Favero, David; Kaplan, Daniel; Kelly, Tyler L. A maximally-graded invertible cubic threefold that does not admit a full exceptional collection of line bundles. (English) Zbl 1460.14044 Forum Math. Sigma 8, Paper No. e56, 8 p. (2020). Summary: 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. Cited in 2 Documents MSC: 14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry Keywords:exceptional collections; derived categories; tilting object PDFBibTeX XMLCite \textit{D. Favero} et al., Forum Math. Sigma 8, Paper No. e56, 8 p. (2020; Zbl 1460.14044) Full Text: DOI arXiv References: [1] Chiodo, A. and Ruan, Y., ‘LG/CY correspondence: the state space isomorphism’, Adv. Math.227(6): 2157-2188 (2011). · Zbl 1245.14038 [2] Efimov, A. I., ‘Maximal lengths of exceptional collections of line bundles’, J. Lond. Math. Soc.90(2): 350-372 (2014). · Zbl 1318.14047 [3] Favero, D., Kaplan, D., and Kelly, T. L., ‘Exceptional collections for mirrors of invertible polynomials’, arxiv:2001:06500. [4] 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. · Zbl 1250.53076 [5] Futaki, M. and Ueda, K., ‘Homological mirror symmetry for Brieskorn-Pham singularities’, Selecta Math. (N.S.)17(2): 435-452 (2011). · Zbl 1250.53076 [6] Griffiths, P., ‘On periods of certain rational integrals: I’, Ann of Math.90: 460-495 (1969). · Zbl 0215.08103 [7] Halpern-Leistner, D. and Pomerleano, D., ‘Equivariant Hodge theory and noncommutative geometry’, arxiv:1507.01924v2. · Zbl 1460.14005 [8] Habermann, M. and Smith, J., ‘Homological Berglund-Hübsch mirror symmetry for curve singularities’, arxiv:1903.01351. · Zbl 1470.53070 [9] Hartshorne, R., ‘Ample subvarieties of algebraic varieties’, Springer Lect. Notes Math. 156 (1970). · Zbl 0208.48901 [10] Hille, L. and Perling, M., ‘A counterexample to King’s conjecture’, Compos. Math.142(6): 1507-1521 (2006). · Zbl 1108.14040 [11] Hirano, Y., ‘Derived Knörrer periodicity and Orlov’s theorem for Gauged Landau-Ginzburg models’, Compos. Math.153: 973-1007 (2017). · Zbl 1370.14019 [12] Hirano, Y. and Ouchi, G., ‘Derived factorization categories of non-Thom-Sebastiani-type sum of potentials’, arxiv:1809.09940. [13] Kajiura, H., Saito, K., and Takahashi, A., ‘Matrix factorizations and representations of quivers II. Type ADE case’, Adv. Math.211(1): 327-362 (2007). · Zbl 1167.16011 [14] 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). · Zbl 1172.18002 [15] Kawamata, Y., ‘Derived categories of toric varieties’, Michigan Math. J.54(3): 517-535 (2006). · Zbl 1159.14026 [16] Kawamata, Y., ‘Derived categories of toric varieties II’, Michigan Math. J.62(2): 353-363 (2013). · Zbl 1322.14075 [17] Krawitz, M., ‘FJRW rings and Landau-Ginzburg mirror symmetry’, arxiv:0906.0796. · Zbl 1250.81087 [18] Kravets, O., ‘Categories of singularities of invertible polynomials’, arxiv:1911.09859. [19] Kreuzer, M. and Skarke, H., ‘On the classification of quasihomogeneous functions’, Comm. Math. Phys.150(1): 137-147 (1992). · Zbl 0767.57019 [20] Lekili, Y. and Ueda, K., ‘Homological mirror symmetry for Milnor fibers via moduli of \({A}_{\infty }\) -structures’, arxiv:1806.04345v2. · Zbl 1476.53104 [21] Takahashi, A., ‘HMS for isolated hypersurface singularities’, Workshop on Homological Mirror Symmetry and Related Topics, 19-24January 2009, University of Miami, http://www-math.mit.edu/ auroux/frg/miami09-notes/. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.