Varieties with $\mathbb {P}$-units
HTML articles powered by AMS MathViewer
- by Andreas Krug PDF
- Trans. Amer. Math. Soc. 370 (2018), 7959-7983 Request permission
Abstract:
We study the class of compact Kähler manifolds with trivial canonical bundle and the property that the cohomology of the trivial line bundle is generated by one element. If the square of the generator is zero, we get the class of strict Calabi–Yau manifolds. If the generator is of degree $2$, we get the class of compact hyperkähler manifolds. We provide some examples and structure results for the cases where the generator is of higher nilpotency index and degree. In particular, we show that varieties of this type are closely related to higher-dimensional Enriques varieties.References
- Roland Abuaf, Homological units, arXiv:1510.01583 (2015).
- Nicolas Addington, New derived symmetries of some hyperkähler varieties, Algebr. Geom. 3 (2016), no. 2, 223–260. MR 3477955, DOI 10.14231/AG-2016-011
- Arnaud Beauville, Some remarks on Kähler manifolds with $c_{1}=0$, Classification of algebraic and analytic manifolds (Katata, 1982) Progr. Math., vol. 39, Birkhäuser Boston, Boston, MA, 1983, pp. 1–26. MR 728605, DOI 10.1007/BF02592068
- Samuel Boissière, Marc Nieper-Wißkirchen, and Alessandra Sarti, Higher dimensional Enriques varieties and automorphisms of generalized Kummer varieties, J. Math. Pures Appl. (9) 95 (2011), no. 5, 553–563 (English, with English and French summaries). MR 2786223, DOI 10.1016/j.matpur.2010.12.003
- Tom Bridgeland, Alastair King, and Miles Reid, The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc. 14 (2001), no. 3, 535–554. MR 1824990, DOI 10.1090/S0894-0347-01-00368-X
- Tom Bridgeland and Antony Maciocia, Fourier–Mukai transforms for quotient varieties, arXiv:math/9811101 (1998).
- Jan Cheah, Cellular decompositions for nested Hilbert schemes of points, Pacific J. Math. 183 (1998), no. 1, 39–90. MR 1616606, DOI 10.2140/pjm.1998.183.39
- S. Cynk and K. Hulek, Higher-dimensional modular Calabi-Yau manifolds, Canad. Math. Bull. 50 (2007), no. 4, 486–503. MR 2364200, DOI 10.4153/CMB-2007-049-9
- Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
- Daniel Huybrechts, Compact hyperkähler manifolds, Calabi-Yau manifolds and related geometries (Nordfjordeid, 2001) Universitext, Springer, Berlin, 2003, pp. 161–225. MR 1963562
- D. Huybrechts, Fourier-Mukai transforms in algebraic geometry, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2006. MR 2244106, DOI 10.1093/acprof:oso/9780199296866.001.0001
- Daniel Huybrechts and Marc Nieper-Wisskirchen, Remarks on derived equivalences of Ricci-flat manifolds, Math. Z. 267 (2011), no. 3-4, 939–963. MR 2776067, DOI 10.1007/s00209-009-0655-z
- Daniel Huybrechts and Richard Thomas, $\Bbb P$-objects and autoequivalences of derived categories, Math. Res. Lett. 13 (2006), no. 1, 87–98. MR 2200048, DOI 10.4310/MRL.2006.v13.n1.a7
- Andreas Krug, On derived autoequivalences of Hilbert schemes and generalized Kummer varieties, Int. Math. Res. Not. IMRN 20 (2015), 10680–10701. MR 3455879, DOI 10.1093/imrn/rnv005
- Andreas Krug and Pawel Sosna, Equivalences of equivariant derived categories, J. Lond. Math. Soc. (2) 92 (2015), no. 1, 19–40. MR 3384503, DOI 10.1112/jlms/jdv014
- Andreas Krug and Pawel Sosna, On the derived category of the Hilbert scheme of points on an Enriques surface, Selecta Math. (N.S.) 21 (2015), no. 4, 1339–1360. MR 3397451, DOI 10.1007/s00029-015-0178-x
- Keiji Oguiso and Stefan Schröer, Enriques manifolds, J. Reine Angew. Math. 661 (2011), 215–235. MR 2863907, DOI 10.1515/CRELLE.2011.077
- Paul Seidel and Richard Thomas, Braid group actions on derived categories of coherent sheaves, Duke Math. J. 108 (2001), no. 1, 37–108. MR 1831820, DOI 10.1215/S0012-7094-01-10812-0
- Claire Voisin, Hodge theory and complex algebraic geometry. I, Reprint of the 2002 English edition, Cambridge Studies in Advanced Mathematics, vol. 76, Cambridge University Press, Cambridge, 2007. Translated from the French by Leila Schneps. MR 2451566
Additional Information
- Andreas Krug
- Affiliation: Universität Marburg, Fachbereich 12 Mathematik und Informatik, Hans-Meerwein-Strasse 6, 35032 Marburg, Germany
- Email: andkrug@outlook.de
- Received by editor(s): July 27, 2016
- Received by editor(s) in revised form: February 3, 2017, and February 21, 2017
- Published electronically: May 30, 2018
- Additional Notes: The early stages of this work were done while the author was financially supported by the research grant KR 4541/1-1 of the DFG (German Research Foundation).
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 7959-7983
- MSC (2010): Primary 14J32; Secondary 14J50, 14F05
- DOI: https://doi.org/10.1090/tran/7218
- MathSciNet review: 3852454