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A maximally-graded invertible cubic threefold that does not admit a full exceptional collection of line bundles

Published online by Cambridge University Press:  16 November 2020

David Favero ,
Daniel Kaplan  and
Tyler L. Kelly
David Favero
Affiliation:
University of Alberta, Department of Mathematical and Statistical Sciences, Central Academic Building 632, Edmonton, AB, Canada T6G 2C7 Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul, Republic of Korea 02455; E-mail: favero@ualberta.ca
Daniel Kaplan
Affiliation:
University of Birmingham, School of Mathematics, Birmingham B15 2TT, United Kingdom; E-mail: d.kaplan@bham.ac.uk
Tyler L. Kelly
Affiliation:
University of Birmingham, School of Mathematics, Birmingham B15 2TT, United Kingdom; E-mail: t.kelly.1@bham.ac.uk

Abstract

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We show that there exists a cubic threefold defined by an invertible polynomial that, when quotiented by the maximal diagonal symmetry group, has a derived category that does not have a full exceptional collection consisting of line bundles. This provides a counterexample to a conjecture of Lekili and Ueda.

Keywords

exceptional collectionsderived categoriestilting object
Type
Algebra
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e56
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

References

Chiodo, A. and Ruan, Y., ‘LG/CY correspondence: the state space isomorphism’, Adv. Math. 227(6): 2157–2188 (2011).CrossRefGoogle Scholar
Efimov, A. I., ‘Maximal lengths of exceptional collections of line bundles’, J. Lond. Math. Soc. 90(2): 350–372 (2014).CrossRefGoogle Scholar
Favero, D., Kaplan, D., and Kelly, T. L., ‘Exceptional collections for mirrors of invertible polynomials’, arxiv:2001:06500.Google Scholar
Futaki, M. and Ueda, K., ‘Homological mirror symmetry for Brieskorn-Pham singularities’, in Proceedings of the 56th Japan Geometry Symposium (Saga University, 2009), 98–107.Google Scholar
Futaki, M. and Ueda, K., ‘Homological mirror symmetry for Brieskorn-Pham singularities’, Selecta Math. (N.S.) 17(2): 435–452 (2011).CrossRefGoogle Scholar
Griffiths, P., ‘On periods of certain rational integrals: I’, Ann of Math. 90: 460–495 (1969).CrossRefGoogle Scholar
Halpern-Leistner, D. and Pomerleano, D., ‘Equivariant Hodge theory and noncommutative geometry’, arxiv:1507.01924v2.Google Scholar
Habermann, M. and Smith, J., ‘Homological Berglund-Hübsch mirror symmetry for curve singularities’, arxiv:1903.01351.Google Scholar
Hartshorne, R., ‘Ample subvarieties of algebraic varieties’, Springer Lect. Notes Math. 156 (1970).CrossRefGoogle Scholar
Hille, L. and Perling, M., ‘A counterexample to King’s conjecture’, Compos. Math. 142(6): 1507–1521 (2006).CrossRefGoogle Scholar
Hirano, Y., ‘Derived Knörrer periodicity and Orlov’s theorem for Gauged Landau-Ginzburg models’, Compos. Math. 153: 973–1007 (2017).CrossRefGoogle Scholar
Hirano, Y. and Ouchi, G., ‘Derived factorization categories of non-Thom–Sebastiani-type sum of potentials’, arxiv:1809.09940.Google Scholar
Kajiura, H., Saito, K., and Takahashi, A., ‘Matrix factorizations and representations of quivers II. Type ADE case’, Adv. Math. 211(1): 327–362 (2007).CrossRefGoogle Scholar
Kajiura, H., Saito, K., and Takahashi, A., ‘Triangulated categories of matrix factorizations for regular systems of weights with $\epsilon =-1$ ’, Adv. Math. 220(5): 1602–1654 (2009).CrossRefGoogle Scholar
Kawamata, Y., ‘Derived categories of toric varieties’, Michigan Math. J. 54(3): 517–535 (2006).CrossRefGoogle Scholar
Kawamata, Y., ‘Derived categories of toric varieties II’, Michigan Math. J. 62(2): 353–363 (2013).CrossRefGoogle Scholar
Krawitz, M., ‘FJRW rings and Landau-Ginzburg mirror symmetry’, arxiv:0906.0796.Google Scholar
Kravets, O., ‘Categories of singularities of invertible polynomials’, arxiv:1911.09859.Google Scholar
Kreuzer, M. and Skarke, H., ‘On the classification of quasihomogeneous functions’, Comm. Math. Phys. 150(1): 137–147 (1992).CrossRefGoogle Scholar
Lekili, Y. and Ueda, K., ‘Homological mirror symmetry for Milnor fibers via moduli of ${A}_{\infty }$ -structures’, arxiv:1806.04345v2.Google Scholar
Takahashi, A., ‘HMS for isolated hypersurface singularities’, Workshop on Homological Mirror Symmetry and Related Topics, 19–24 January 2009, University of Miami, http://www-math.mit.edu/~auroux/frg/miami09-notes/.Google Scholar