Algebraic cobordism rings of wonderful varieties and matroids

Technical Summary: Algebraic Cobordism Rings of Wonderful Varieties and Matroids

Problem Statement
The paper addresses the computation and structural understanding of the algebraic cobordism ring $\Omega^*(X)$ for specific classes of varieties associated with matroids: the toric variety $X_M$ of the Bergman fan of a loopless matroid $M$, and the wonderful compactification $W_H$ of a complex hyperplane arrangement $H$. While the Chow rings $CH^*(X_M)$ and $CH^*(W_H)$ and the K-rings $K_0(X_M)$ and $K_0(W_H)$ have been extensively studied and shown to possess rich combinatorial structures (including the "Kähler package" and exceptional integral isomorphisms between Chow and K-rings), explicit presentations for their algebraic cobordism rings $\Omega^*(X_M)$ and $\Omega^*(W_H)$ were previously unavailable. Algebraic cobordism, constructed by Levine and Morel, is the universal oriented cohomology theory, making its computation significantly more difficult than that of Chow or K-theory, as it involves the universal formal group law over the Lazard ring $L^*$.

Methodology
The authors employ a combination of combinatorial algebra, the theory of formal group laws, and the geometry of toric varieties and blow-ups.

1. Toric Variety Framework: The authors utilize the description of the algebraic cobordism ring of a smooth toric variety $X_\Sigma$ provided by Krishna and Uma. This presentation involves generators corresponding to toric divisors and relations derived from the fan's combinatorics and the universal formal group law $F_U$.
2. Combinatorial Presentations: They adapt the Feichtner–Yuzvinsky presentation for Chow rings of matroidal toric varieties to the setting of algebraic cobordism. This involves defining a ring $\Omega^*(M)$ generated by variables indexed by the non-empty flats of the matroid, modulo relations arising from incomparable flats and the formal group law.
3. Simplicial Presentation: A key methodological step is the derivation of a "simplicial" presentation for $\Omega^*(X_M)$. This involves a change of variables using Möbius inversion on the lattice of flats, lifting the "simplicial generators" known from the Chow and K-theory contexts to the cobordism setting.
4. Comparison Theorems: The authors leverage comparison theorems between algebraic cobordism, Chow theory, and complex cobordism ($MU^*$). Specifically, they use the fact that for certain varieties, the natural map $\Omega^*(X) \to CH^*(X)$ allows for the recovery of $\Omega^*(X)$ as a free module over $L^*$ if $CH^*(X)$ is free over $\mathbb{Z}$.
5. Embedding Analysis: For the wonderful variety $W_H$, the authors analyze the canonical inclusion $\iota: W_H \hookrightarrow X_{M_H}$. They prove that this inclusion induces isomorphisms on the relevant cohomology rings by studying the cycle class map and the structure of $W_H$ as an iterated blow-up of projective space.

Key Contributions and Results

1. Simplicial Presentation for $\Omega^*(X_M)$:
   The primary result is a combinatorial presentation of the algebraic cobordism ring of the toric variety $X_M$ associated with a loopless matroid $M$. The ring is presented as an $L^*$-algebra generated by elements $h_G$ (lifts of simplicial generators) for non-empty flats $G$, subject to relations involving the formal group law $F_U$:
   $$\Omega^*(X_M) \cong \frac{L^*[h_G : G \neq \emptyset]}{\langle (h_G - h_{G \vee H})(h_H - h_{G \vee H}) \rangle + \langle h_G : \text{rank}(G)=1 \rangle}$$
   Crucially, this presentation yields an isomorphism of $L^*$-algebras:
   $$\Omega^*(X_M) \cong CH^*(X_M) \otimes_{\mathbb{Z}} L^*$$
   This result implies that for any free oriented cohomology theory $h^*$, $h^*(X_M) \cong \Omega^*(X_M) \otimes_{L^*} h^*(pt)$.

2. Generalization of the Exceptional Isomorphism:
   The isomorphism $\Omega^*(X_M) \cong CH^*(X_M) \otimes_{\mathbb{Z}} L^*$ generalizes the "exceptional isomorphism" $K_0(X_M) \cong CH^*(X_M)$ recently established by Berget–Eur–Spink–Tseng and Larson–Li–Payne–Proudfoot. The authors note that this property is rare; they cite Speyer's example of a smooth projective toric variety where $CH^*(X) \not\cong K_0(X)$, demonstrating that the isomorphism for matroidal toric varieties is a special structural feature.

3. Isomorphism for Wonderful Varieties:
   The authors prove that the inclusion $\iota: W_H \hookrightarrow X_{M_H}$ induces an isomorphism of algebraic cobordism rings:
   $$\iota^*: \Omega^*(X_{M_H}) \xrightarrow{\sim} \Omega^*(W_H)$$
   Furthermore, they establish that the natural map from algebraic cobordism to complex cobordism induces an isomorphism:
   $$\Omega^*(W_H) \xrightarrow{\sim} MU^*(W_H(\mathbb{C}))$$
   Consequently, the algebraic cobordism ring of the wonderful variety coincides with its complex cobordism ring and is isomorphic to the tensor product of its Chow ring with the Lazard ring.

4. Dependence on Matroid Structure:
   The paper demonstrates that the cobordism class $[W_H \to pt]$ in $\Omega^*(pt)$ (and $MU^*(pt)$) depends only on the underlying matroid $M_H$ of the hyperplane arrangement $H$. This is shown by verifying that the Chern numbers of $W_H$, which determine the cobordism class, are determined entirely by the combinatorics of $M_H$ via Cheng's formula for the total Chern class of the tangent bundle.

Significance and Claims
The paper claims to provide the first explicit combinatorial presentations for the algebraic cobordism rings of toric varieties of Bergman fans and wonderful varieties of hyperplane arrangements. By establishing the isomorphism $\Omega^*(X_M) \cong CH^*(X_M) \otimes_{\mathbb{Z}} L^*$, the authors show that the complex structure of these rings is entirely governed by their Chow ring structure and the universal formal group law.

The significance lies in:

• Unification: It unifies the study of Chow rings, K-rings, and algebraic cobordism rings for matroidal varieties under a single framework, showing that the "exceptional isomorphism" between Chow and K-theory is a shadow of a deeper isomorphism involving algebraic cobordism.
• Computational Access: It provides a concrete method for computing algebraic cobordism classes for these varieties, which are generally difficult to compute.
• Matroidal Invariants: It confirms that deep geometric invariants (like the cobordism class in $\Omega^*(pt)$) of the wonderful compactification are purely combinatorial invariants of the underlying matroid, extending this property from Chow and K-theory to the universal oriented cohomology theory.

The authors explicitly state that while they provide a recursive formula for the class of $M_{0,n}$ in a related context (via Ellis-Bloor), the explicit recursive formula for $[W_H \to pt]$ in terms of Lazard ring generators remains a direction for future work. They also note that the isomorphism $\Omega^*(X) \cong CH^*(X) \otimes L^*$ does not hold for all smooth projective toric varieties, highlighting the special nature of matroidal varieties.