Seminar on $K_0(Var_k)$, motivic integration and characteristic classes

Tuesdays WSC-N-U-4.05 16-18 Uhr

This will be an introduction to the Grothendieck ring of varieties and how it has been used in a number of applications, including constructions of motivic nearby fibers and Milnor fibers, relations for Betti numbers and Hodge numbers, as well as constuctions of characteristic classes for singular varieties.

Program of Lectures

1. (16.04-Alessandro D'Angelo) Introduction to $K_0(Var_k)$. Present the material in [§1]{Mustata}. State but do not prove Bittner's theorem, the Theorem of Larsen-Lunt, Poonen's theorem {Poonen} and Borisov's theorem {Borisov}. See also {Blickle}.

2. (23.04-Enzo Serandon) First structure results. Prove Bittner's theorem {Bittner}, the Larsen-Lunt theorem, Poonen's theorem {Poonen} and Borisov's theorem {Borisov}.

3. (30.04-Ran Azouri) Kapranov's motivic zeta function. Follow the outline of [§ 2]{Mustata} , with more details on Totaro's argument [Lemma 4.4]{Go}.

4. (07.05-Chirantan Choudhury) Kapranov's motivic zeta function for curves and surfaces. [§ 3-4]{Mustata}, {LL1}, {LL2}.

5.-6. (14./21.05) Applications to Betti numbers and Hodge numbers: Arc spaces, motivic measures, change of variables formula. This is a survey (without many details) of the results of [§ 2]{Loeser}, [§1-4]{DL00}, [§1-4]{Looij}, [§1-3]{Craw} on this topic. See also {Blickle}.

7. (28.05-Fangzhou Jin) The motivic nearby fiber/Milnor fiber {Bittner1}, [§5]{Looij}, [§ 3]{Loeser}

8. (04.06-Matteo Tamiozzo) MacPherson's Chern classes for singular varieties {MacP}

9. (18.06-Louis-Clément Lefèvre) Chern-Schwartz-MacPherson classes in the Chow ring {Kennedy}. Discuss as well the formula of Gonzalez-Sprinberg, Verdier for the obstruction class.

10, 11. (02.07-Enzo Serandon) $K$-theory of assemblers and $K(Var_k)$ {Zakharevich1}, {Zakharevich2}, see also {Z}


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{Bittner} Bittner, Franziska, The universal Euler characteristic for varieties of characteristic zero. Compos. Math. 140 (2004), no. 4, 1011--1032.

{Bittner1} Bittner, Franziska, On motivic zeta functions and the motivic nearby fiber. Math. Z. 249 (2005), no. 1, 63–83.

{Blickle} Blickle, Manuel, A short course on geometric motivic integration. Motivic integration and its interactions with model theory and non-Archimedean geometry. Volume I, 189--243, London Math. Soc. Lecture Note Ser., 383, Cambridge Univ. Press, Cambridge, 2011.

{Borisov} Borisov, Lev A., The class of the affine line is a zero divisor in the Grothendieck ring. J. Algebraic Geom. 27 (2018), no. 2, 203--209.

{BSY} Brasselet, Jean-Paul; Schürmann, Jörg; Yokura, Shoji, Hirzebruch classes and motivic Chern classes for singular spaces. J. Topol. Anal. 2 (2010), no. 1, 1--55.

{Craw} Craw, Alastair, An introduction to motivic integration. Strings and geometry, 203--225, Clay Math. Proc., 3, Amer. Math. Soc., Providence, RI, 2004.

{DL00} Denef, Jan; Loeser, Francois, Geometry on arc spaces of algebraic varieties. European Congress of Mathematics, Vol. I (Barcelona, 2000), 327--348, Progr. Math., 201, Birkhäuser, Basel, 2001.

{deFernex} de Fernex, Tommaso, Lectures on Relative Motivic Integration, MacPherson's Transformation, and Stringy Chern Classes.

{deFernexEtAl} de Fernex, Tommaso; Lupercio, Ernesto; Nevins, Thomas; Uribe, Bernardo, Stringy Chern classes of singular varieties. Adv. Math. 208 (2007), no. 2, 597–621.

{Go} Göttsche, Lothar,On the motive of the Hilbert scheme of points on a surface. Math. Res. Lett. 8 (2001), no. 5-6, 613--627.

{Kennedy} Kennedy, Gary, MacPherson's Chern classes of singular algebraic varieties. Comm. Algebra 18 (1990), no. 9, 2821--2839.

Larsen, Michael; Lunts, Valery A.,Rationality criteria for motivic zeta functions. Compos. Math. 140 (2004), no. 6, 1537--1560.

Larsen, Michael; Lunts, Valery A., Motivic measures and stable birational geometry. Mosc. Math. J. 3 (2003), no. 1, 85--95, 259.

Loeser, Francois, Seatle Lectures on motivic integration.

{Looij} Looijenga, Eduard, Motivic measures. Séminaire Bourbaki, Vol. 1999/2000. Astérisque No. 276 (2002), 267--297.

MacPherson, R. D., Chern classes for singular algebraic varieties. Ann. of Math. (2) 100 (1974), 423--432.

Mustata, Mircea, Lecture 8. The Grothendieck ring of varieties and Kapranov's motivic zeta function.

{MustataZeta} Mustata, Mircea, Zeta functions in algebraic geometry

Parusinski, Adam; Pragacz, Piotr, Characteristic classes of hypersurfaces and characteristic cycles. J. Algebraic Geom. 10 (2001), no. 1, 63--79.

{Poonen} Poonen, Bjorn, The Grothendieck ring of varieties is not a domain. Math. Res. Lett. 9 (2002), no. 4, 493--497.

{Z} Zakharevich, Inna, Perspectives on scissors congruence. Bull. Amer. Math. Soc. (N.S.) 53 (2016), no. 2, 269--294.

{Zakharevich1} Zakharevich, Inna, The K-theory of assemblers. Adv. Math. 304 (2017), 1176--1218.

{Zakharevich2} Zakharevich, Inna, The annihilator of the Lefschetz motive. Duke Math. J. 166 (2017), no. 11, 1989--2022.


Program notes