Derived categories view on rationality problems. (English) Zbl 1368.14029
Pardini, Rita (ed.) et al., Rationality problems in algebraic geometry. Levico Terme, Italy, June 22–27 2015. Lectures of the CIME-CIRM course. Cham: Springer; Florence: Fondazione CIME (ISBN 978-3-319-46208-0/pbk; 978-3-319-46209-7/ebook). Lecture Notes in Mathematics 2172. CIME Foundation Subseries, 67-104 (2016).
Summary: We discuss a relation between the structure of derived categories of smooth projective varieties and their birational properties. We suggest a possible definition of a birational invariant, the derived category analogue of the intermediate Jacobian, and discuss its possible applications to the geometry of prime Fano threefolds and cubic fourfolds.
For the entire collection see [Zbl 1364.14001].
For the entire collection see [Zbl 1364.14001].
MSC:
| 14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |
| 14E08 | Rationality questions in algebraic geometry |
References:
| [1] | N. Addington, On two rationality conjectures for cubic fourfolds. Math. Res. Lett. 23 (1), 1–13 (2016) · Zbl 1375.14134 · doi:10.4310/MRL.2016.v23.n1.a1 |
| [2] | N. Addington, R. Thomas, Hodge theory and derived categories of cubic fourfolds. Duke Math. J 163 (10), 1885–1927 (2014) · Zbl 1309.14014 · doi:10.1215/00127094-2738639 |
| [3] | M. Artin, D. Mumford, Some elementary examples of unirational varieties which are not rational. Proc. Lond. Math. Soc. 3, 75–95 (1972) · Zbl 0244.14017 · doi:10.1112/plms/s3-25.1.75 |
| [4] | A. Beauville, Determinantal hypersurfaces. Mich. Math. J. 48 (1), 39–64 (2000) · Zbl 1076.14534 · doi:10.1307/mmj/1030132707 |
| [5] | A. Beauville, The Lüroth problem, in Rationality Problems in Algebraic Geometry, vol. 2172, ed. by R. Pardini, G.P. Pirola. Lecture Notes in Mathematics (Springer, Berlin, 2016). doi: 10.1007/978-3-319-46209-7_1 · Zbl 1374.14040 · doi:10.1007/978-3-319-46209-7_1 |
| [6] | A. Beilinson, Coherent sheaves on
\[
\mathbb{P}^{n}
\] and problems of linear algebra. Funct. Anal. Appl. 12 (3), 214–216 (1978) · Zbl 0424.14003 · doi:10.1007/BF01681436 |
| [7] | M. Bernardara, M. Bolognesi, Categorical representability and intermediate Jacobians of Fano threefolds, in Derived Categories in Algebraic Geometry (European Mathematical Society, Zürich, 2013), pp. 1–25 · Zbl 1287.18010 · doi:10.4171/115-1/1 |
| [8] | M. Bolognesi, F. Russo, G. Staglianò, Some loci of rational cubic fourfolds. arXiv preprint. arXiv:1504.05863 |
| [9] | A. Bondal, M. Kapranov, Representable functors, Serre functors, and mutations. Izv. RAN. Ser. Mat. 53 (6) 1183–1205 (1989) · Zbl 0703.14011 |
| [10] | A. Bondal, D. Orlov, Semiorthogonal decompositions for algebraic varieties. Preprint math.AG/9506012 |
| [11] | A. Bondal, D. Orlov, Derived categories of coherent sheaves, in Proceedings of the International Congress of Mathematicians, vol. II (Higher Education Press, Beijing, 2002), pp. 47–56 · Zbl 0996.18007 |
| [12] | T. Bridgeland, Equivalences of triangulated categories and Fourier–Mukai transforms. Bull. Lond. Math. Soc. 31 (1), 25–34 (1999) · Zbl 0937.18012 |
| [13] | O. Debarre, A. Iliev, L. Manivel, Special prime Fano fourfolds of degree 10 and index 2, in Recent Advances in Algebraic Geometry, vol. 417 (Cambridge University Press, Cambridge, 2014), p. 123 · Zbl 1326.14094 |
| [14] | O. Debarre, A. Kuznetsov, Gushel–Mukai varieties: classification and birationalities. arXiv preprint math.AG/1510.05448 |
| [15] | S. Gelfand, Yu. Manin, Homological Algebra (Springer, Berlin, 1999) · Zbl 0787.18008 |
| [16] | R. Hartshorne, Residues and Duality, Lecture Notes of a Seminar on the Work of A. Grothendieck, Given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Mathematics, vol. 20 (Springer, Berlin, 1966) |
| [17] | B. Hassett, Special cubic fourfolds. Compos. Math. 120 (1), 1–23 (2000) · Zbl 0956.14031 · doi:10.1023/A:1001706324425 |
| [18] | Sh. Hosono, H. Takagi, Derived categories of Artin-Mumford double solids. Preprint arXiv:1506.02744 |
| [19] | D. Huybrechts, Fourier Mukai Transforms (Oxford University Press, Oxford, 2006) · Zbl 1095.14002 · doi:10.1093/acprof:oso/9780199296866.001.0001 |
| [20] | D. Huybrechts, The K3 category of a cubic fourfold. Preprint arXiv:1505.01775. Appear in Compositio · Zbl 1440.14180 |
| [21] | C. Ingalls, A. Kuznetsov, On nodal Enriques surfaces and quartic double solids. Math. Ann. 361 (1–2), 107–133 (2015) · Zbl 1408.14069 · doi:10.1007/s00208-014-1066-y |
| [22] | K. Kawatani, S. Okawa, Derived categories of smooth proper Deligne-Mumford stacks with p g >0. Preprint 2012 |
| [23] | B. Keller, On differential graded categories, in International Congress of Mathematicians, vol. II (European Mathematical Society, Zürich, 2006), pp. 151–190 · Zbl 1140.18008 |
| [24] | O. Küchle, On Fano 4-folds of index 1 and homogeneous vector bundles over Grassmannians. Math. Z. 218 (1), 563–575 (1995) · Zbl 0826.14024 · doi:10.1007/BF02571923 |
| [25] | A. Kuznetsov, An exceptional set of vector bundles on the varieties V22. Moscow Univ. Math. Bull. 51 (3), 35–37 (1996) |
| [26] | A. Kuznetsov, Derived categories of cubic and V 14 threefolds. Proc. V.A.Steklov Inst. Math. 246, 183–207 (2004) |
| [27] | A. Kuznetsov, Derived categories of Fano threefolds V 12. Mat. Zametki 78 (4), 579–594 (2005); translation in Math. Notes 78 (4), 537–550 (2005) |
| [28] | A. Kuznetsov, Hyperplane sections and derived categories. Izvestiya RAN: Ser. Mat. 70 (3), 23–128 (2006) (in Russian); translation in Izv. Math. 70 (3), 447–547 |
| [29] | A. Kuznetsov, Homological projective duality for Grassmannians of lines. Preprint math.AG/0610957 |
| [30] | A. Kuznetsov, Homological projective duality. Publ. Math. L’IHES, 105 (1), 157–220 (2007) · Zbl 1131.14017 · doi:10.1007/s10240-007-0006-8 |
| [31] | A. Kuznetsov, Derived categories of quadric fibrations and intersections of quadrics. Adv. Math. 218 (5), 1340–1369 (2008) · Zbl 1168.14012 · doi:10.1016/j.aim.2008.03.007 |
| [32] | A. Kuznetsov, Derived categories of Fano threefolds. Proc. V.A. Steklov Inst. Math. 264, 110–122 (2009) · Zbl 1312.14055 · doi:10.1134/S0081543809010143 |
| [33] | A. Kuznetsov, Hochschild homology and semiorthogonal decompositions. Preprint arxiv:0904.4330 |
| [34] | A. Kuznetsov, Derived categories of cubic fourfolds, in Cohomological and Geometric Approaches to Rationality Problems. New Perspectives Progress in Mathematics, vol. 282 (Birkhäuser, Basel, 2010) |
| [35] | A. Kuznetsov, Height of exceptional collections and Hochschild cohomology of quasiphantom categories. J. Reine Angew. Math. (Crelles Journal) 708, 213243 (2015) · Zbl 1331.14024 · doi:10.1515/crelle-2013-0077 |
| [36] | A. Kuznetsov, A simple counterexample to the Jordan-Hölder property for derived categories (2013). arXiv preprint arXiv:1304.0903 |
| [37] | A. Kuznetsov, Semiorthogonal decompositions in algebraic geometry, in Proceedings of the International Congress of Mathematicians (Seoul, 2014), vol. II (2014), pp. 635–660. Preprint math.AG/1404.3143 · Zbl 1373.18009 |
| [38] | A. Kuznetsov, Küchle fivefolds of type c5, Preprint Math.AG/1603.03161, to appear in Math. Z. · Zbl 1352.14029 |
| [39] | A. Kuznetsov, Calabi-Yau and fractional Calabi-Yau categories. Preprint math.AG/1509.07657 |
| [40] | A. Kuznetsov, V. Lunts, Categorical resolutions of irrational singularities. Int. Math. Res. Not. (13), 2015, 4536–4625 (2015) · Zbl 1338.14020 |
| [41] | A. Kuznetsov, A. Perry, Derived categories of Gushel–Mukai varieties. Preprint Math.AG/1605.06568 · Zbl 1401.14181 |
| [42] | J. Lipman, Notes on derived functors and Grothendieck duality, in Foundations of Grothendieck Duality for Diagrams of Schemes. Lecture Notes in Mathematics, vol. 1960 (Springer, Berlin, 2009), pp. 1–259 |
| [43] | N. Markarian, Poincaré–Birkoff–Witt isomorphism, Hochshild homology and Riemann–Roch theorem. Preprint MPIM2001-52 |
| [44] | N. Markarian, The Atiyah class, Hochschild cohomology and the Riemann–Roch theorem J. Lond. Math. Soc. 79, 129–143 (2009) · Zbl 1167.14005 · doi:10.1112/jlms/jdn064 |
| [45] | R. Moschetti, The derived category of a non generic cubic fourfold containing a plane, PreprintMath.AG/1607.06392 |
| [46] | S. Okawa, Semi-orthogonal decomposability of the derived category of a curve. Adv. Math. 228 (5), 2869–2873 (2011) · Zbl 1230.14020 · doi:10.1016/j.aim.2011.06.043 |
| [47] | D. Orlov, Exceptional set of vector bundles on the variety V 5. Vestnik Moskov. Univ. Ser. I Mat. Mekh 5, 69–71 (1991) · Zbl 0753.14012 |
| [48] | J.-L. Verdier, Catégories dérivées: quelques résultats (Etat O) (Institut des hautes ètudes scientifiques, Paris, 1965) |
| [49] | J.-L. Verdier, Des catégories dérivées des catégories abéliennes (French) [On derived categories of abelian categories] With a preface by Luc Illusie. Edited and with a note by Georges Maltsiniotis. Astrisque No. 239, xii+253 pp. (1996/1997) |
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.
