Journal of Noncommutative Geometry

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Volume 10, Issue 3, 2016, pp. 981–1042
DOI: 10.4171/JNCG/253

Matrix factorizations and motivic measures

Valery A. Lunts[1] and Olaf M. Schnürer[2]

(1) Department of Mathematics, Indiana University, Rawles Hall, 831 East 3rd St, IN 47405, Bloomington, USA
(2) Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115, Bonn, Germany

This article is the continuation of [17]. We use categories of matrix factorizations to define a morphism of rings (= a Landau–Ginzburg motivic measure) from the (motivic) Grothendieck ring of varieties over $\mathbb A^1$ to the Grothendieck ring of saturated dg categories (with relations coming from semi-orthogonal decompositions into admissible subcategories). Our Landau–Ginzburg motivic measure is the analog for matrix factorizations of the motivic measure in [5] whose definition involved bounded derived categories of coherent sheaves. On the way we prove smoothness and a Thom–Sebastiani theorem for enhancements of categories of matrix factorizations.

Keywords: Matrix factorization, motivic measure, Grothendieck ring of saturated dg categories, smoothness, Thom–Sebastiani theorem

Lunts Valery, Schnürer Olaf: Matrix factorizations and motivic measures. J. Noncommut. Geom. 10 (2016), 981-1042. doi: 10.4171/JNCG/253