Complexes of stable birational invariants

TL;DR

This paper introduces a new stable birational invariant based on homotopy types of chain complexes, providing a categorification of motivic volume and a motivic obstruction to retract rationality.

Contribution

It develops a universal stable birational invariant framework, extending existing invariants like unramified cohomology to stratified degenerations.

Findings

Defines a functorial stable birational invariant as a chain complex homotopy type.

Establishes a motivic obstruction to retract rationality in a Grothendieck group quotient.

Constructs a broad class of invariants applying any stable birational invariant to degeneration strata.

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.