On motivic vanishing cycles of critical loci
Vittoria Bussi, Dominic Joyce, Sven Meinhardt

TL;DR
This paper establishes that motivic vanishing cycles depend only on third-order thickenings, behave compatibly under embeddings, and can be globally defined on oriented algebraic d-critical loci, advancing motivic Donaldson-Thomas theory.
Contribution
It proves the local dependence of motivic vanishing cycles on third-order thickenings, describes their behavior under embeddings, and constructs global motives on d-critical loci, linking to Donaldson-Thomas invariants.
Findings
Dependence of motivic vanishing cycles on third-order thickenings
Compatibility of motivic vanishing cycles under embeddings with twists
Existence of global motives on oriented algebraic d-critical loci
Abstract
Let be a smooth scheme over an algebraically closed field of characteristic zero and a regular function, and write Crit, as a closed subscheme of . The motivic vanishing cycle is an element of the -equivariant motivic Grothendieck ring defined by Denef and Loeser math.AG/0006050 and Looijenga math.AG/0006220, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants, arXiv:0811.2435. We prove three main results: (a) depends only on the third-order thickenings of . (b) If is another smooth scheme, is regular, Crit, and is an embedding with and an isomorphism, then equals "twisted" by a…
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