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The class of the affine line is a zero divisor in the Grothendieck ring: via \(G_2\)-Grassmannians. (English) Zbl 1420.14019

Let \(k\) be an algebraically closed field of characteristic zero. The Grothendieck ring \(K_0({\mathcal{V}}/k)\) of algebraic varieties over \(k\) is generated (as an abelian group) by the isomorphism classes of schemes of finite type over \(k\) subject to the relations \([X]=[X\backslash Z] + [Z],\) where \(Z\subset X\) is a closed subscheme with the reduced structure. The product is defined as \([X]\cdot [Y]=[X\times Y].\) The main result of the paper asserts that for a pair of closed subschemes cut out (in certain way depending on a non-zero global section \(s\) of the appropriate homogenous variety) from the pair of Grassmanians of type \(G_2\) one has \(([X]-[Y])\cdot {\mathbb L} =0.\) Moreover, for the general choice of \(s\) one has \([X]\neq [Y] \) and both \(X\) and \(Y\) are smooth Calabi-Yau \(3\)-folds.

MSC:

14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
19A99 Grothendieck groups and \(K_0\)
14A10 Varieties and morphisms
14M15 Grassmannians, Schubert varieties, flag manifolds
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
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[1] Borisov, Lev; C\u{a}ld\u{a}raru, Andrei, The Pfaffian-Grassmannian derived equivalence, J. Algebraic Geom., 18, 2, 201-222 (2009) · Zbl 1181.14020 · doi:10.1090/S1056-3911-08-00496-7
[2] Borisov, Lev A., The class of the affine line is a zero divisor in the Grothendieck ring, J. Algebraic Geom., 27, 2, 203-209 (2018) · Zbl 1415.14006
[3] Sergey Galkin and Evgeny Shinder, The Fano variety of lines and rationality problem for a cubic hypersurface, arXiv:1405.5154 (2014). · Zbl 1408.14068
[4] 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. · Zbl 1421.14011
[5] Atsushi Ito, Makoto Miura, Shinnosuke Okawa, and Kazushi Ueda, Calabi-Yau complete intersections in homogeneous spaces of \(G_2\), arXiv:1606.04076 (2016). · Zbl 1420.14019
[6] Kawamata, Yujiro, Flops connect minimal models, Publ. Res. Inst. Math. Sci., 44, 2, 419-423 (2008) · Zbl 1145.14014 · doi:10.2977/prims/1210167332
[7] Kapustka, Grzegorz; Kapustka, Micha\l, Calabi-Yau threefolds in \(\mathbb{P}^6\), Ann. Mat. Pura Appl. (4), 195, 2, 529-556 (2016) · Zbl 1350.14030 · doi:10.1007/s10231-015-0476-0
[8] Larsen, Michael; Lunts, Valery A., Motivic measures and stable birational geometry, Mosc. Math. J., 3, 1, 85-95, 259 (2003) · Zbl 1056.14015
[9] Liu, Qing; Sebag, Julien, The Grothendieck ring of varieties and piecewise isomorphisms, Math. Z., 265, 2, 321-342 (2010) · Zbl 1195.14003 · doi:10.1007/s00209-009-0518-7
[10] Martin, Nicolas, The class of the affine line is a zero divisor in the Grothendieck ring: an improvement, C. R. Math. Acad. Sci. Paris, 354, 9, 936-939 (2016) · Zbl 1378.14009 · doi:10.1016/j.crma.2016.05.016
[11] Mukai, Shigeru, Polarized \(K3\) surfaces of genus \(18\) and \(20\). Complex projective geometry, Trieste, 1989/Bergen, 1989, London Math. Soc. Lecture Note Ser. 179, 264-276 (1992), Cambridge Univ. Press, Cambridge · Zbl 0774.14035 · doi:10.1017/CBO9780511662652.019
[12] Nicaise, Johannes, A trace formula for varieties over a discretely valued field, J. Reine Angew. Math., 650, 193-238 (2011) · Zbl 1244.14017 · doi:10.1515/CRELLE.2011.008
[13] R\o dland, Einar Andreas, The Pfaffian Calabi-Yau, its mirror, and their link to the Grassmannian \(G(2,7)\), Compositio Math., 122, 2, 135-149 (2000) · Zbl 0974.14026 · doi:10.1023/A:1001847914402
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