The motivic Thom-Sebastiani theorem for regular and formal functions

TL;DR

This paper provides a model-theoretic proof of the motivic Thom-Sebastiani theorem for regular functions and extends it to formal functions, impacting the theory of motivic Donaldson-Thomas invariants.

Contribution

It introduces a new proof approach for the motivic Thom-Sebastiani theorem and extends it to formal functions, linking to motivic Donaldson-Thomas invariants.

Findings

Proof of the motivic Thom-Sebastiani theorem for regular functions

Extension of the theorem to formal functions

Application to motivic Donaldson-Thomas invariants

Abstract

Thanks to Hrushovski-Loeser's work on motivic Milnor fibers, we give a model-theoretic proof for the motivic Thom-Sebastiani theorem in the case of regular functions. Moreover, slightly extending of Hrushovski-Loeser's construction adjusted to Sebag, Loeser and Nicaise's motivic integration for formal schemes and rigid varieties, we formulate and prove an analogous result for formal functions. The latter is meaningful as it has been a crucial element of constructing Kontsevich-Soibelman's theory of motivic Donaldson-Thomas invariants.