[Paper Review] Cobordism ring of toric varieties

Amalendu Krishna, V. Uma|arXiv (Cornell University)|Nov 2, 2010

Homotopy and Cohomology in Algebraic Topology18 references3 citations

TL;DR

This paper computes the algebraic cobordism ring of smooth toric varieties by first determining their equivariant cobordism ring using a decomposition theorem inspired by Vezzosi and Vistoli, then deriving the ordinary cobordism ring via a quotient construction. The key result is an explicit isomorphism between the ordinary cobordism ring and a quotient of a power series ring over the Lazard ring, with relations determined by the fan structure and formal group law.

ABSTRACT

We describe the equivariant cobordism ring of smooth toric varieties. This equivariant description is used to compute the ordinary cobordism ring of such varieties.

Motivation & Objective

• To compute the ordinary algebraic cobordism ring of smooth toric varieties, a class of algebraic varieties with rich combinatorial structure.
• To extend equivariant cobordism techniques—previously used for Chow rings—to the setting of algebraic cobordism.
• To establish a structural description of the equivariant cobordism ring of smooth toric varieties analogous to the equivariant Chow ring case.
• To derive the ordinary cobordism ring from the equivariant one using a known quotient formula from [14, Theorem 3.4].
• To provide explicit presentations of the cobordism ring in terms of generators and relations using the fan data and the formal group law.

Proposed method

• Use the equivariant cobordism theory of smooth varieties developed in [13], which generalizes Totaro and Edidin-Graham's equivariant Chow theory.
• Adapt techniques from Vezzosi and Vistoli on equivariant K-theory to construct a decomposition theorem for the equivariant cobordism ring of smooth toric varieties.
• Define the equivariant cobordism ring as a quotient of the power series ring ${\mathbb{L}}[[t_{\rho}]]$ over the Lazard ring, modulo an ideal $I_\Delta$ generated by monomials corresponding to non-spanning sets of one-dimensional cones.
• Apply [14, Theorem 3.4] to express the ordinary cobordism ring as a quotient of the equivariant ring by the ideal generated by the equivariant first Chern classes of characters.
• Use the formal group law $F(u,v)$ on the Lazard ring to express equivariant Chern classes as $\sum [n_i]_F x_i$, incorporating the torus action via the pairing $<\chi, v_\rho>$.
• Establish isomorphisms between the ordinary cobordism ring and a polynomial quotient by showing that higher-degree monomials vanish due to dimension constraints, leading to a finite presentation.

Experimental results

Research questions

• RQ1 How can the equivariant cobordism ring of a smooth toric variety be described in terms of its fan data?
• RQ2 What is the precise structure of the ordinary cobordism ring of a smooth toric variety, and how does it relate to the equivariant cobordism ring?
• RQ3 Can the formal group law of the Lazard ring be used to encode the torus action in the cobordism ring via Chern classes?
• RQ4 How do the combinatorial data of the fan—specifically the non-spanning sets of rays—determine the relations in the cobordism ring?
• RQ5 To what extent does the cobordism ring of a smooth toric variety resemble the Chow ring in structure, and what are the key differences introduced by the formal group law?

Key findings

• The equivariant cobordism ring $\Omega^*_T(X)$ of a smooth toric variety $X=X(\Delta)$ is isomorphic to $\mathbb{L}[[t_\rho]]/I_\Delta$, where $I_\Delta$ is the ideal generated by monomials $\prod_{\rho \in S} t_\rho$ for $S \in \Delta^0_1$, the set of non-spanning subsets of rays.
• The ordinary cobordism ring $\Omega^*(X)$ is isomorphic to $\mathbb{L}[t_\rho]/(\overline{I}_\Delta, \sum_{\rho} [\langle \chi, v_\rho \rangle]_F t_\rho)$, where $\overline{I}_\Delta$ includes both the fan ideal and the $n+1$st power relations $t_\rho^{n+1}$.
• For projective space $\mathbb{P}^n$, the cobordism ring is isomorphic to $\mathbb{L}[t_{\rho_{n+1}}]/(t_{\rho_{n+1}}^{n+1})$, recovering the known result via the formal group law and fan relations.
• For a toric variety associated to a single cone, the cobordism ring is isomorphic to the Lazard ring $\mathbb{L}$, since all generators are killed by the relations.
• The Chern class of a $T$-equivariant line bundle $L_\chi$ is given by $c^T_1(L_\chi) = \sum_{\rho} [\langle \chi, v_\rho \rangle]_F t_\rho$, showing the role of the formal group law in encoding torus weights.
• The inclusion $\mathbb{L}[t_\rho] \to \mathbb{L}[[t_\rho]]$ descends to an isomorphism after quotienting by the appropriate ideals, confirming the finiteness of the ordinary cobordism ring.