The class of the affine line is a zero divisor in the Grothendieck ring
Author:
Lev A. Borisov
Journal:
J. Algebraic Geom. 27 (2018), 203-209
DOI:
https://doi.org/10.1090/jag/701
Published electronically:
June 1, 2017
MathSciNet review:
3764275
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Abstract |
References |
Additional Information
Abstract: We show that the class of the affine line is a zero divisor in the Grothendieck ring of algebraic varieties over complex numbers. The argument is based on the Pfaffian-Grassmannian double mirror correspondence.
References
- Dan Abramovich, Kalle Karu, Kenji Matsuki, and Jarosław Włodarczyk, Torification and factorization of birational maps, J. Amer. Math. Soc. 15 (2002), no. 3, 531–572. MR 1896232, DOI https://doi.org/10.1090/S0894-0347-02-00396-X
- Lev Borisov and Andrei Căldăraru, The Pfaffian-Grassmannian derived equivalence, J. Algebraic Geom. 18 (2009), no. 2, 201–222. MR 2475813, DOI https://doi.org/10.1090/S1056-3911-08-00496-7
- A. Chambert-Loir, private communication.
- Jan Denef and François Loeser, On some rational generating series occurring in arithmetic geometry, Geometric aspects of Dwork theory. Vol. I, II, Walter de Gruyter, Berlin, 2004, pp. 509–526. MR 2099079
- S. Galkin and E. Shinder, The Fano variety of lines and rationality problem for a cubic hypersurface, preprint, arXiv:1405.5154.
- I. Karzhemanov, On the cut-and-paste property of algebraic varieties, preprint, arXiv:1411.6084.
- János Kollár, Yoichi Miyaoka, and Shigefumi Mori, Rationally connected varieties, J. Algebraic Geom. 1 (1992), no. 3, 429–448. MR 1158625
- Alexander Kuznetsov, Lefschetz decompositions and categorical resolutions of singularities, Selecta Math. (N.S.) 13 (2008), no. 4, 661–696. MR 2403307, DOI https://doi.org/10.1007/s00029-008-0052-1
- A. Kuznetsov, private communication.
- Michael Larsen and Valery A. Lunts, Motivic measures and stable birational geometry, Mosc. Math. J. 3 (2003), no. 1, 85–95, 259 (English, with English and Russian summaries). MR 1996804, DOI https://doi.org/10.17323/1609-4514-2003-3-1-85-95
- M. Larsen and V. Lunts, Rationality of motivic zeta function and cut-and-paste problem, preprint, arXiv:1410.7099.
- Daniel Litt, Symmetric powers do not stabilize, Proc. Amer. Math. Soc. 142 (2014), no. 12, 4079–4094. MR 3266979, DOI https://doi.org/10.1090/S0002-9939-2014-12155-1
- Nicolas Martin, The class of the affine line is a zero divisor in the Grothendieck ring: an improvement, C. R. Math. Acad. Sci. Paris 354 (2016), no. 9, 936–939 (English, with English and French summaries). MR 3535349, DOI https://doi.org/10.1016/j.crma.2016.05.016
- Bjorn Poonen, The Grothendieck ring of varieties is not a domain, Math. Res. Lett. 9 (2002), no. 4, 493–497. MR 1928868, DOI https://doi.org/10.4310/MRL.2002.v9.n4.a8
- Einar Andreas Rødland, The Pfaffian Calabi-Yau, its mirror, and their link to the Grassmannian $G(2,7)$, Compositio Math. 122 (2000), no. 2, 135–149. MR 1775415, DOI https://doi.org/10.1023/A%3A1001847914402
References
- Dan Abramovich, Kalle Karu, Kenji Matsuki, and Jarosław Włodarczyk, Torification and factorization of birational maps, J. Amer. Math. Soc. 15 (2002), no. 3, 531–572. MR 1896232, DOI https://doi.org/10.1090/S0894-0347-02-00396-X
- Lev Borisov and Andrei Căldăraru, The Pfaffian-Grassmannian derived equivalence, J. Algebraic Geom. 18 (2009), no. 2, 201–222. MR 2475813, DOI https://doi.org/10.1090/S1056-3911-08-00496-7
- A. Chambert-Loir, private communication.
- Jan Denef and François Loeser, On some rational generating series occurring in arithmetic geometry, Geometric aspects of Dwork theory. Vol. I, II, Walter de Gruyter, Berlin, 2004, pp. 509–526. MR 2099079
- S. Galkin and E. Shinder, The Fano variety of lines and rationality problem for a cubic hypersurface, preprint, arXiv:1405.5154.
- I. Karzhemanov, On the cut-and-paste property of algebraic varieties, preprint, arXiv:1411.6084.
- János Kollár, Yoichi Miyaoka, and Shigefumi Mori, Rationally connected varieties, J. Algebraic Geom. 1 (1992), no. 3, 429–448. MR 1158625
- Alexander Kuznetsov, Lefschetz decompositions and categorical resolutions of singularities, Selecta Math. (N.S.) 13 (2008), no. 4, 661–696. MR 2403307, DOI https://doi.org/10.1007/s00029-008-0052-1
- A. Kuznetsov, private communication.
- Michael Larsen and Valery A. Lunts, Motivic measures and stable birational geometry, Mosc. Math. J. 3 (2003), no. 1, 85–95, 259 (English, with English and Russian summaries). MR 1996804
- M. Larsen and V. Lunts, Rationality of motivic zeta function and cut-and-paste problem, preprint, arXiv:1410.7099.
- Daniel Litt, Symmetric powers do not stabilize, Proc. Amer. Math. Soc. 142 (2014), no. 12, 4079–4094. MR 3266979, DOI https://doi.org/10.1090/S0002-9939-2014-12155-1
- Nicolas Martin, The class of the affine line is a zero divisor in the Grothendieck ring: an improvement, C. R. Math. Acad. Sci. Paris 354 (2016), no. 9, 936–939 (English, with English and French summaries). MR 3535349, DOI https://doi.org/10.1016/j.crma.2016.05.016
- Bjorn Poonen, The Grothendieck ring of varieties is not a domain, Math. Res. Lett. 9 (2002), no. 4, 493–497. MR 1928868, DOI https://doi.org/10.4310/MRL.2002.v9.n4.a8
- Einar Andreas Rødland, The Pfaffian Calabi-Yau, its mirror, and their link to the Grassmannian $G(2,7)$, Compositio Math. 122 (2000), no. 2, 135–149. MR 1775415, DOI https://doi.org/10.1023/A%3A1001847914402
Additional Information
Lev A. Borisov
Affiliation:
Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854
Email:
borisov@math.rutgers.edu
Received by editor(s):
January 6, 2015
Received by editor(s) in revised form:
April 30, 2015, and December 10, 2016
Published electronically:
June 1, 2017
Additional Notes:
The author was partially supported by NSF grant DMS-1201466
Article copyright:
© Copyright 2017
University Press, Inc.
