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Cellular $\mathbb{A}^1$-Homology of Smooth Toric Varieties

Haoyang Liu, Keyao Peng

TL;DR

The paper develops a cellular ^1-homology framework for smooth toric varieties, enabling explicit computations via cubical cells and toric actions. For pure shellable fans, it yields a Milnor-Witt motivic decomposition and a precise decomposition of cellular ^1-homology into sums of Milnor-Witt K-theory twisted by l-eta, along with a basis for Chow-Witt groups in general. The moment-angle complex and toric quotient provide the combinatorial and motivic scaffolding to translate fan data into explicit chain complexes, while shellability ensures strong, computable results; non-pure or exotic cases still allow Chow-group bases. Collectively, the results give actionable, algebraic-topological tools to compute motivic invariants and Chow groups of a wide class of toric varieties, including complete toric surfaces, with clear decompositions and explicit examples.

Abstract

In this paper, we present the calculations of cellular $\mathbb{A}^1$-homology for smooth toric varieties, along with an explicit description of pure shellable cases. Consequently, we derive the (Milnor-Witt) motivic decomposition for these pure shellable cases. Furthermore, we obtain an additive basis for the Chow groups of general smooth toric varieties.

Cellular $\mathbb{A}^1$-Homology of Smooth Toric Varieties

TL;DR

The paper develops a cellular ^1-homology framework for smooth toric varieties, enabling explicit computations via cubical cells and toric actions. For pure shellable fans, it yields a Milnor-Witt motivic decomposition and a precise decomposition of cellular ^1-homology into sums of Milnor-Witt K-theory twisted by l-eta, along with a basis for Chow-Witt groups in general. The moment-angle complex and toric quotient provide the combinatorial and motivic scaffolding to translate fan data into explicit chain complexes, while shellability ensures strong, computable results; non-pure or exotic cases still allow Chow-group bases. Collectively, the results give actionable, algebraic-topological tools to compute motivic invariants and Chow groups of a wide class of toric varieties, including complete toric surfaces, with clear decompositions and explicit examples.

Abstract

In this paper, we present the calculations of cellular -homology for smooth toric varieties, along with an explicit description of pure shellable cases. Consequently, we derive the (Milnor-Witt) motivic decomposition for these pure shellable cases. Furthermore, we obtain an additive basis for the Chow groups of general smooth toric varieties.
Paper Structure (11 sections, 30 theorems, 160 equations)

This paper contains 11 sections, 30 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Cellular $\mathbb{A}^1$-homology
  3. Cubical Cells
  4. Toric Action on Cubical Cells
  5. Cellular Structures of Smooth Toric Varieties
  6. Moment-Angle Complex and Toric Quotient
  7. Shellable Toric Variety
  8. The Cellular Chain Complexes of $K_\omega$
  9. Examples of Toric Surfaces
  10. Complete Toric Surfaces
  11. Exotic Examples

Key Result

Theorem 1.1

In the category $D(Ab_{\mathbb{A}^1}(k))$, for a smooth pure shellable toric variety $X_{\Sigma}$, we have a quasi-isomorphism: for a certain subset $B(1) \subset K_{\max}$, which can be derived from the complex $\bigoplus_{\omega \in \mathrm{row}\lambda} \overline{C}^{cri}_i(K_\omega)$. Specifically, we establish the correspondences between: As a result, the cellular $\mathbb{A}^1$-homology of

Theorems & Definitions (72)

  • Theorem 1.1: Corollary \ref{['mainA1cellular']}
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Proposition 1.5: Proposition \ref{['chowDecomp']}
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • ...and 62 more