# Motives Seminar-SS-2019

### 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}

### References

{Aluffi} Aluffi, Paolo, Limits of Chow groups, and a new construction of Chern-Schwartz-MacPherson classes. Pure Appl. Math. Q. 2 (2006), no. 4, Special Issue: In honor of Robert D. MacPherson. Part 2, 915--941.

{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. https://www.math.utah.edu/~defernex/Utah-VIGRE05-ln.06.0116.pdf

{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. https://webusers.imj-prg.fr/~francois.loeser/notes_seattle_09_04_2008.pdf

{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.http://www.math.lsa.umich.edu/~mmustata/lecture8.pd

{MustataZeta} Mustata, Mircea, Zeta functions in algebraic geometry http://www-personal.umich.edu/~mmustata/zeta_book.pdf

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