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On the tangent bundle and the divisor theory of a general matroid

Ronnie Cheng

TL;DR

This work defines a canonical K-class $T_M$ for any loopless matroid $M$, extending the geometric tangent-bundle perspective from realizable matroids to the general combinatorial setting. It furnishes two explicit formulas for the total Chern class of $T_M$, proves the Todd class of $T_M$ coincides with the Todd class in matroid Hirzebruch--Riemann--Roch, and derives a Chow-polynomial interpretation via Euler characteristics of exterior cotangent powers. By introducing a combinatorial fake effective cone and studying big-and-nef divisors, the paper establishes a framework for positivity in matroid Chow rings and formulates a matroid Kawamata--Viehweg vanishing theory, at least in low rank. The beta-classes are developed as Cremona-conjugate nef-divisor generators with favorable computability and vanishing properties, enriching the nef-test toolkit beyond the alpha-classes. Together, these results extend key algebro-geometric notions to nonrealizable matroids, yielding valuative methods to transfer structural and vanishing phenomena from realizable to general matroids and opening new avenues in matroid positivity and divisor theory.

Abstract

For a loopless matroid $M$, we construct a $K$-class $T_M\in K(X_{M})$. When $M$ is realizable, $T_M$ recovers the $K$-class of the tangent bundle of the wonderful compactification $W_L$. We derive two formulas for the total Chern class of $T_M$ and prove that the associated Todd class agrees with the Todd class appearing in the matroid Hirzebruch--Riemann--Roch formula. We define a "fake effective cone" so that big and nef divisors in a matroid can be characterized in a manner analogous to how the effective cone characterizes big and nef divisors in classical algebraic geometry. Finally, we define the classes $β_S$ and study their properties.

On the tangent bundle and the divisor theory of a general matroid

TL;DR

This work defines a canonical K-class for any loopless matroid , extending the geometric tangent-bundle perspective from realizable matroids to the general combinatorial setting. It furnishes two explicit formulas for the total Chern class of , proves the Todd class of coincides with the Todd class in matroid Hirzebruch--Riemann--Roch, and derives a Chow-polynomial interpretation via Euler characteristics of exterior cotangent powers. By introducing a combinatorial fake effective cone and studying big-and-nef divisors, the paper establishes a framework for positivity in matroid Chow rings and formulates a matroid Kawamata--Viehweg vanishing theory, at least in low rank. The beta-classes are developed as Cremona-conjugate nef-divisor generators with favorable computability and vanishing properties, enriching the nef-test toolkit beyond the alpha-classes. Together, these results extend key algebro-geometric notions to nonrealizable matroids, yielding valuative methods to transfer structural and vanishing phenomena from realizable to general matroids and opening new avenues in matroid positivity and divisor theory.

Abstract

For a loopless matroid , we construct a -class . When is realizable, recovers the -class of the tangent bundle of the wonderful compactification . We derive two formulas for the total Chern class of and prove that the associated Todd class agrees with the Todd class appearing in the matroid Hirzebruch--Riemann--Roch formula. We define a "fake effective cone" so that big and nef divisors in a matroid can be characterized in a manner analogous to how the effective cone characterizes big and nef divisors in classical algebraic geometry. Finally, we define the classes and study their properties.

Paper Structure

This paper contains 27 sections, 54 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The wonderful variety
  4. Matroids and their geometric realizations
  5. Chow ring of a matroid
  6. K-theory and Chern character
  7. Maps between Chow ring of matroids
  8. Valuative Invariants
  9. Tangent bundle and its Chern class
  10. Hirzebruch-Riemann-Roch formula
  11. Properties of Chow ring
  12. Valuativeness
  13. Serre duality for matroids
  14. The Chow (Poincaré) polynomial
  15. A geometric derivation via blow-ups
  16. ...and 12 more sections

Key Result

Theorem 1.1

For every loopless matroid $M$ there exists a canonical $K$-class $\widetilde{T}_M\in K(X_E)$ whose restriction $T_M := i_M^*\widetilde{T}_M\in K(X_M)$ (Definition tangent bundle def) satisfies the following properties:

Theorems & Definitions (108)

  • Theorem 1.1: Tangent class, Todd class, and Chow polynomial
  • Theorem 1.2: big and nef classes and the fake effective cone
  • Theorem 1.3: Properties of the $\beta$-classes
  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6: Matroid Hirzebruch–Riemann–Roch, cf. Section \ref{['subsec:HRR']}
  • Lemma 2.7: Pullback map
  • ...and 98 more