$\mathbb {A}^1$-equivalence of zero cycles on surfaces
HTML articles powered by AMS MathViewer
- by Yi Zhu PDF
- Trans. Amer. Math. Soc. 370 (2018), 6735-6749 Request permission
Abstract:
In this paper, we study $\mathbb {A}^1$-equivalence classes of zero cycles on open algebraic surfaces. We prove the logarithmic version of Mumford’s theorem on zero cycles. We also prove that the log Bloch conjecture holds for surfaces with log Kodaira dimension $-\infty$.References
- Dan Abramovich and Qile Chen, Stable logarithmic maps to Deligne-Faltings pairs II, Asian J. Math. 18 (2014), no. 3, 465–488. MR 3257836, DOI 10.4310/AJM.2014.v18.n3.a5
- S. Bloch, A. Kas, and D. Lieberman, Zero cycles on surfaces with $p_{g}=0$, Compositio Math. 33 (1976), no. 2, 135–145. MR 435073
- Spencer Bloch, Lectures on algebraic cycles, Duke University Mathematics Series, IV, Duke University, Mathematics Department, Durham, N.C., 1980. MR 558224
- Qile Chen, Stable logarithmic maps to Deligne-Faltings pairs I, Ann. of Math. (2) 180 (2014), no. 2, 455–521. MR 3224717, DOI 10.4007/annals.2014.180.2.2
- Qile Chen and Yi Zhu, $\mathbb {A}^1$-curves on log smooth varieties, submitted, Journal für die reine und angewandte Mathematik, DOI 10.1515/crelle-2017-0028. arXiv:1407.5476.
- Qile Chen and Yi Zhu, Very free curves on Fano complete intersections, Algebr. Geom. 1 (2014), no. 5, 558–572. MR 3296805, DOI 10.14231/AG-2014-024
- Thomas Geisser, On Suslin’s singular homology and cohomology, Doc. Math. Extra vol.: Andrei A. Suslin sixtieth birthday (2010), 223–249. MR 2804255
- Mark Gross, Tropical geometry and mirror symmetry, CBMS Regional Conference Series in Mathematics, vol. 114, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2011. MR 2722115, DOI 10.1090/cbms/114
- Mark Gross and Bernd Siebert, Logarithmic Gromov-Witten invariants, J. Amer. Math. Soc. 26 (2013), no. 2, 451–510. MR 3011419, DOI 10.1090/S0894-0347-2012-00757-7
- Shigeru Iitaka, Logarithmic forms of algebraic varieties, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), no. 3, 525–544. MR 429884
- Kazuya Kato, Logarithmic structures of Fontaine-Illusie, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988) Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 191–224. MR 1463703
- János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR 1658959, DOI 10.1017/CBO9780511662560
- Seán Keel and James McKernan, Rational curves on quasi-projective surfaces, Mem. Amer. Math. Soc. 140 (1999), no. 669, viii+153. MR 1610249, DOI 10.1090/memo/0669
- Masayoshi Miyanishi and Shuichiro Tsunoda, Logarithmic del Pezzo surfaces of rank one with noncontractible boundaries, Japan. J. Math. (N.S.) 10 (1984), no. 2, 271–319. MR 884422, DOI 10.4099/math1924.10.271
- D. Mumford, Rational equivalence of $0$-cycles on surfaces, J. Math. Kyoto Univ. 9 (1968), 195–204. MR 249428, DOI 10.1215/kjm/1250523940
- Carlo Mazza, Vladimir Voevodsky, and Charles Weibel, Lecture notes on motivic cohomology, Clay Mathematics Monographs, vol. 2, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2006. MR 2242284
- Arthur Ogus, Lectures on logarithmic algebraic geometry, TeXed notes (2006).
- Michael Rosen, Number theory in function fields, Graduate Texts in Mathematics, vol. 210, Springer-Verlag, New York, 2002. MR 1876657, DOI 10.1007/978-1-4757-6046-0
- Jean-Pierre Serre, Algebraic groups and class fields, Graduate Texts in Mathematics, vol. 117, Springer-Verlag, New York, 1988. Translated from the French. MR 918564, DOI 10.1007/978-1-4612-1035-1
- Michael Spieß and Tamás Szamuely, On the Albanese map for smooth quasi-projective varieties, Math. Ann. 325 (2003), no. 1, 1–17. MR 1957261, DOI 10.1007/s00208-002-0359-8
- The Stacks Project Authors, stacks project, http://stacks.math.columbia.edu, 2015.
- Claire Voisin, Hodge theory and complex algebraic geometry. II, Cambridge Studies in Advanced Mathematics, vol. 77, Cambridge University Press, Cambridge, 2003. Translated from the French by Leila Schneps. MR 1997577, DOI 10.1017/CBO9780511615177
- Yi Zhu, Log rationally connected surfaces, Math. Res. Lett. 23 (2016), no. 5, 1527–1536. MR 3601077, DOI 10.4310/MRL.2016.v23.n5.a13
Additional Information
- Yi Zhu
- Affiliation: Department of Pure Mathematics, Univeristy of Waterloo, Waterloo, Ontario N2L3G1, Canada
- MR Author ID: 1094131
- Email: yi.zhu@uwaterloo.ca
- Received by editor(s): October 28, 2015
- Received by editor(s) in revised form: January 5, 2017, and January 6, 2017
- Published electronically: April 4, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 6735-6749
- MSC (2010): Primary 14C15, 14C25, 19E15
- DOI: https://doi.org/10.1090/tran/7178
- MathSciNet review: 3814346