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Complexes of stable birational invariants

James Hotchkiss, David Stapleton

TL;DR

The paper develops a categorified stable birational invariant by constructing a functorial categorical motivic volume that assigns to semistable degenerations a chain complex in $K^b( extbf{Z}[ extbf{RE}_k])$, unifying motivic volume with stable birational data. This volume factors through the universal Kahn--Sujatha stable birational category $ extbf{RE}_k$ and yields a motivic obstruction to retract rationality via Vol_{re}, with corollaries connecting to universal $ ext{R}$-triviality and $ ext{A}^1$-connectedness. The framework extends to a general class of invariants $F$ valued in additive categories, giving rise to Vol_{F} and a suite of homological invariants (e.g., unramified cohomology, Picard, Brauer, CH_0) through the subdivision/Čech complexes and their functorial behavior under toroidal modifications. Core technical contributions include the construction of coherent morphisms between strata, the demonstration that star and barycentric subdivisions preserve the homotopy type of the associated complexes, and a robust proof of specialization/variation for universal $ ext{R}$-triviality in families. Together, these results provide new tools for obstructing (or confirming) retract rationality and stable rationality in degenerations, with concrete implications for questions such as the retract irrationality of very general quartic fivefolds.

Abstract

We introduce a new stable birational invariant, which takes the form of a functor sending a degenerating variety to the homotopy type of a chain complex. Our invariant is a categorification of the motivic volume of Nicaise and Shinder. From the class of the chain complex in a Grothendieck group, we obtain a motivic obstruction to retract rationality, valued in a quotient of the Grothendieck ring of varieties. In addition, we construct a general class of stable birational invariants, with the invariant above as the universal example, given by applying any chosen stable birational invariant (e.g., unramified cohomology) to the strata of a semistable degeneration.

Complexes of stable birational invariants

TL;DR

The paper develops a categorified stable birational invariant by constructing a functorial categorical motivic volume that assigns to semistable degenerations a chain complex in , unifying motivic volume with stable birational data. This volume factors through the universal Kahn--Sujatha stable birational category and yields a motivic obstruction to retract rationality via Vol_{re}, with corollaries connecting to universal -triviality and -connectedness. The framework extends to a general class of invariants valued in additive categories, giving rise to Vol_{F} and a suite of homological invariants (e.g., unramified cohomology, Picard, Brauer, CH_0) through the subdivision/Čech complexes and their functorial behavior under toroidal modifications. Core technical contributions include the construction of coherent morphisms between strata, the demonstration that star and barycentric subdivisions preserve the homotopy type of the associated complexes, and a robust proof of specialization/variation for universal -triviality in families. Together, these results provide new tools for obstructing (or confirming) retract rationality and stable rationality in degenerations, with concrete implications for questions such as the retract irrationality of very general quartic fivefolds.

Abstract

We introduce a new stable birational invariant, which takes the form of a functor sending a degenerating variety to the homotopy type of a chain complex. Our invariant is a categorification of the motivic volume of Nicaise and Shinder. From the class of the chain complex in a Grothendieck group, we obtain a motivic obstruction to retract rationality, valued in a quotient of the Grothendieck ring of varieties. In addition, we construct a general class of stable birational invariants, with the invariant above as the universal example, given by applying any chosen stable birational invariant (e.g., unramified cohomology) to the strata of a semistable degeneration.

Paper Structure

This paper contains 40 sections, 42 theorems, 187 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. The rationality problem
  3. The categorical motivic volume
  4. A motivic obstruction to retract rationality
  5. Stable rationality and retract rationality
  6. A more general class of invariants
  7. Organization of the paper
  8. Notation and conventions
  9. Acknowledgements
  10. Toroidal embeddings
  11. Toroidal embeddings
  12. Categories of toroidal embeddings
  13. Functoriality of strata
  14. Toroidal modifications
  15. Star subdivision
  16. ...and 25 more sections

Key Result

Theorem 1.3

Let $k$ be an algebraically closed field of characteristic $0$, and let $\bar{K}$ be the field of Puiseux series over $k$. There is an additive functor satisfying the following property: Given a semistable model $X_{\bar{K}}$ over $\mathop{\mathrm{Spec}}\nolimits k\llbracket t^{1/d} \rrbracket$ for some $d > 0$ with special fiber $X_0 = D_0 \cup \dots \cup D_n$, $\mathscr{V}\mathrm{ol}_{\mathrm{r

Figures (1)

  • Figure 4.1: On the left, the decomposition of $\partial \circ h|_{\mathrm{F}(\bm{\sigma})}$ into $A, B, C, D, UD$, and $LD$. On the right, the decomposition of $h \circ \partial|_{\mathrm{F}(\bm{\sigma})}$ into $LT$ and $UT$. The case $n = 6$ is shown.

Theorems & Definitions (112)

  • Definition 1.1: Manin
  • Remark 1.2: The universal stable birational invariant
  • Theorem 1.3: Categorical motivic volume
  • Corollary 1.4
  • Corollary 1.5: The motivic obstruction to retract rationality
  • Remark 1.6: Specialization of universal $\mathrm{R}$-triviality
  • Example 1.8: Quartic fivefolds
  • Remark 1.9
  • Theorem 1.10
  • Corollary 1.11
  • ...and 102 more