On the Chern numbers of a smooth threefold
HTML articles powered by AMS MathViewer
- by Paolo Cascini and Luca Tasin PDF
- Trans. Amer. Math. Soc. 370 (2018), 7923-7958 Request permission
Abstract:
We study the behaviour of Chern numbers of three-dimensional terminal varieties under divisorial contractions.References
- A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171 (French). MR 751966
- Cinzia Bisi, Paolo Cascini, and Luca Tasin, A remark on the Ueno-Campana’s threefold, Michigan Math. J. 65 (2016), no. 3, 567–572. MR 3542766, DOI 10.1307/mmj/1472066148
- Mirel Caibăr, Minimal models of canonical 3-fold singularities and their Betti numbers, Int. Math. Res. Not. 26 (2005), 1563–1581. MR 2148264, DOI 10.1155/IMRN.2005.1563
- Ivan Cheltsov, Nonrational nodal quartic threefolds, Pacific J. Math. 226 (2006), no. 1, 65–81. MR 2247856, DOI 10.2140/pjm.2006.226.65
- Jungkai A. Chen and Christopher D. Hacon, Factoring 3-fold flips and divisorial contractions to curves, J. Reine Angew. Math. 657 (2011), 173–197. MR 2824787, DOI 10.1515/CRELLE.2011.056
- Jungkai Alfred Chen, Birational maps of 3-folds, Taiwanese J. Math. 19 (2015), no. 6, 1619–1642. MR 3434269, DOI 10.11650/tjm.19.2015.5337
- H.-K. Chen, Betti numbers in three dimensional minimal model program, ArXiv e-prints: 1605.04372 (2016).
- Paolo Cascini and De-Qi Zhang, Effective finite generation for adjoint rings, Ann. Inst. Fourier (Grenoble) 64 (2014), no. 1, 127–144 (English, with English and French summaries). MR 3330543, DOI 10.5802/aif.2841
- Pierre Deligne, Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5–77 (French). MR 498552
- I. M. Gel′fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1264417, DOI 10.1007/978-0-8176-4771-1
- Friedrich Hirzebruch, Some problems on differentiable and complex manifolds, Ann. of Math. (2) 60 (1954), 213–236. MR 66013, DOI 10.2307/1969629
- Christopher D. Hacon and James McKernan, Boundedness of pluricanonical maps of varieties of general type, Invent. Math. 166 (2006), no. 1, 1–25. MR 2242631, DOI 10.1007/s00222-006-0504-1
- Masayuki Kawakita, Three-fold divisorial contractions to singularities of higher indices, Duke Math. J. 130 (2005), no. 1, 57–126. MR 2176548, DOI 10.1215/S0012-7094-05-13013-7
- Yujiro Kawamata, On the plurigenera of minimal algebraic $3$-folds with $K\rlap {\raise 3.25pt\hbox {$∼$}}\rlap {$≈$}\phantom {\approx }0$, Math. Ann. 275 (1986), no. 4, 539–546. MR 859328, DOI 10.1007/BF01459135
- János Kollár and Shigefumi Mori, Classification of three-dimensional flips, J. Amer. Math. Soc. 5 (1992), no. 3, 533–703. MR 1149195, DOI 10.1090/S0894-0347-1992-1149195-9
- János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR 1658959, DOI 10.1017/CBO9780511662560
- János Kollár, Flops, Nagoya Math. J. 113 (1989), 15–36. MR 986434, DOI 10.1017/S0027763000001240
- János Kollár, Effective base point freeness, Math. Ann. 296 (1993), no. 4, 595–605. MR 1233485, DOI 10.1007/BF01445123
- János Kollár, Shafarevich maps and plurigenera of algebraic varieties, Invent. Math. 113 (1993), no. 1, 177–215. MR 1223229, DOI 10.1007/BF01244307
- D. Kotschick, Chern numbers and diffeomorphism types of projective varieties, J. Topol. 1 (2008), no. 2, 518–526. MR 2399142, DOI 10.1112/jtopol/jtn007
- D. Kotschick, Topologically invariant Chern numbers of projective varieties, Adv. Math. 229 (2012), no. 2, 1300–1312. MR 2855094, DOI 10.1016/j.aim.2011.10.020
- Serge Lang, Fundamentals of Diophantine geometry, Springer-Verlag, New York, 1983. MR 715605, DOI 10.1007/978-1-4757-1810-2
- Robert Lazarsfeld, Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. MR 2095471, DOI 10.1007/978-3-642-18808-4
- Claude LeBrun, Topology versus Chern numbers for complex $3$-folds, Pacific J. Math. 191 (1999), no. 1, 123–131. MR 1725466, DOI 10.2140/pjm.1999.191.123
- Anatoly S. Libgober and John W. Wood, Uniqueness of the complex structure on Kähler manifolds of certain homotopy types, J. Differential Geom. 32 (1990), no. 1, 139–154. MR 1064869
- Yoichi Miyaoka, The Chern classes and Kodaira dimension of a minimal variety, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 449–476. MR 946247, DOI 10.2969/aspm/01010449
- Shigefumi Mori, On $3$-dimensional terminal singularities, Nagoya Math. J. 98 (1985), 43–66. MR 792770, DOI 10.1017/S0027763000021358
- Ch. Okonek and A. Van de Ven, Cubic forms and complex $3$-folds, Enseign. Math. (2) 41 (1995), no. 3-4, 297–333. MR 1365849
- Miles Reid, Young person’s guide to canonical singularities, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 345–414. MR 927963
- Morihiko Saito, Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci. 24 (1988), no. 6, 849–995 (1989) (French). MR 1000123, DOI 10.2977/prims/1195173930
- Stefan Schreieder and Luca Tasin, Algebraic structures with unbounded Chern numbers, J. Topol. 9 (2016), no. 3, 849–860. MR 3551840, DOI 10.1112/jtopol/jtw011
- Stefan Schreider and Luca Tasin, Kähler structures on spin 6-manifolds, Math. Ann., DOI 10.1007/s00208-017-1615-2
- J. H. M. Steenbrink, Mixed Hodge structures associated with isolated singularities, Singularities, Part 2 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 513–536. MR 713277, DOI 10.1016/0022-4049(92)90033-c
- Bernd Sturmfels, Algorithms in invariant theory, Texts and Monographs in Symbolic Computation, Springer-Verlag, Vienna, 1993. MR 1255980, DOI 10.1007/978-3-7091-4368-1
Additional Information
- Paolo Cascini
- Affiliation: Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK
- MR Author ID: 674262
- Email: p.cascini@imperial.ac.uk
- Luca Tasin
- Affiliation: Università Roma Tre, Dipartimento di Matematica e Fisica, Largo San Leonardo Murialdo I-00146 Roma, Italy
- Address at time of publication: Mathematical Institute of the University of Bonn, Endenicher Allee 60, D-53115 Bonn, Germany
- MR Author ID: 899050
- Email: tasin@math.uni-bonn.de
- Received by editor(s): June 23, 2016
- Received by editor(s) in revised form: February 18, 2017, and February 28, 2017
- Published electronically: May 17, 2018
- Additional Notes: The first author was partially supported by an EPSRC Grant.
The second author was supported by the DFG Emmy Noether-Nachwuchsgruppe “Gute Strukturen in der höherdimensionalen birationalen Geometrie” and was funded by the Max Planck Institute for Mathematics in Bonn during part of the realization of this project. - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 7923-7958
- MSC (2010): Primary 14E30, 14J30, 51-XX
- DOI: https://doi.org/10.1090/tran/7216
- MathSciNet review: 3852453