### Seminar on 0-cycles and related topics

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

The main thrust of this motives seminar is a historical review of the role of 0-cycles on surfaces in the overall development of the modern theory of algebraic cycles, motives and motivic homotopy theory. The story begins with the paper of Mumford on infinite dimensionality, the papers of Roitman on finite dimensionality, the Albanese map and torsion in the group of 0-cycles, then is taken over by Bloch, who introduced both motivic and $K$-theoretic methods. Bloch used these methods to extend the Albanese map to a version defined on the torsion subgroup of cycles (of arbitrary dimension) to an appropriate étale cohomology group, which was studied by Murre, who proved an analog of Roitman's theorem on the torsion 0-cycles for codimension two cycles.

Roitman's theorem does not cover the $p$-torsion in characteristic $p>0$; this gap was filled in by Milne using $K$-theory and work of Kato. Murthy proved an affine version of Roitman's theorem and used this to prove his striking theorem on splitting rank $d$ vector bundles on a $d$-dimensional smooth affine variety over an algebraically closed field. This has been vastly extended by work of Srinivas and Krishna to cover arbitrary affine varieties over an algebraically closed field, and by Schlichting, Asok-Fasel for affine varieties over an arbitrary field.

Coming back to 0-cycles on surfaces, Bloch stated his famous conjecture complementing Mumford's infinite dimensionality results, namely, that a smooth projective surface with $p_g=0$ should have group of degree zero 0-cycles modulo rational equivalence isomorphic to the Albanese variety. With Kas and Lieberman, Bloch proved this for all surfaces not of general type, which gave the study and classification of surfaces of general type with $p_g=0$ an added impetus. A whole series of classification results followed, starting with the case of Godeaux surfaces ($K^2=1$) and becoming gradually more difficult with increasing $K^2$; this includes Mumford's famous example of the ``fake $\mathbb{P}^2$''. It seems that a complete classification and description is not possible. Recently Joseph Ayoub has announced a conservativity result that would imply Bloch's conjecture without resulting to classification; the status of this is at present unclear.

### Program of Lectures

1. 16.10. (Marc Levine) 0-cycles on surfaces and algebraic cycles: an overview

2. 23.10. (Fangzhou Jin) Mumford's infinite dimensionality theorem {Mumford}.

3. 30.10 (Enzo Serandon) Roitman's finite dimensionality theorem on the Albanese map {RoitmanFinite}.

4. 06.11 (Matteo Tamiozzo) Roitman's theorem on torsion 0-cycles {RoitmanTorsion}.

5. 13.11 (Ran Azouri) Milne's theorem on 0-cycles in characteristic $p>0$ {Milne}

6. 20.11 (Daniel Harrer) Bloch's conjecture {BlochK2}

7. 27.11 (Chirantan Chowdhury) Bloch's treatment of Mumford's theorem and the Bloch-Srinivas decomposition of the diagonal [Chap. 1, Appendix]{BlochAlgCyc}, {BlochSrinivas}.

8. 04.12. (Marc Levine) Surfaces with $\kappa<2$ with $p_g=0$ {BlochKasLieberman}

9. 11.12 (special lecture: Christian Haesemeyer-Univ. Melbourne). K-theory of line bundles and singularity.

10 18.12. (Maria Yakeson) Bloch's torsion cycle map and Roitman's theorem {BlochTorsion}

11. 08.01.2019. (Enzo Serandon) Murre's theorem on codimension 2 cycles {Murre}

12.14.01. (Monday, 16 Uhr, WSC-S.14, note special day and room) (special lecture: Elden Elmanto-Univ. Copenhagen) Motivic contractibility of the space of rational maps

*Abstract* I will sketch the proof of an enhancement of a theorem of Gaitsgory concerning the stack of rational maps from a curve to a quasi-affine variety to the level of motives. The motive of this stack is null. I will also explain the context of this result within the study of the moduli stack of G-bundles on curve.

12. 22.01. (Gabriela Guzman) 0-cycles on affine varieties {LevineWeibel, SrinivasTorsion}

13. 29.01. Vector bundles on affine varieties {MKMurthy, Murthy}.

### References

{BlochKasLieberman} Bloch, S.; Kas, A.; Lieberman, D. Zero cycles on surfaces with $p_g=0$. Compositio Math. 33 (1976), no. 2, 135--145.

{BlochSrinivas} Bloch, S.; Srinivas, V. Remarks on correspondences and algebraic cycles. Amer. J. Math. 105 (1983), no. 5, 1235–1253.

{BauerEtAl} Bauer, Ingrid; Catanese, Fabrizio; Pignatelli, Roberto, Surfaces of general type with geometric genus zero: a survey. Complex and differential geometry, 1--48, Springer Proc. Math., 8, Springer, Heidelberg, 2011

{BlochEtAlIdeal} Bloch, Spencer; Murthy, M. Pavaman; Szpiro, Lucien, Zero cycles and the number of generators of an ideal. Colloque en l'honneur de Pierre Samuel (Orsay, 1987). M\'em. Soc. Math. France (N.S.) No. 38 (1989), 51--74.

{Murthy} Murthy, M. Pavaman, Zero cycles and projective modules. Ann. of Math. (2) 140 (1994), no. 2, 405--434.

{MKMurthy} Kumar, N. Mohan; Murthy, M. Pavaman, Algebraic cycles and vector bundles over affine three-folds. Ann. of Math. (2) 116 (1982), no. 3, 579--591.

{LevineWeibel} Levine, Marc; Weibel, Chuck, Zero cycles and complete intersections on singular varieties. J. Reine Angew. Math. 359 (1985), 106--120.

{PedriniWeibel} Pedrini, C.; Weibel, C. Some surfaces of general type for which Bloch's conjecture holds. Recent advances in Hodge theory, 308--329, London Math. Soc. Lecture Note Ser., 427, Cambridge Univ. Press, Cambridge, 2016.

{Levine0Cycles} Levine, Marc, Zero-cycles and K-theory on singular varieties. Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 451--462, Proc. Sympos. Pure Math., 46, Part 2, Amer. Math. Soc., Providence, RI, 1987.

{RoitmanTorsion} Rojtman, A. A. The torsion of the group of 0-cycles modulo rational equivalence. Ann. of Math. (2) 111 (1980), no. 3, 553--569.

{RoitmanFinite} Rojtman, A. A., Rational equivalence of zero-dimensional cycles. Mat. Zametki 28 (1980), no. 1, 85--90, 169; Mat. Sb. (N.S.) 89(131) (1972), 569--585, 671.

{Mumford} Mumford, D., Rational equivalence of 0-cycles on surfaces. J. Math. Kyoto Univ. 9 1968 195--204.

{MumfordFakeP2} Mumford, D. An algebraic surface with $K$ ample, $K^2=9$, $p_g=q=0$. Amer. J. Math. 101 (1979), no. 1, 233--244.

{BlochAlgCyc} Bloch, Spencer, **Lectures on algebraic cycles** Second edition. New Mathematical Monographs, 16. Cambridge University Press, Cambridge, 2010. xxiv+130 pp.

{BlochK2} Bloch, Spencer, $K_2$ of Artinian $\Q$-algebras, with application to algebraic cycles. Comm. Algebra 3 (1975), 405--428.

{BlochTorsion} Bloch, S., Torsion algebraic cycles and a theorem of Roitman. Compositio Math. 39 (1979), no. 1, 107--127.

{BlochMumford} Bloch, S. On an argument of Mumford in the theory of algebraic cycles. Journées de Géometrie Algébrique d'Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, pp. 217--221, Sijthoff \& Noordhoff, Alphen aan den Rijn--Germantown, Md., 1980.

{Murre} Murre, J. P. Applications of algebraic K-theory to the theory of algebraic cycles. in **Algebraic geometry, Sitges (Barcelona), 1983**, 216--261, Lecture Notes in Math., 1124, Springer, Berlin, 1985

{SrinivasTorsion} Srinivas, V. Torsion 0-cycles on affine varieties in characteristic $p$. J. Algebra 120 (1989), no. 2, 428--432.

{Milne} Milne, J. S., Zero cycles on algebraic varieties in nonzero characteristic: Rojtman's theorem. Compositio Math. 47 (1982), no. 3, 271--287.

{BeauvilleVoisin} A. Beauville, C. Voisin. On the Chow ring of a K3 surface, J. Algebraic Geom. 13 (2004), no. 3, 417--426.

{Barlow1} R. Barlow, A simply connected surface of general type with $p_g = 0$, Invent. Math. 79 (1985), no. 2, 293--301.

{Barlow2} R. Barlow, Rational equivalence of zero cycles for some more surfaces with $p_g = 0$, Invent. Math. 79 (1985), no. 2, 303--308.