j ď k whereas cl K p B j q “ B j for j ě k ` . For ď j ď k , B j is not a flat in M K , so B j “ B j Y i “ cl K p B j q Now, notice that B k and B k ` are flats of consecutive ranks in p M { i q K “ M K ´ i , so the flats B k Y i “ cl K p B k q and B k ` “ cl K p B k ` q of M K also have consecutive ranks. Therefore B k ` Y i ,which is strictly between them, cannot be a flat. Thus we must have B k ` “ B k ` , and hence B j “ B j “ cl K p B j q for j ě k ` as well. We conclude that A | B “ p A Y i q| cl K p B q as desired. (cid:3) Now we can give an intersection-theoretic interpretation of the beta invariant of a flag.
Proposition 4.17.
Let F “ { F Ĺ ¨ ¨ ¨ Ĺ F k } be a strictly increasing flag of flats of M . We have deg p x F | F K δ n ´ k ´ q “ β p M r F sq . Proof.
We proceed by induction on k . The case k “ is Proposition 4.8. For k ě , let F | F K bethe first biflat in F | F K , and write F | F K “ F | F K Y G | G K . Then G ´ F – { G ´ F : G P G} is aflag of flats in M { F . It leads to the flag of biflats of M { F p G ´ F q|p G ´ F q K – { p G ´ F q|p G K ´ F q : G P G} , where the notation is justified by the fact that G K ´ F “ cl p M { F q K pp E ´ F q ´ p G ´ F qq for G Ě F .We have β p M r F sq “ β p M | F q ¨ β pp M { F qr G ´ F sq because M r G j ´ , G j s – p M { F qr G j ´ ´ F, G j ´ F s for j “ , . . . , k ´ . We consider two cases:Case 1. F “ { i } for some i P E .Since β p M rH , i sq “ , we have β p M r F sq “ β pp M { i qr G ´ i sq“ deg M { i (cid:0) x G ´ i |p G ´ i q K δ p n ´ q´p k ´ q´ { i (cid:1) by the inductive hypothesis “ deg st (cid:0) ψ ˚ p x G ´ i |p G ´ i q K q δ n ´ k ´ (cid:1) by Lemma 4.16(3) and (4) “ deg M (cid:0) x i | E ψ ˚ p x G ´ i |p G ´ i q K q δ n ´ k ´ q (cid:1) since x i | E x F ,G “ for i R F “ deg M (cid:0) x i | E (cid:89) G P G p x G |p G K ´ i q ` x G | G K q δ n ´ k ´ (cid:1) by Lemma 4.16.2 “ deg M (cid:0) x i | E x G | G K δ n ´ k ´ (cid:1) by the Vanishing Lemma 4.14 “ deg M (cid:0) x F | F K δ n ´ k ´ (cid:1) . Case 2. | F | ą . AGRANGIAN GEOMETRY OF MATROIDS 47
By the Vanishing Lemma 4.14, we may assume the interval rH , F s is coloopless. This meansthat the flat F is cyclic ; that is, E ´ F is a coflat, and F K “ E ´ F . Then we have bijections φ : { biflats of M | F } ÝÑ { biflats F | G of M with F Ď F and G Ě E ´ F } φ : { biflats of M { F } ÝÑ { biflats F | G of M with F Ě F and G Ď E ´ F } given by φ p A | B q “ A |p B Y p E ´ F qq and φ p A | B q “ p A Y F q| B . These extend to bijections φ (resp. φ ) between the biflags of M | F (resp. M { F ) and the biflags of M that are supported onthe corresponding set of biflats, and have a gap to the left (resp. to the right) of F |p E ´ F q .Now let us compute deg p x F | F K δ n ´ k ´ q using the following variant of the canonical expan-sion of Definition 4.1, which proceeds in two stages:Stage 1. At each step, choose e to be the largest gap element that is in F , if there is one.Stage 2. At each step, choose e to be the largest gap element in E ´ F .The first | F | ´ steps of this computation will give x F | F K times the image under φ of thecanonical expansion of δ | F |´ M | F . By Proposition 4.8, there will be β p M | F q squarefree monomials.Each such monomial will have a unique non-empty gap before F ; say it is D j , between biflats F j | G j and F j ` | G j ` of M , where F j and F j ` (resp. G j and G j ` ) have consecutive ranks (resp.coranks). In step | F | ´ of the computation, this gap D j will be filled in a unique way by thebiflat F j ` | G j . There will no longer be gap elements in F .In step | F | , the computation will enter Stage 2 for each of the resulting β p M | F q monomials.The following p| E ´ F | ´ q ´ p k ´ q ´ steps will compute the image under φ of the canonicalexpansion of x p G ´ F q|p G ´ F q K δ | F |´ M { F . This expansion has β pp M { F qr G ´ F sq squarefree monomials,by the inductive hypothesis.Since r| F | ´ s ` ` rp| E ´ F | ´ q ´ p k ´ q ´ s “ n ´ k ´ , this will conclude the computationof x F | F K δ n ´ k ´ . The result will be the sum of β p M | F q β pp M { F qr G ´ F sq “ β p M r F sq squarefreemonomials, as we wished to prove. (cid:3) Proposition 4.18.
Let F “ { F Ĺ ¨ ¨ ¨ Ĺ F k } be a strictly increasing flag of flats of M . We have deg p π ˚ p x F q δ n ´ k ´ q “ β p M r F sq . Proof.
Since π ˚ p x F q “ (cid:80) F | G biflag x F | G , this follows from Lemma 4.14 and Proposition 4.17. (cid:3)
5. A
CONORMAL INTERPRETATION OF THE C HERN –S CHWARTZ –M AC P HERSON CYCLES
Recall that the k -dimensional Chern–Schwartz–MacPherson cycle of M is the Minkowskiweight csm k p M q on the Bergman fan of M defined by the formula csm k p M qp σ F q “ p´ q r ´ k β p M r F sq , where σ F is the k -dimensional cone corresponding to a flag of flats F of M . Theorem 1.1.
When M has no loops and no coloops, for every nonnegative integer k ď r , csm k p M q “ p´ q r ´ k π ˚ p δ n ´ k ´ X M , M K q . Proof.
We have β p M r F sq “ deg Σ M , M K p π ˚ p x F q δ n ´ k ´ q by Proposition 4.18 , “ p π ˚ p x F q δ n ´ k ´ q X M , M K by Definition 3.4 , “ x F X π ˚ p δ n ´ k ´ X M , M K q by the projection formula , “ π ˚ p δ n ´ k ´ X M , M K qp σ F q . The result then follows by the definition of the Chern–Schwartz-MacPherson cycle of M . (cid:3) The following property of the Chern–Schwartz–MacPherson cycles of matroids generalizes[Alu13, Theorem 1.2].
Proposition 5.1 ([LdMRS20], Thm. 5.8) . For each ď k ď r , we have α k X csm k p M q “ p´ q r ´ k h r ´ k p BC p M qq Theorem 1.2.
When M has no loops and no coloops, we have χ M p q ` q “ r (cid:88) k “ p´ q r ´ k deg p γ k δ n ´ k ´ q q k . Proof.
For each ď k ď r , h r ´ k p BC p M qq “ p´ q r ´ k α k X csm k p M q by Proposition 5.1 , “ α k X π ˚ p δ n ´ k ´ X M , M K q by Theorem 1.1 , “ π ˚ (cid:0) π ˚ α k X p δ n ´ k ´ X M , M K q (cid:1) by the projection formula , “ π ˚ (cid:0) γ k δ n ´ k ´ X M , M K (cid:1) , as desired. (cid:3)
6. T
ROPICAL H ODGE THEORY
Lefschetz fans.
For a simplicial fan Σ in a vector space N , we continue to let K p Σ q Ď A p Σ q denote the cone of strictly convex piecewise linear functions on Σ . We recall that the Lefschetzproperty for Σ (Definition 1.5) involves five conditions. A Lefschetz fan has (1) a fundamentalweight w P MW d p Σ q which induces Poincaré duality (2). We shall abbreviate the latter by PD.The Hard Lefschetz property (3) and Hodge–Riemann relations (4) are statements that hold forall ď k ď d { and all (cid:96) P K p Σ q : we will call those statements HL k p (cid:96) q or HR k p (cid:96) q , respectively,and say that Σ satisfies HL k if HL k p (cid:96) q is true for all (cid:96) P K p Σ q , and that Σ satisfies HL if it satisfiesHL k for all k . We will use HR k and HR k p (cid:96) q analogously. If K p Σ q is empty, then of course the AGRANGIAN GEOMETRY OF MATROIDS 49
HL and HR properties hold vacuously. The hereditary property (5) says that stars of cones inLefschetz fans are also Lefschetz.
Definition 6.1 (Mixed Lefschetz) . We say that(3 ) Σ has the mixed Hard Lefschetz property if, for all ď k ď d { and (cid:96) , . . . , (cid:96) d ´ k P K p Σ q , themultiplication map L ¨ : A k p Σ q Ñ A d ´ k p Σ q is an isomorphism, where L – (cid:96) ¨ ¨ ¨ (cid:96) d ´ k , and(4 ) Σ satisfies the mixed Hodge–Riemann relations if, for all ď k ď d { , all (cid:96) P K p Σ q and all L as above, the bilinear form on A k p Σ q defined by (cid:104) u , u (cid:105) L – p´ q k deg p L ¨ u u q is positive-definite when restricted to the subspace PA k p Σ , L, (cid:96) q – ker p (cid:96)L ¨q .Clearly the mixed properties imply the ordinary ones. Cattani showed that the converse istrue as well in [Cat08] using the results from [CKS87]. Since the mixed HR property is partic-ularly convenient for applications such as Theorem 1.4, we include a self-contained proof thatLefschetz fans also possess the “mixed” properties p q and p q ; see Theorem 6.20. Example . If Σ is a complete, unimodular, polyhedral simplicial fan, then Σ is Lefschetz. Inthis case, A k p Σ q – H k p X Σ , R q for all k , where X Σ is the normal projective toric variety con-structed from the fan Σ . Here, the Lefschetz properties follow because X Σ is a smooth projectivevariety, and K p Σ q is the cone of Kähler forms on X Σ .6.2. The ample cone.
Let us look at the cone K p Σ q in more detail. We say a piecewise linearfunction φ : Σ Ñ R is positive on Σ if φ p x q ą for all non-zero x P | Σ | , and say an equivalenceclass (cid:96) P A p Σ q is positive if it has a positive representative. The (open) effective cone , is definedto be the set Eff ˝ p Σ q Ď A p Σ q of positive classes.For each cone σ of Σ , the subfan st Σ p σ q Ď Σ maps to the star st Σ p σ q under the linear projection N Ñ N { span p σ q . This map is a Chow equivalence, so we will identify the Chow rings of st Σ p σ q and st Σ p σ q , and let ι ˚ σ : A p Σ q Ñ A p st Σ p σ qq denote pullback along the inclusion. Definition 6.3. If Σ is a Lefschetz fan, the Kähler (or ample) cone of Σ is defined recursively: if Σ is -dimensional, then K p Σ q “ Eff ˝ p Σ q . Otherwise, K p Σ q – (cid:8) (cid:96) P A p Σ q : (cid:96) P Eff ˝ p Σ q and ι ˚ σ p (cid:96) q P K p st Σ p σ qq for all σ P Σ (cid:9) . Clearly, (cid:96) P K p Σ q if and only if ι ˚ σ p (cid:96) q P Eff ˝ p st Σ p σ qq for all σ P Σ . Geometrically, this meansthat (cid:96) is in the Kähler cone if and only if, for each cone σ , (cid:96) has a piecewise linear representative φ supported on st Σ p σ q which is zero on σ and positive on the cones containing σ . That is, (cid:96) isthe class of a piecewise linear function which is strictly convex around each σ . Proposition 6.4.
The set K p Σ q is an open polyhedral cone, and ι ˚ σ K p Σ q Ď K p st Σ p σ qq for all σ P Σ . Proof.
The property of being a polyhedral cone is preserved under finite intersections. Thesecond claim follows from the definition. (cid:3)
A fan Σ is quasiprojective if it is a subfan of the normal fan of a (strictly convex) polytope. If Σ is quasiprojective, the cone K p Σ q is nonempty.If we replace strict inequalities with weak ones above, we arrive instead at the nef cone L Σ de-fined in [GM12]. This is a nonempty, closed polyhedral cone in A p Σ q . If L Σ is full-dimensional,then K p Σ q is the interior of L Σ . Otherwise, K p Σ q is empty.6.3. Stellar subdivisions.
Now we focus on the effect of a single blowup of a toric variety alonga torus orbit closure. On the level of fans, this is realized by a stellar subdivision. More precisely,suppose Σ is simplicial, σ P Σ is a cone, and V p σ q denotes the corresponding closed orbit. Werecall (see [CLS11, §3.3]) that Bl V p σ q p X Σ q “ X (cid:101) Σ , where the fan (cid:101) Σ – stellar σ Σ . Let ρ be theunique element of (cid:101) Σ p q ´ Σ p q : then e ρ “ (cid:88) η P σ p q e η , (6.3.1)where e ν denotes a primitive vector generating the ray ν . Let p : (cid:101) Σ Ñ Σ be the map of fansinduced by the identity map on N . Definition 6.5.
At this point, we distinguish two possibilities. In the first, every closed orbit in X Σ meets V p σ q . In terms of fans, this means Σ “ st Σ p σ q , and (cid:101) Σ “ st (cid:101) Σ p ρ q . In this case, A p Σ q – A p st Σ p σ qq and A p (cid:101) Σ q – A p st (cid:101) Σ p ρ qq , which are Chow rings of fans of dimensions dim p Σ q ´ d and dim p Σ q ´ , respectively, where d “ dim p σ q . We will call this a star-shaped subdivision (obtainedby blowing up a star.) (cid:101) Σ . Otherwise, we will say the stellar subdivision is ordinary .The star-shaped subdivision has an alternative interpretation. The stars st Σ p σ q and st (cid:101) Σ p ρ q arefans in N { (cid:104) σ (cid:105) and N { (cid:104) ρ (cid:105) , respectively. The quotient map N { (cid:104) ρ (cid:105) Ñ N { (cid:104) σ (cid:105) induces a map offans st (cid:101) Σ p ρ q Ñ st Σ p σ q . The corresponding map of toric varieties is a P d ´ -bundle. We refer to[CLS11, §3.3] for details. For trivial reasons, Σ and (cid:101) Σ cannot be Lefschetz; however, st (cid:101) Σ p ρ q and st Σ p σ q may be.Now we relate the Chow rings of Σ and (cid:101) Σ . We continue to let ρ denote the ray that subdividesthe cone σ . Recall that p ˚ : A p Σ q Ñ A p (cid:101) Σ q gives A p (cid:101) Σ q the structure of a A p Σ q -module. Since p isa proper map of fans (see [CLS11, Thm. 3.4.11]), there is a Gysin pushforward map p ˚ : A p (cid:101) Σ q Ñ A p Σ q which is a homomorphism of A p Σ q -modules. A special case of Brion’s formula [Bri96,Thm. 2.3] states that, for each cone (cid:101) τ , p ˚ p x (cid:101) τ q “ x τ if p p (cid:101) τ q Ď τ and dim (cid:101) τ “ dim τ ; otherwise. (6.3.2) AGRANGIAN GEOMETRY OF MATROIDS 51
Lemma 6.6.
The pullback homomorphism p ˚ : A p Σ q Ñ A p (cid:101) Σ q is defined in degree by theformula x ν ÞÑ x ν if ν R σ p q ; x ν ` x ρ if ν P σ p q . (6.3.3) Proof.
Since A p Σ q is generated in degree , it is sufficient to check that p induces the map ofpiecewise linear functions in (6.3.3). To avoid confusion, we temporarily denote the Courantfunctions on (cid:101) Σ by (cid:110)(cid:101) x ν : ν P (cid:101) Σ p q (cid:111) .Consider the function p ˚ x ν on the fan (cid:101) Σ . For rays ν not in σ , clearly p ˚ x ν “ (cid:101) x ν . For ν P σ p q ,we check that the functions p ˚ x ν and (cid:101) x ν ` (cid:101) x ρ agree on each ray µ P (cid:101) Σ p q : since they are piecewiselinear, this implies they are equal. Indeed, for µ ‰ ρ , we have p ˚ x ν p e µ q “ (cid:101) x ν p e µ q “ δ µ,ν and (cid:101) x ρ p µ q “ . For µ “ ρ , (cid:101) x ν p e ρ q ` (cid:101) x ρ p e ρ q “ ` “ x ν p e ρ q , because x ν is linear on σ and the coefficient of e ν in e ρ equals , by (6.3.1). (cid:3) Proposition 6.7. If (cid:101) Σ is a stellar subdivision of Σ , then p ˚ : A k p Σ q Ñ A k p (cid:101) Σ q is injective for all k ,and an isomorphism for k “ d . Proof.
Using the formula (6.3.2) for pushforward and Lemma 6.6 for pullback, we see p ˚ p ˚ isthe identity function on A p Σ q . Now A p Σ q is generated by as an A p Σ q -module, so p ˚ p ˚ “ inall degrees.It follows that p ˚ is injective. To check that it is also surjective in top degree, we check thedual statement instead, that p ˚ : MW d p (cid:101) Σ q Ñ MW d p Σ q is injective. For this, let w P MW d p Σ q be a non-zero Minkowski weight on the maximal cones of (cid:101) Σ . For σ P Σ p d q , we have p ˚ p w qp σ q “ w p (cid:101) σ q , provided that p p (cid:101) σ q Ď σ . Clearly p ˚ p w q ‰ , so p ˚ is injective. (cid:3) Our goal in the next few pages is to understand how the Lefschetz property behaves underedge subdivisions, so our first step is the Chow ring.
Theorem 6.8.
Let p : (cid:101) Σ Ñ Σ be the map of fans given by subdividing an edge σ P Σ p q with aray ρ P (cid:101) Σ p q . There is an isomorphism of graded A p Σ q -modules A i p (cid:101) Σ q – A i p Σ q ‘ x ρ ¨ A i ´ p st Σ p σ qq , (6.3.4)for all i ě . To prove Theorem 6.8, we consider the subdivision of Σ , restricted to the star of σ . It is nothard to see that p : (cid:101) Σ Ñ Σ restricts to a star-shaped subdivision: the star of σ within Σ : st (cid:101) Σ p ρ q (cid:101) Σst Σ p σ q Σ p σ j pi σ (6.3.5)Keel [Kee92, Thm. 1 (Appendix)] relates the Chow rings of the star-shaped subdivision. Lemma 6.9.
For any cone σ P Σ p k q with k ě , let (cid:101) Σ “ stellar σ p Σ q . Then there is an algebraisomorphism A p st (cid:101) Σ p ρ qq – A p st Σ p σ qqr t s{p t k ` c t k ´ ` ¨ ¨ ¨ ` c k q induced by the pullback p ˚ σ , where c i P A i p st Σ p σ qq are the Chern classes of the normal bundle of V p σ q in X st Σ p σ q , for ď i ď k . Under the isomorphism, x ρ ÞÑ ´ t . Proof of Theorem 6.8.
Let p : (cid:101) Σ Ñ Σ be an edge subdivision, and apply the Chow functor to thesquare (6.3.5). The maps i ˚ σ and j ˚ are surjective, since they are clearly surjective in degree .The vertical maps are injective, by Proposition 6.7. We obtain short exact sequences of A p Σ q -modules, where J i and C i for i “ , denote the respective kernels and cokernels: C C A p st (cid:101) Σ p ρ qq A p (cid:101) Σ q J A p st Σ p σ qq A p Σ q J – j ˚ p ˚ p ˚ σ i ˚ σ p ˚ – From [Kee92], it follows that J – J , so C – C by the Snake Lemma. Lemma 6.9 says C – x ρ ¨ A p st Σ p σ qq . To see that p ˚ is a split injection, we recall that p ˚ is a left inverse to p ˚ . (cid:3) Remark . We have used a pushforward of Minkowski weights for any map f : Σ Ñ Σ offans, and a Gysin pushforward of Chow groups which is defined only for proper maps. If Σ and Σ have the same dimension, these can be shown to agree. More generally, if Σ and Σ havePoincaré duality in dimensions d and d , respectively, the pushforward f ˚ : MW p Σ q Ñ MW p Σ q gives a map f ˚ : A p Σ q Ñ A p Σ qr d ´ d s via the Poincaré duality isomorphisms MW i p Σ q – A d ´ i p Σ q and MW i p Σ q – A d ´ i p Σ q for all i .In particular, if σ is a k -dimensional cone in a Lefschetz fan Σ , then by definition both st Σ p σ q and Σ satisfy PD. So we obtain a pushforward map i σ ˚ : A p st Σ p σ qq Ñ A p Σ qr k s . AGRANGIAN GEOMETRY OF MATROIDS 53
It has the property that i σ ˚ i ˚ σ : A p Σ q Ñ A p Σ qr s is given by multiplication by x σ .Finally, we note that the pullback of K p Σ q lies in the boundary of K p (cid:101) Σ q along an edge subdi-vision. Lemma 6.11. If (cid:96) P p ˚ p K p Σ qq , then (cid:96) ´ (cid:15) ¨ x ρ P K p (cid:101) Σ q for sufficiently small values of (cid:15) ą . Proof. If τ is a cone of (cid:101) Σ which belongs to Σ , then st (cid:101) Σ p τ q Ď st Σ p τ q . So (cid:96) is the class of a strictlyconvex function φ around τ . Strict convexity is an open condition, so φ (cid:15) – φ ´ (cid:15) ¨ x ρ has thesame property for (cid:15) sufficiently close to .Otherwise, τ contains the ray ρ , so τ is not a cone of Σ , and (cid:96) is the class of a linear function φ on the closed star of τ . In that case, φ (cid:15) agrees with φ on the link of τ , and is strictly smallerinside τ , provided (cid:15) ą . That is, φ (cid:15) is strictly convex around τ .Combining the conditions, (cid:96) ´ (cid:15) ¨ x ρ P K p (cid:101) Σ q for sufficiently small, positive (cid:15) . (cid:3) Signatures of Hodge–Riemann forms.
Suppose that multiplication by some element L P A d ´ k p Σ q is an isomorphism in degree k . One can check directly that the real bilinear form hr k p Σ , L q is nondegenerate, which is to say that it has b ` i positive eigenvalues and b ´ i negativeeigenvalues, where b ` i ` b ´ i “ b i p Σ q – dim R A i p Σ q . Its signature, b ` i ´ b ´ i , can be used tocharacterize the HR property. This useful fact appears as [AHK18, Prop. 7.6], as well as [McM93,Thm. 8.6] in the case when L “ (cid:96) d ´ k . Proposition 6.12.
Suppose Σ satisfies PD, and U Ď A d ´ k p Σ q is a connected set in the Euclideantopology. For a fixed k ď d { , if hr k p Σ , L q is nondegenerate on A k p Σ q for all L P U , then thesignature of hr k p Σ , L q is constant for all L P U . Proof.
The eigenvalues of hr k p Σ , L q are real, and they vary continuously with L . By hypothesis,they are all non-zero for L P U , so their signs (taken as a multiset) are constant on U , because U is connected. (cid:3) Theorem 6.13 (The HR signature test) . Suppose Σ satisfies PD and k ď d { is an integer forwhich(1) hr i p Σ , L q is nondegenerate for all ď i ď k and all L P Sym d ´ i K p Σ q , and(2) hr i p Σ , L q is positive-definite on P A i p Σ , L, (cid:96) q , for all (cid:96) P K p Σ q and i ă k .Then, for any L P Sym d ´ k K p Σ q , the form hr k p Σ , L q is positive-definite on PA k p Σ , L, (cid:96) q for all (cid:96) P K p Σ q if and only if its signature on A k p Σ q equals k (cid:88) i “ p´ q k ´ i (cid:0) b i p Σ q ´ b i ´ p Σ q (cid:1) . (6.4.1) Proof.
In the special case where L “ (cid:96) d ´ k , we refer to [AHK18, Prop. 7.6]. Since K p Σ q is con-nected, and U – Sym d ´ k K p Σ q is a quotient of K p Σ q ˆp d ´ k q , the set U is also connected. So ageneral Lefschetz element has the same signature as (cid:96) d ´ k , by Proposition 6.12, so hr k p Σ , L q ispositive-definite on its space of primitives because hr k p Σ , (cid:96) d ´ k q is. (cid:3) We note that, if L passes the signature test above, then b i p Σ q ´ b i ´ p Σ q “ dim PA k p Σ , L, (cid:96) q ,for any (cid:96) . Schematically, A p Σ q looks like ` `` `´ ´ ¨ ¨ ¨ A PA ¨ ¨ ¨ A PA ¨ ¨ ¨ A PA Corollary 6.14.
Let Σ be a fan of dimension d satisfying Poincaré duality. Let k ď d { . Suppose Σ satisfies mixed HR i for all i ă k , mixed HL k , as well as HR k p L q for some L P Sym d ´ k K p Σ q .Then Σ satisfies HR k . Proof.
Let L P Sym d ´ k K p Σ q be any element. By the Hard Lefschetz hypothesis, hr k p Σ , L q isnondegenerate. By Proposition 6.12, it has the same signature as hr k p Σ , L q . Since we assume Σ satisfies mixed HR i for i ă k , Theorem 6.13 shows HR k p L q ô HR k p L q . (cid:3) In the special case of a star-shaped blowup, the signature test simplifies slightly. Let ∆ “ st Σ p σ q and (cid:101) ∆ “ st (cid:101) Σ p ρ q , where ∆ has dimension d . Corollary 6.15.
An element L of degree k ď p d ` q{ has the HR property for (cid:101) ∆ if and only ifthe signature of hr k p (cid:101) ∆ , L q equals b k p ∆ q ´ b k ´ p ∆ q . Proof.
By Theorem 6.8, we have b k p (cid:101) ∆ q “ b k p ∆ q ` b k ´ p ∆ q for all k ď p d ` q{ . Substituting into(6.4.1) simplifies as shown. (cid:3) Lefschetz properties under edge subdivision I.
With these preparations, we now set outto show that the Lefschetz property of a fan is unaffected by codimension- blowups and blow-downs. The precise statement and its proof appear in Section 6.7 as Theorems 6.26 and 6.27.Here, we get started with Poincaré duality, and we do so for star-shaped subdivisions first. Proposition 6.16.
Let Σ be a simplicial fan, σ P Σ p q , and (cid:101) Σ – stellar σ p Σ q . Then PD holds for st Σ p σ q if and only if it holds for st (cid:101) Σ p ρ q . Proof.
Let ∆ “ st Σ p σ q and (cid:101) ∆ “ st (cid:101) Σ p ρ q . Assume that PD holds for at least one of them, and let d “ dim (cid:101) ∆ . By Theorem 6.8, for all i ě , A i p (cid:101) ∆ q – A i p ∆ q ‘ x ρ A i ´ p ∆ q . AGRANGIAN GEOMETRY OF MATROIDS 55
We see that A d ´ p ∆ q – A d p (cid:101) ∆ q , so if one of ∆ or (cid:101) ∆ has a fundamental weight, they both do. Byinspection, b i p ∆ q “ b d ´ ´ i p ∆ q for all i if and only if b i p (cid:101) ∆ q “ b d ´ i p (cid:101) ∆ q for all i . So we may assumeboth sets of equalities hold.For any u P A i p (cid:101) ∆ q and v P A d ´ i p (cid:101) ∆ q , we write u “ u ` u x ρ and v “ v ` v x ρ where u , u , v , v are elements of A p ∆ q of degrees i , i ´ , d ´ i , and d ´ ´ i , respectively. Then u v P A d p ∆ q “ , and x ρ “ c ¨ x ρ ` c for some c , c P A p ∆ q . With respect to the decompositionabove, the matrix of the multiplication pairing has the form M i p (cid:101) ∆ q “ (cid:32) ´ M i ´ p ∆ q´ M i p ∆ q ˚ (cid:33) , (6.5.1)where M i p ∆ q denotes the matrix of the pairing A i p ∆ q ˆ A d ´ ´ i p ∆ q Ñ R . Thus if each matrix M i p (cid:101) ∆ q is invertible, so is each matrix M i p ∆ q , and conversely. If either ∆ or (cid:101) ∆ has PD, then theyboth do. (cid:3) Proposition 6.17.
Let Σ be a simplicial fan, σ P Σ p q , and (cid:101) Σ – stellar σ p Σ q . Suppose that PDholds for st Σ p σ q . Then PD holds for (cid:101) Σ if and only if it holds for Σ . Proof.
For dimensional reasons, if either fan has Poincaré duality, (cid:101) Σ is an ordinary subdivision.Let d – dim Σ “ dim (cid:101) Σ . By Proposition 6.7, we have A d p (cid:101) Σ q – A d p Σ q , and they have a commondegree map.Using the decomposition (6.3.4) and Poincaré duality in st Σ p σ q , we have b i p Σ q “ b d ´ i p Σ q and b i p (cid:101) Σ q “ b d ´ i p (cid:101) Σ q for all ď i ď d . Since A s p Σ q ˆ A t p st Σ p σ qq Ñ A s ` t p st Σ p σ qq is the zero mapwhen s ` t ą d ´ , ordering bases compatibly with (6.3.4) gives a block-diagonal matrix: M i p (cid:101) Σ q “ (cid:32) M i p Σ q M i p st Σ p σ qq (cid:33) Clearly M i p (cid:101) Σ q has full rank if and only if M i p Σ q and M i ´ p st Σ p σ qq both do as well, whichcompletes the proof. (cid:3) Lemma 6.18. If Σ is a d -dimensional simplicial fan with PD, let I Ď Σ p q be a subset for which { x ν : ν P I } spans A p Σ q . Then the map ‘ i ˚ ν : A i p Σ q Ñ (cid:77) ν P S A i p st Σ p ν qq is injective for all ď i ă d . Proof.
Suppose i ˚ ν p u q “ for each ray ν . Then i ν ˚ i ˚ ν p u q “ x ν ¨ u “ for a set of generators x ν of A p Σ q . Since A p Σ q is Gorenstein, this implies u P A d p Σ q . (cid:3) Proposition 6.19.
Let Σ be a simplicial fan satisfying PD in degree d . Suppose that the fan st Σ p ν q satisfies mixed HR for each ray ν P Σ p q . Then Σ satisfies mixed HL. Proof.
Let L – (cid:96) ¨ ¨ ¨ (cid:96) d ´ k be a Lefschetz element, and consider the map L ¨ : A k p Σ q Ñ A d ´ k p Σ q .By PD, we know b k p Σ q “ b d ´ k p Σ q , so it is enough to show that L ¨ is injective. Suppose, then,that L ¨ u “ for some u P A k p Σ q .Let L – (cid:96) ¨ ¨ ¨ (cid:96) d ´ k . For each ray ν P Σ p q , the pullback i ˚ ν p L q is a Lefschetz element for st Σ p ν q by Proposition 6.4. Since L ¨ u “ , each pullback of u is primitive; that is, i ˚ ν p u q P PA k p st Σ p ν q , i ˚ ν p (cid:96) qq .We may write (cid:96) “ (cid:80) ν P Σ p q c ν x ν where each coefficient c ν ą , since we can represent (cid:96) bya PL function which is strictly positive on each ray. Degree commutes with pullback: “ deg Σ p L ¨ u ¨ u q“ deg Σ p (cid:88) ν P Σ p q c ν x ν L ¨ u ¨ u q“ (cid:88) ν c ν deg st Σ p ν q p i ˚ ν p L q ¨ i ˚ ν p u q ¨ i ˚ ν p u qq“ p´ q k ´ (cid:88) ν P Σ p q c ν (cid:104) i ˚ ν p u q , i ˚ ν p u q (cid:105) i ˚ ν p L q . Since the c ν ’s are strictly positive, each summand is zero, and the mixed HR property in st Σ p ν q implies i ˚ ν p u q “ , for each ν . By Lemma 6.18, we have u “ , and L ¨ is injective. (cid:3) As an application, we see that the mixed Lefschetz properties in Definition 6.1 are actuallyno stronger than the pure ones. See [Cat08] for a discussion in a more general context.
Theorem 6.20. If Σ is a Lefschetz fan, then it also has the mixed HL and mixed HR properties. Proof.
We use induction on dimension. If dim Σ “ , the mixed and pure properties are identical,so let us suppose the claim is true for all Lefschetz fans of dimension less than d , for some d ą .Let Σ be a Lefschetz fan of dimension d . By induction, st Σ p ν q satisfies mixed HR for all rays ν P Σ p q . By Proposition 6.19, then Σ satisfies mixed HL.Now we establish mixed HR for Σ . For any (cid:96) P K p Σ q and ď k ď d { , the “pure” propertyHR k p L q holds for L “ (cid:96) d ´ k . Corollary 6.14 states that mixed HL and mixed HR i for i ă k implies mixed HR k . Setting k “ , we see Σ has the mixed HR property. Arguing by inductionon k , we obtain mixed HR k for all k ď d { . (cid:3) Lefschetz properties under edge subdivision II.
Now we examine how the Hodge–Riemannforms fare under stellar subdivisions. We begin with a technical lemma, then the case of an or-dinary subdivision.
Lemma 6.21.
Suppose p : (cid:101) Σ Ñ Σ is an edge subdivision. Then p ˚ p x ρ q “ , and p ˚ p x ρ q “ ´ x σ . Proof.
The first claim follows from the pushforward formula (6.3.2). Now let x , x be theCourant functions for the rays ν , ν of the cone σ P Σ p q , so x σ “ x x . By Lemma 6.6, we AGRANGIAN GEOMETRY OF MATROIDS 57 compute for i “ , that “ p ˚ p x ρ q x i “ p ˚ (cid:0) x ρ p x i ` x ρ q (cid:1) , so p ˚ p x ρ x i q “ ´ p ˚ p x ρ q . Since { ν , ν } is not contained in a cone of (cid:101) Σ , we have x σ “ p ˚ p ˚ p x x q “ p ˚ (cid:0) p x ` x ρ qp x ` x ρ q (cid:1) “ p ´ ` q p ˚ p x ρ q . (cid:3) Lemma 6.22.
Let Σ be a d -dimensional fan with PD. Suppose σ P Σ p q and (cid:101) Σ “ stellar σ p Σ q is an ordinary subdivision. Then, for all ď k ď d { and all L P Sym d ´ k K p Σ q , we have hr k p (cid:101) Σ , p ˚ L q “ hr k p Σ , L q ‘ hr k ´ p st Σ p σ q , i ˚ σ p L qq , an orthogonal direct sum. Proof.
We consider hr k p (cid:101) Σ , p ˚ L q under the direct sum decomposition (6.3.4). Given elements p a, q and p , b q P A k p Σ q ‘ A k ´ p st Σ p σ qq , we calculate as follows.Since i ˚ σ is surjective, we may write b “ i ˚ σ p b q for some b P A p Σ q . Then p´ q k (cid:104) p a, q , p , b q (cid:105) “ deg (cid:101) Σ (cid:0) p ˚ p L q ¨ p ˚ p a q ¨ j ˚ p ˚ σ p b q (cid:1) “ deg (cid:101) Σ (cid:0) p ˚ p L q p ˚ p a q ¨ p ˚ i ˚ p b q ¨ x ρ (cid:1) “ deg Σ (cid:0) L ¨ ab ¨ p ˚ p x ρ q (cid:1) “ , because p ˚ p x ρ q “ .If a, b P A k p Σ q , the equality (cid:104) p a, q , p b, q (cid:105) p ˚ p L q “ (cid:104) a, b (cid:105) L is straightforward. If a, b P A k ´ p st Σ p σ qq ,again write a “ ι ˚ σ p a q and b “ ι ˚ σ p b q for some a , b P A k ´ p Σ q . Then, calculating as above, (cid:104) p a, q , p , b q (cid:105) “ p´ q k deg (cid:101) Σ (cid:0) p ˚ p L q ¨ p ˚ p a q p ˚ p b q ¨ x ρ (cid:1) “ p´ q k deg Σ (cid:0) L ¨ a b ¨ p ˚ p x ρ q (cid:1) “ ´p´ q k deg Σ (cid:0) L ¨ a b ¨ x σ (cid:1) “ p´ q k ´ deg st Σ p σ q (cid:0) i ˚ σ p L q ¨ i ˚ σ p a q i ˚ σ p b q (cid:1) “ (cid:104) a, b (cid:105) i ˚ σ p L q The result follows. (cid:3)
Next we address star-shaped subdivisions. Let d “ dim st Σ p σ q “ dim Σ ´ . Lemma 6.23.
Suppose P and Q are n ˆ n matrices with real entries and Q “ Q T . Let M – (cid:32) PP T Q (cid:33) . If P is nonsingular, then M has signature zero. Proof.
Assume first that Q is invertible, and let S “ ´ P Q ´ P T (the Schur complement.) Thenit is easily seen that M is congruent to a block-diagonal matrix: M “ (cid:32) I n P Q ´ I n (cid:33) (cid:32) S Q (cid:33) (cid:32) I n Q ´ P T I n (cid:33) , and the signature of S is the negative of the signature of Q . It follows that M has signature zero.Now suppose Q is singular. We replace Q by Q p (cid:15) q to define M p (cid:15) q as above, for some real,invertible symmetric matrices Q p (cid:15) q with lim (cid:15) Ñ Q p (cid:15) q “ Q . Then det p M p (cid:15) qq “ p´ q n det p P q ‰ ,regardless of (cid:15) , so the argument above shows M p (cid:15) q has n positive eigenvalues and n negativeeigenvalues. By continuity, so does M . (cid:3) The last result in this section relates HL and HR along an edge subdivision.
Proposition 6.24.
Suppose that at least one of st Σ p σ q and st (cid:101) Σ p ρ q has Poincaré duality, and that (cid:96) P K p st Σ p σ qq has the Hard Lefschetz property. Then ‚ (cid:96) (cid:15) – (cid:96) ´ (cid:15) ¨ x ρ P K p st (cid:101) Σ p ρ qq has the HL property for sufficiently small (cid:15) ą , and ‚ For such (cid:15) , the fan st (cid:101) Σ p ρ q satisfies HR p (cid:96) (cid:15) q if st Σ p σ q satisfies HR p (cid:96) q . Proof.
Let ∆ “ st Σ p σ q and (cid:101) ∆ “ st (cid:101) Σ p ρ q . By Proposition 6.17, we may assume both ∆ and (cid:101) ∆ havePoincaré duality. By Lemma 6.11, we have (cid:96) (cid:15) P K p (cid:101) ∆ q for small enough positive (cid:15) .If k ă p d ` q{ , we use the HR property of (cid:96) P K p ∆ q and (6.3.4) to obtain a decomposition A k p (cid:101) ∆ q “ PA k p ∆ , (cid:96) q ‘ (cid:96)A k ´ p ∆ q ‘ x ρ A k ´ p ∆ q , with respect to which hr k p (cid:101) ∆ , (cid:96) (cid:15) q is represented by a block matrix hr k p ∆ , (cid:96) (cid:15) q “ H p (cid:15) q H p (cid:15) q H p (cid:15) q H p (cid:15) q H p (cid:15) q H p (cid:15) q H p (cid:15) q H p (cid:15) q H p (cid:15) q . For any (cid:15) ą , the matrix above is congruent to the matrix hr k p (cid:15) q – (cid:15) ´ H p (cid:15) q (cid:15) ´ H p (cid:15) q H p (cid:15) q (cid:15) ´ H p (cid:15) q (cid:15) ´ H p (cid:15) q H p (cid:15) q H p (cid:15) q H p (cid:15) q (cid:15)H p (cid:15) q , (6.6.1)the entries of which we will see are polynomial in (cid:15) . For elements p , p P PA k p ∆ , (cid:96) q , we have (cid:104) p , p (cid:105) (cid:96) (cid:15) “ p´ q k deg (cid:101) ∆ (cid:0) p (cid:96) ´ (cid:15)x ρ q d ` ´ k p p (cid:1) “ ´p´ q k ¨ (cid:15) ¨ deg (cid:101) ∆ (cid:0) (cid:96) d ´ k p d ` ´ k q p p x ρ (cid:1) ` O p (cid:15) q“ p´ q k (cid:15) p d ` ´ k q deg ∆ (cid:0) (cid:96) d ´ k p p (cid:1) ` O p (cid:15) q“ p d ` ´ k q (cid:15) ¨ (cid:104) p , p (cid:105) (cid:96) ` O p (cid:15) q . so the block H p (cid:15) q represents a positive multiple of the pairing hr k p ∆ , (cid:96) q , modulo (cid:15) . AGRANGIAN GEOMETRY OF MATROIDS 59
Similar computations show that the block H p (cid:15) q is the matrix of the pairing p d ` ´ k q (cid:15) ¨ hr k ´ p ∆ , (cid:96) q , modulo (cid:15) , and H p (cid:15) q “ H p (cid:15) q “ ´ hr k ´ p ∆ , (cid:96) q modulo (cid:15) . Along the same lines,we see H p (cid:15) q “ H p (cid:15) q are divisible by (cid:15) , and H p (cid:15) q “ H p (cid:15) q is divisible by (cid:15) . Returning to(6.6.1), we have hr k p (cid:15) q “ p d ` ´ k q hr k p ∆ , (cid:96) q | PA k ´p d ` ´ k q hr k ´ p ∆ , (cid:96) q ´ hr k ´ p ∆ , (cid:96) q ´ hr k ´ p ∆ , (cid:96) q ` O p (cid:15) q . Given our assumption that k ă p d ` q{ , the matrix hr k p q is invertible, because each non-zero block is nondegenerate (since (cid:96) has the HL property). It follows that (cid:96) (cid:15) has the HL k propertyfor all ď k ă p d ` q{ , for some sufficiently small (cid:15) ą . Using Lemma 6.23, we see thesignature of hr k p (cid:15) q agrees with that of the top-left block. By hypothesis, hr k p ∆ , (cid:96) q is positive-definite on PA k p ∆ , (cid:96) q . Now dim PA k p ∆ , (cid:96) q “ b k p ∆ q ´ b k ´ p ∆ q , which by Corollary 6.15 is theexpected signature for hr k p (cid:15) q ; that is, HR k p (cid:96) (cid:15) q holds for sufficiently small (cid:15) .It remains to consider the case where d is odd and k “ p d ` q{ . In this case we have A k p (cid:101) ∆ q “ A k ´ p ∆ q ‘ x ρ A k ´ p ∆ q , and (up to a sign) the pairing is just the Poincaré pairing M k p (cid:101) ∆ q . In the middle dimension, M k p ∆ q “ M k ´ p ∆ q , so we have a block decomposition from(6.5.1): M k p (cid:101) ∆ q “ (cid:32) ´ M k p ∆ q´ M k p ∆ q Q (cid:33) for some square matrix Q . The matrix M k p ∆ q is nonsingular, by HL k , so M k p (cid:101) ∆ q has signaturezero by Lemma 6.23, which shows (cid:96) (cid:15) has HR k for any (cid:15) by Corollary 6.15 again. (cid:3) Proofs of the main results.
We are now ready to prove the main result of this section. Wetreat the star-shaped and ordinary cases separately, beginning with the former.We will need to use a result of Włodarczyk [Wło97, Theorem A]:
Theorem 6.25. If Σ and Σ are two smooth, simplicial fans and | Σ | “ | Σ | , there exists a sequenceof simplicial fans Σ , Σ , . . . , Σ N for which Σ “ Σ , Σ N “ Σ , and Σ i is obtained from Σ i ´ byan edge subdivision or an inverse edge subdivision, for all ď i ď N . Proof.
By [Wło97, Theorem A], there is a sequence of simplicial fans as above, where either Σ i is a stellar subdivision of Σ i ´ , or vice-versa.If we regard Σ and Σ as cones over geometric simplicial complexes, then stellar subdivisionscorrespond to barycentric subdivisions. Alexander proved [Ale30, Corollary 10:2c] that we mayrefine the chain of fans above in such a way that each step is the subdivision of an edge, whichis to say a cone of codimension . (cid:3) Theorem 6.26.
Let (cid:101) Σ “ stellar σ p Σ q be a star-shaped subdivision of a simplicial fan Σ , for somecone σ P Σ p q . Then st Σ p σ q is a Lefschetz fan if and only if st (cid:101) Σ p ρ q is a Lefschetz fan, where ρ isthe ray subdividing σ . Proof.
Let ∆ “ st Σ p σ q and (cid:101) ∆ “ st (cid:101) Σ p ρ q . First, suppose ∆ is Lefschetz, and let ν , ν denote thetwo extreme rays of σ . First, we check that the star of each cone τ P (cid:101) ∆ is Lefschetz. This is easy if τ does not contain ν or ν , since then τ is a cone of ∆ . Otherwise, τ contains (exactly) one suchray, say ν . The remaining rays of τ span a cone τ of ∆ , and by inspection, st (cid:101) ∆ p τ q “ st ∆ p τ q ,which is again Lefschetz by hypothesis.The PD property for (cid:101) ∆ follows from Proposition 6.17. To establish HL, we use Proposi-tion 6.19. For this, we need to know that the star of each ray satisfies mixed HR, but the starof a ray of (cid:101) ∆ is also a star in ∆ , so HL for (cid:101) ∆ follows. Finally, we use Proposition 6.24: for any (cid:96) P K p ∆ q , there exists some (cid:96) (cid:15) P K p (cid:101) ∆ q with the HR property. By Corollary 6.14, (cid:101) ∆ has HR.The converse is trivial: if (cid:101) ∆ is Lefschetz, then st (cid:101) ∆ p ν q “ ∆ , so ∆ is Lefschetz too. (cid:3) We note that, in this case, K p ∆ q is nonempty if and only if K p (cid:101) ∆ q is nonempty. The forwardimplication follows immediately from Lemma 6.11. The converse holds by Proposition 6.4,since ∆ is a star in (cid:101) ∆ . For arbitrary subdivisions, however, K p (cid:101) Σ q can be nonempty while K p Σ q is empty. Theorem 6.27.
Let Σ be a simplicial fan and σ P Σ p q . If Σ is a Lefschetz fan and K p Σ q ‰ H ,then (cid:101) Σ – stellar σ p Σ q is Lefschetz. Conversely, if (cid:101) Σ is a Lefschetz fan, then Σ is Lefschetz. Proof of “ ñ ”: Let d “ dim Σ : we argue by induction on d . The statement is vacuously true if d “ , so let us assume it holds for all Lefschetz fans of dimension less than d .First we check that the star of every cone τ P (cid:101) Σ is Lefschetz, for which we consider two cases.First suppose τ P Σ . If σ R st Σ p τ q , then st (cid:101) Σ p τ q “ st Σ p τ q , which is Lefschetz. If, on the other hand, σ P st Σ p τ q , then st (cid:101) Σ p τ q “ stellar σ p st Σ p τ qq , which is a star-shaped subdivision. Since st Σ p τ q isLefschetz, so is st (cid:101) Σ p τ q , by Theorem 6.26.Now suppose τ R Σ , and let ρ denote the subdividing ray. Then ρ P τ , so st (cid:101) Σ p τ q Ď st (cid:101) Σ p ρ q : infact, st (cid:101) Σ p τ q “ st Σ p τ q , where Σ “ st (cid:101) Σ p ρ q . Since Σ “ stellar σ p st Σ p σ qq , a star-shaped subdivision, Σ is Lefschetz by Theorem 6.26, and it follows that st (cid:101) Σ p τ q is Lefschetz too.By Propositions 6.17 and 6.19, respectively, the fan (cid:101) Σ satisfies PD and HL. It remains to checkthat (cid:101) Σ satisfies HR as well.Consider any ď k ď d { and (cid:96) P K p Σ q . By Lemma 6.22, we have hr k p (cid:101) Σ , p ˚ (cid:96) q “ hr k p Σ , (cid:96) q ‘ hr k ´ p st Σ p σ q , i ˚ σ p (cid:96) qq . The summands are nondegenerate, because Σ and st Σ p σ q satisfy HL p (cid:96) q andHL p i ˚ σ (cid:96) q , respectively, so hr k p (cid:101) Σ , p ˚ (cid:96) q is nondegenerate as well. AGRANGIAN GEOMETRY OF MATROIDS 61
By the HR signature test (Theorem 6.13) the signature of hr k p (cid:101) Σ , p ˚ (cid:96) q equals k (cid:88) i “ p´ q k ´ i (cid:0) b i p Σ q ´ b i ´ p Σ q (cid:1) ` k ´ (cid:88) i “ p´ q k ´p i ´ q (cid:0) b i ´ p st Σ p σ qq ´ b i ´ p st Σ p σ qq (cid:1) “ k (cid:88) i “ p´ q k ´ i (cid:0) b i p Σ q ` b i p st Σ p σ qq ´ b i ´ p Σ q ´ b i ´ p st Σ p σ qq (cid:1) “ k (cid:88) i “ p´ q k ´ i (cid:0) b i p (cid:101) Σ q ´ b i ´ p (cid:101) Σ q (cid:1) . (6.7.1)Lemma 6.11 states p ˚ (cid:96) P cl K p (cid:101) Σ q . Then there exists an open ball U Ď A p (cid:101) Σ q containing p ˚ (cid:96) onwhich hr k p (cid:101) Σ , ´q is nondegenerate. Choosing any (cid:96) P U X K p (cid:101) Σ q , we can use Corollary 6.14 toconclude that (cid:101) Σ satisfies HR k . (cid:3) The converse is similar in spirit:
Proof of “ ð ”. Again, we argue by induction on dimension. The base case being trivial, we as-sume that, if (cid:101) Σ is Lefschetz and has dimension less than d , then Σ is Lefschetz as well. Nowassume (cid:101) Σ is a Lefschetz fan of dimension d , and we show Σ is as well.PD for Σ follows from Proposition 6.17. Next, consider a ray ν P Σ p q . If ν R st Σ p σ qp q , then st Σ p ν q “ st (cid:101) Σ p ν q , which is Lefschetz. If, on the other hand, ν P st Σ p σ qp q , then σ P st Σ p ν qp q , and st (cid:101) Σ p ν q “ stellar σ p st Σ p ν qq . Since st (cid:101) Σ p ν q is Lefschetz, so is st Σ p ν q , by Theorem 6.26. Either way, st Σ p ν q has the HR property for each ray ν , so Σ has the HL property (by Proposition 6.19).A similar argument shows that st Σ p τ q is Lefschetz for all cones τ of Σ : if the star remains astar in (cid:101) Σ , it is Lefschetz by hypothesis. Otherwise, a subdivision of it is a star in (cid:101) Σ . If τ “ σ , thesubdivided edge, we invoke Theorem 6.26. Otherwise, we note the dimension is less than d , so st Σ p τ q is Lefschetz by induction.It remains to establish HR k for Σ , for ď k ď d { . The condition is vacuous if K p Σ q “ H .Otherwise, choose any (cid:96) P K p Σ q . By Lemma 6.22, hr k p (cid:101) Σ , p ˚ (cid:96) q “ hr k p Σ , (cid:96) q ‘ hr k ´ p st Σ p σ q , i ˚ σ p (cid:96) qq . Since the second factor is the blowdown of st (cid:101) Σ p ρ q , it is Lefschetz by Theorem 6.26, and the firstfactor is Lefschetz by the argument above. So both summands are nondegenerate, and so is hr k p (cid:101) Σ , p ˚ (cid:96) q .By HR, the bilinear form hr k p (cid:101) Σ , (cid:101) (cid:96) q has the expected signature for all (cid:101) (cid:96) P K p (cid:101) Σ q . It follows byProposition 6.12 that hr k p (cid:101) Σ , p ˚ (cid:96) q also has that signature, since it is nondegenerate and p ˚ (cid:96) lies inthe boundary of K p (cid:101) Σ q .The HR property for st Σ p σ q determines the signature of hr k ´ p st Σ p σ q , i ˚ σ p (cid:96) qq , and we obtainthe signature of hr k p Σ , (cid:96) q by subtraction. Using the calculation (6.7.1) again, we find that itequals (cid:80) ki “ p´ q k ´ i (cid:0) b i p Σ q ´ b i ´ p Σ q (cid:1) , and we conclude Σ has the HR k property. (cid:3) Putting the pieces together gives a proof that the Lefschetz property is an invariant of thesupport of a fan.
Theorem 1.6.
Let Σ and Σ be simplicial fans that have the same support | Σ | “ | Σ | . If K p Σ q and K p Σ q are nonempty, then Σ is Lefschetz if and only if Σ is Lefschetz. Proof of Theorem 1.6.
Suppose | Σ | “ | Σ | . According to Theorem 6.25, there is a sequence of fans p Σ , Σ , ¨ ¨ ¨ , Σ N q with Σ “ Σ , Σ N “ Σ , and for which either Σ i Ñ Σ i ` or Σ i ` Ñ Σ i is anedge subdivision, for each i . It is implicit in the argument of [Wło97] that edge subdivisions canbe chosen in such a way that, whenever Σ i “ stellar σ p Σ i ` q , if K p Σ i q is nonempty, then so is K p Σ i ` q : see, for example, the discussion around [AKMW02, Theorem 0.3.1]. By Theorem 6.27,if any one of these fans is Lefschetz, then they all are. (cid:3) In our terminology, the main result of [AHK18] says that the Bergman fan of M is Lefschetz.We use the result to show that the conormal fan of M is Lefschetz. Lemma 6.28. If Σ and Σ are Lefschetz fans, then so is Σ ˆ Σ . Proof.
It was shown in [AHK18, Section 7.2] that, if Σ and Σ have PD, HL, and HR, then sodoes Σ ˆ Σ . Since stars of cones in a product are products of stars in the factors, we concludethat Σ ˆ Σ is a Lefschetz fan by induction on dimension. (cid:3) AGRANGIAN GEOMETRY OF MATROIDS 63
Theorem 6.29.
The conormal fan Σ M , M K of a loopless and coloopless matroid M is Lefschetz. Proof.
Since the Bergman fan is Lefschetz, from Lemma 6.28 we see the fan Σ M ˆ Σ M K is Lef-schetz. Moreover, its support is equal to that of Σ M , M K . Bergman fans are quasiprojective,since they are subfans of the permutohedral fan, so K p Σ M ˆ Σ M K q is nonempty. We saw thatthe bipermutohedral fan Σ E,E is the normal fan of the bipermutohedron, so the conormal fanis also quasiprojective, and K p Σ M , M K q is nonempty as well. By Theorem 1.6, then, Σ M , M K isLefschetz. (cid:3) The extra structure present in the Chow rings of Lefschetz fans leads easily to an Aleksandrov–Fenchel-type inequality.
Theorem 6.30.
Let Σ be a Lefschetz fan of dimension d , and (cid:96) , (cid:96) , . . . , (cid:96) d elements of cl K p Σ q .Then for any (cid:96) P A p Σ q , deg p (cid:96) (cid:96) ¨ ¨ ¨ (cid:96) d q ě deg p (cid:96) (cid:96) (cid:96) ¨ ¨ ¨ (cid:96) d q ¨ deg p (cid:96) (cid:96) (cid:96) ¨ ¨ ¨ (cid:96) d q . (6.7.2) Proof.
We first verify the inequality when (cid:96) i P K p Σ q for each ď i ď d . For this, let L “ (cid:96) ¨ ¨ ¨ (cid:96) d ,a Lefschetz element, and consider (cid:104) ´ , ´ (cid:105) – (cid:104) ´ , ´ (cid:105) L on A p Σ q .If (cid:104) (cid:96) , (cid:96) (cid:105) ‰ , let (cid:96) “ (cid:96) ´ (cid:104) (cid:96) ,(cid:96) (cid:105)(cid:104) (cid:96) ,(cid:96) (cid:105) (cid:96) , so that (cid:104) (cid:96) , (cid:96) (cid:105) “ . This means (cid:96) P P A p Σ , (cid:96) q , so by HR, ď (cid:10) (cid:96) , (cid:96) (cid:11) “ (cid:10) (cid:96) , (cid:96) (cid:11) “ (cid:104) (cid:96) , (cid:96) (cid:105) ´ (cid:104) (cid:96) , (cid:96) (cid:105)(cid:104) (cid:96) , (cid:96) (cid:105) (cid:104) (cid:96) , (cid:96) (cid:105) . By the signature test, (cid:104) ´ , ´ (cid:105) is negative-definite on the orthogonal complement of (cid:96) . Therefore (cid:104) (cid:96) , (cid:96) (cid:105) ă , and we see (cid:104) (cid:96) , (cid:96) (cid:105) ě (cid:104) (cid:96) , (cid:96) (cid:105) ¨ (cid:104) (cid:96) , (cid:96) (cid:105) , which is equivalent to (6.7.2). (If, on the other hand, (cid:104) (cid:96) , (cid:96) (cid:105) “ , this inequality is obvious.)Now we relax the hypothesis to consider (cid:96) , . . . , (cid:96) d P cl K p Σ q . The inequality (6.7.2) continuesto hold by continuity, as in [AHK18, Theorem 8.8]. (cid:3) Theorem 1.4.
For any matroid M , the h -vector of the broken circuit complex of M is log-concave. Proof.
It suffices to assume that M is loopless and coloopless. The classes γ “ γ i and δ “ δ i are pullbacks of the nef classes α “ α i P A p Σ M q and α “ α i P A p ∆ E q , along the two maps π : Σ M , M K Ñ Σ M and µ : Σ M , M K Ñ ∆ E , respectively. The pullback of a convex function ona fan is convex, so both γ and δ represent nef classes on the conormal fan. Since K p Σ M , M K q isnonempty, we see that γ, δ P cl K p Σ M , M K q , following the discussion at the end of Section 6.2. By Theorem 1.2, we have h r ´ k p BC p M qq “ deg Σ M , M K p γ k δ n ´ k ´ q“ (cid:104) γ, δ (cid:105) L , where L “ γ k ´ δ n ´ k ´ . Since Σ M , M K is Lefschetz (Theorem 6.29) the log-concave inequalitiesfollow from Theorem 6.30. (cid:3) R EFERENCES[AHK18] Karim Adiprasito, June Huh, and Eric Katz,
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