Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

On motivic vanishing cycles of critical loci


Authors: Vittoria Bussi, Dominic Joyce and Sven Meinhardt
Journal: J. Algebraic Geom.
DOI: https://doi.org/10.1090/jag/737
Published electronically: March 13, 2019
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Abstract | References | Additional Information

Abstract: Let $ U$ be a smooth scheme over an algebraically closed field $ \mathbin {\mathbb{K}}$ of characteristic zero and let $ f:U\rightarrow \mathbin {\mathbb{A}}^1$ be a regular function, and write $ X=\mathop {\rm Crit}(f)$, as a closed $ \mathbin {\mathbb{K}}$-subscheme of $ U$. The motivic vanishing cycle $ MF_{U,f}^{\rm mot,\phi }$ is an element of the $ \hat \mu $-equivariant motivic Grothendieck ring $ \mathbin {\mathcal M}^{\hat \mu }_X$, defined by Denef and Loeser, and Looijenga, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants.

We prove three main results:

(a) $ MF_{U,f}^{\rm mot,\phi }$ depends only on the third-order thickenings $ U^{(3)},f^{(3)}$ of $ U,f$.

(b) If $ V$ is another smooth $ \mathbin {\mathbb{K}}$-scheme, $ g:V\rightarrow \mathbin {\mathbb{A}}^1$ is regular, $ Y=\mathop {\rm Crit}(g)$, and $ \Phi :U\rightarrow V$ is an embedding with $ f=g\circ \Phi $ and $ \Phi \vert _X:X\rightarrow Y$ an isomorphism, then $ \Phi \vert _X^*\bigl (MF^{\rm mot, \phi }_{V,g}\bigr )=MF^{\rm mot, \phi }_{U,f}\odot \Upsilon (P_\Phi )$ in a certain quotient ring $ \mathbin {\smash {\,\,\overline {\!\!\mathcal M\!}\,}}^{\hat \mu }_X$ of $ \mathbin {\mathcal M}^{\hat \mu }_X$, where $ P_\Phi \rightarrow X$ is a principal $ \mathbin {\mathbb{Z}}_2$-bundle associated to $ \Phi $ and $ \Upsilon :\{$principal $ \mathbin {\mathbb{Z}}_2$-bundles on $ X\}\rightarrow \mathbin {\smash {\,\,\overline {\!\!\mathcal M\!}\,}}^{\hat \mu }_X$ a natural morphism.

(c) If $ (X,s)$ is an oriented algebraic d-critical locus in the sense of Joyce, there is a natural motive $ MF_{X,s}\in \mathbin {\smash {\,\,\overline {\!\!\mathcal M\!}\,}}^{\hat \mu }_X$, such that if $ (X,s)$ is locally modelled on $ \mathop {\rm Crit}(f:U\rightarrow \mathbin {\mathbb{A}}^1)$, then $ MF_{X,s}$ is locally modelled on  $ MF_{U,f}^{\rm mot,\phi }$.

Using results of Pantev, Toën, Vaquié, and Vezzosi, these imply the existence of natural motives on moduli schemes of coherent sheaves on a Calabi-Yau 3-fold equipped with ``orientation data'', as required in Kontsevich and Soibelman's motivic Donaldson-Thomas theory, and on intersections $ L\cap M$ of oriented Lagrangians $ L,M$ in an algebraic symplectic manifold $ (S,\omega )$.


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Additional Information

Vittoria Bussi
Affiliation: ICTP, Strada Costiera 11, Trieste, Italy
Email: vittoria87b@gmail.com

Dominic Joyce
Affiliation: The Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom
Email: joyce@maths.ox.ac.uk

Sven Meinhardt
Affiliation: University of Sheffield, School of Mathematics and Statistics, Hicks Building, Hounsfield Road, Sheffield, S3 7RH, United Kingdom
Email: S.Meinhardt@shef.ac.uk

DOI: https://doi.org/10.1090/jag/737
Received by editor(s): April 14, 2014
Received by editor(s) in revised form: November 20, 2018
Published electronically: March 13, 2019
Additional Notes: This research was supported by EPSRC Programme Grant EP/I033343/1 on ‘Motivic invariants and categorification’.
Article copyright: © Copyright 2019 University Press, Inc.