The class of the affine line is a zero divisor in the Grothendieck ring: Via $G _{ 2 }$-Grassmannians
Authors:
Atsushi Ito, Makoto Miura, Shinnosuke Okawa and Kazushi Ueda
Journal:
J. Algebraic Geom. 28 (2019), 245-250
DOI:
https://doi.org/10.1090/jag/731
Published electronically:
December 6, 2018
MathSciNet review:
3912058
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Abstract |
References |
Additional Information
Abstract: Motivated by [J. Algebraic Geom. 27 (2018), pp. 203–209] and [C. R. Math. Acad. Sci. Paris 354 (2016), pp. 936–939], we show the equality $\left ( [ X ] - [ Y ] \right ) \cdot [ \mathbb {A} ^{ 1 } ] = 0$ in the Grothendieck ring of varieties, where $( X, Y )$ is a pair of Calabi-Yau 3-folds cut out from the pair of Grassmannians of type $G _{ 2 }$.
References
- 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
- Lev A. Borisov, The class of the affine line is a zero divisor in the Grothendieck ring, J. Algebraic Geom. 27 (2018), no. 2, 203–209. MR 3764275, DOI https://doi.org/10.1090/S1056-3911-2017-00701-X
- Sergey Galkin and Evgeny Shinder, The Fano variety of lines and rationality problem for a cubic hypersurface, arXiv:1405.5154 (2014).
- Atsushi Ito, Daisuke Inoue, and Makoto Miura, Complete intersection Calabi–Yau manifolds with respect to homogeneous vector bundles on Grassmannians, Math. Z. (2018), https://doi.org/10.1007/s00209-018-2163-5.
- Atsushi Ito, Makoto Miura, Shinnosuke Okawa, and Kazushi Ueda, Calabi–Yau complete intersections in homogeneous spaces of $G_2$, arXiv:1606.04076 (2016).
- Yujiro Kawamata, Flops connect minimal models, Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 419–423. MR 2426353, DOI https://doi.org/10.2977/prims/1210167332
- Grzegorz Kapustka and Michał Kapustka, Calabi-Yau threefolds in $\Bbb {P}^6$, Ann. Mat. Pura Appl. (4) 195 (2016), no. 2, 529–556. MR 3476687, DOI https://doi.org/10.1007/s10231-015-0476-0
- 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
- Qing Liu and Julien Sebag, The Grothendieck ring of varieties and piecewise isomorphisms, Math. Z. 265 (2010), no. 2, 321–342. MR 2609314, DOI https://doi.org/10.1007/s00209-009-0518-7
- 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
- Shigeru Mukai, Polarized $K3$ surfaces of genus $18$ and $20$, Complex projective geometry (Trieste, 1989/Bergen, 1989) London Math. Soc. Lecture Note Ser., vol. 179, Cambridge Univ. Press, Cambridge, 1992, pp. 264–276. MR 1201388, DOI https://doi.org/10.1017/CBO9780511662652.019
- Johannes Nicaise, A trace formula for varieties over a discretely valued field, J. Reine Angew. Math. 650 (2011), 193–238. MR 2770561, DOI https://doi.org/10.1515/CRELLE.2011.008
- 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
- 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
- Lev A. Borisov, The class of the affine line is a zero divisor in the Grothendieck ring, J. Algebraic Geom. 27 (2018), no. 2, 203–209. MR 3764275
- Sergey Galkin and Evgeny Shinder, The Fano variety of lines and rationality problem for a cubic hypersurface, arXiv:1405.5154 (2014).
- Atsushi Ito, Daisuke Inoue, and Makoto Miura, Complete intersection Calabi–Yau manifolds with respect to homogeneous vector bundles on Grassmannians, Math. Z. (2018), https://doi.org/10.1007/s00209-018-2163-5.
- Atsushi Ito, Makoto Miura, Shinnosuke Okawa, and Kazushi Ueda, Calabi–Yau complete intersections in homogeneous spaces of $G_2$, arXiv:1606.04076 (2016).
- Yujiro Kawamata, Flops connect minimal models, Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 419–423. MR 2426353, DOI https://doi.org/10.2977/prims/1210167332
- Grzegorz Kapustka and MichałKapustka, Calabi-Yau threefolds in $\mathbb {P}^6$, Ann. Mat. Pura Appl. (4) 195 (2016), no. 2, 529–556. MR 3476687, DOI https://doi.org/10.1007/s10231-015-0476-0
- 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
- Qing Liu and Julien Sebag, The Grothendieck ring of varieties and piecewise isomorphisms, Math. Z. 265 (2010), no. 2, 321–342. MR 2609314, DOI https://doi.org/10.1007/s00209-009-0518-7
- 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
- Shigeru Mukai, Polarized $K3$ surfaces of genus $18$ and $20$, Complex projective geometry (Trieste, 1989/Bergen, 1989) London Math. Soc. Lecture Note Ser., vol. 179, Cambridge Univ. Press, Cambridge, 1992, pp. 264–276. MR 1201388, DOI https://doi.org/10.1017/CBO9780511662652.019
- Johannes Nicaise, A trace formula for varieties over a discretely valued field, J. Reine Angew. Math. 650 (2011), 193–238. MR 2770561, DOI https://doi.org/10.1515/CRELLE.2011.008
- 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
Atsushi Ito
Affiliation:
Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan
MR Author ID:
1019212
Email:
aito@math.kyoto-u.ac.jp
Makoto Miura
Affiliation:
Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul, 130-722, Republic of Korea
MR Author ID:
1050384
Email:
miura@kias.re.kr
Shinnosuke Okawa
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, Machikaneyama 1-1, Toyonaka, Osaka, 560-0043, Japan
MR Author ID:
953215
Email:
okawa@math.sci.osaka-u.ac.jp
Kazushi Ueda
Affiliation:
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan
MR Author ID:
772510
Email:
kazushi@ms.u-tokyo.ac.jp
Received by editor(s):
July 23, 2016
Published electronically:
December 6, 2018
Additional Notes:
The first author was supported by the Grant-in-Aid for JSPS fellows, No. 26–1881. A part of this work was done when the second author was supported by Frontiers of Mathematical Sciences and Physics at University of Tokyo. The second author was also supported by Korea Institute for Advanced Study. The third author was partially supported by Grants-in-Aid for Scientific Research (16H05994, 16K13746, 16H02141, 16K13743, 16K13755, 16H06337) and the Inamori Foundation. The fourth author was partially supported by Grants-in-Aid for Scientific Research (24740043, 15KT0105, 16K13743, 16H03930).
Article copyright:
© Copyright 2018
University Press, Inc.
