A finiteness theorem for Lagrangian fibrations
Author:
Justin Sawon
Journal:
J. Algebraic Geom. 25 (2016), 431-459
DOI:
https://doi.org/10.1090/jag/673
Published electronically:
August 7, 2015
MathSciNet review:
3493589
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We consider (holomorphic) Lagrangian fibrations $\pi :X\rightarrow \mathbb {P}^n$ that satisfy some natural hypotheses. We prove that there are only finitely many such Lagrangian fibrations up to deformation.
References
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- T. Matsusaka and D. Mumford, Two fundamental theorems on deformations of polarized varieties, Amer. J. Math. 86 (1964), 668–684. MR 171778, DOI https://doi.org/10.2307/2373030
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- Daisuke Matsushita, On singular fibres of Lagrangian fibrations over holomorphic symplectic manifolds, Math. Ann. 321 (2001), no. 4, 755–773. MR 1872528, DOI https://doi.org/10.1007/s002080100251
- Daisuke Matsushita, Higher direct images of dualizing sheaves of Lagrangian fibrations, Amer. J. Math. 127 (2005), no. 2, 243–259. MR 2130616
- Daisuke Matsushita, A canonical bundle formula for projective Lagrangian fibrations, preprint arXiv:0701.0122.
- David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay; by Hindustan Book Agency, New Delhi, 2008. With appendices by C. P. Ramanujam and Yuri Manin; Corrected reprint of the second (1974) edition. MR 2514037
- Yukihiko Namikawa, Toroidal compactification of Siegel spaces, Lecture Notes in Mathematics, vol. 812, Springer, Berlin, 1980. MR 584625
- Kieran G. O’Grady, Desingularized moduli spaces of sheaves on a $K3$, J. Reine Angew. Math. 512 (1999), 49–117. MR 1703077, DOI https://doi.org/10.1515/crll.1999.056
- Keiji Oguiso, Shioda-Tate formula for an abelian fibered variety and applications, J. Korean Math. Soc. 46 (2009), no. 2, 237–248. MR 2494474, DOI https://doi.org/10.4134/JKMS.2009.46.2.237
- Antonio Rapagnetta, On the Beauville form of the known irreducible symplectic varieties, Math. Ann. 340 (2008), no. 1, 77–95. MR 2349768, DOI https://doi.org/10.1007/s00208-007-0139-6
- Justin Sawon, Derived equivalence of holomorphic symplectic manifolds, Algebraic structures and moduli spaces, CRM Proc. Lecture Notes, vol. 38, Amer. Math. Soc., Providence, RI, 2004, pp. 193–211. MR 2096146, DOI https://doi.org/10.1090/crmp/038/09
- Justin Sawon, On the discriminant locus of a Lagrangian fibration, Math. Ann. 341 (2008), no. 1, 201–221. MR 2377475, DOI https://doi.org/10.1007/s00208-007-0189-9
- Justin Sawon, Twisted Fourier-Mukai transforms for holomorphic symplectic four-folds, Adv. Math. 218 (2008), no. 3, 828–864. MR 2414323, DOI https://doi.org/10.1016/j.aim.2008.01.013
- Justin Sawon, Deformations of holomorphic Lagrangian fibrations, Proc. Amer. Math. Soc. 137 (2009), no. 1, 279–285. MR 2439451, DOI https://doi.org/10.1090/S0002-9939-08-09473-2
- Justin Sawon, On Lagrangian fibrations by Jacobians I, J. Reine Angew. Math. 701 (2015), 127–151. MR 3331728, DOI https://doi.org/10.1515/crelle-2013-0023
- N. I. Shepherd-Barron, Perfect forms and the moduli space of abelian varieties, Invent. Math. 163 (2006), no. 1, 25–45. MR 2208417, DOI https://doi.org/10.1007/s00222-005-0453-0
References
- Wolf P. Barth, Klaus Hulek, Chris A. M. Peters, and Antonius Van de Ven, Compact complex surfaces, 2nd ed., 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. 4, Springer-Verlag, Berlin, 2004. MR 2030225 (2004m:14070)
- Mark Gross, A finiteness theorem for elliptic Calabi-Yau threefolds, Duke Math. J. 74 (1994), no. 2, 271–299. MR 1272978 (95c:14047), DOI https://doi.org/10.1215/S0012-7094-94-07414-0
- Samuel Grushevsky, Geometry of $\mathcal {A}_g$ and its compactifications, Algebraic geometry—Seattle 2005. Part 1, Proc. Sympos. Pure Math., vol. 80, Amer. Math. Soc., Providence, RI, 2009, pp. 193–234. MR 2483936 (2010h:14071), DOI https://doi.org/10.1090/pspum/080.1/2483936
- Christopher D. Hacon, A derived category approach to generic vanishing, J. Reine Angew. Math. 575 (2004), 173–187. MR 2097552 (2005m:14026), DOI https://doi.org/10.1515/crll.2004.078
- Hongyu He and Jerome William Hoffman, Picard groups of Siegel modular 3-folds and $\theta$-liftings, J. Lie Theory 22 (2012), no. 3, 769–801. MR 3012154
- Klaus Hulek, Constantin Kahn, and Steven H. Weintraub, Moduli spaces of abelian surfaces: compactification, degenerations, and theta functions, de Gruyter Expositions in Mathematics, vol. 12, Walter de Gruyter & Co., Berlin, 1993. MR 1257185 (95e:14034)
- Klaus Hulek and Gregory Kumar Sankaran, The nef cone of toroidal compactifications of $\mathcal {A}_4$, Proc. London Math. Soc. (3) 88 (2004), no. 3, 659–704. MR 2044053 (2005a:14061), DOI https://doi.org/10.1112/S0024611503014564
- Daniel Huybrechts, Finiteness results for compact hyperkähler manifolds, J. Reine Angew. Math. 558 (2003), 15–22. MR 1979180 (2004b:53066), DOI https://doi.org/10.1515/crll.2003.038
- Jun-Muk Hwang, Base manifolds for fibrations of projective irreducible symplectic manifolds, Invent. Math. 174 (2008), no. 3, 625–644. MR 2453602 (2010a:14076), DOI https://doi.org/10.1007/s00222-008-0143-9
- Jun-Muk Hwang and Keiji Oguiso, Characteristic foliation on the discriminant hypersurface of a holomorphic Lagrangian fibration, Amer. J. Math. 131 (2009), no. 4, 981–1007. MR 2543920 (2010g:32028), DOI https://doi.org/10.1353/ajm.0.0062
- Jun-Muk Hwang and Keiji Oguiso, Multiple fibers of holomorphic Lagrangian fibrations, Commun. Contemp. Math. 13 (2011), no. 2, 309–329. MR 2794489 (2012g:32021), DOI https://doi.org/10.1142/S0219199711004269
- Jun-Muk Hwang and Keiji Oguiso, Local structure of principally polarized stable Lagrangian fibrations, preprint arXiv:1007.2043.
- Yujiro Kawamata, Subadjunction of log canonical divisors. II, Amer. J. Math. 120 (1998), no. 5, 893–899. MR 1646046 (2000d:14020)
- János Kollár, Higher direct images of dualizing sheaves. I, Ann. of Math. (2) 123 (1986), no. 1, 11–42. MR 825838 (87c:14038), DOI https://doi.org/10.2307/1971351
- János Kollár, Kodaira’s canonical bundle formula and adjunction, Flips for 3-folds and 4-folds, Oxford Lecture Ser. Math. Appl., vol. 35, Oxford Univ. Press, Oxford, 2007, pp. 134–162. MR 2359346, DOI https://doi.org/10.1093/acprof%3Aoso/9780198570615.003.0008
- János Kollár, Hulls and husks, preprint arXiv:0805:0576.
- János Kollár, Exercises in the birational geometry of algebraic varieties, Analytic and algebraic geometry, IAS/Park City Math. Ser., vol. 17, Amer. Math. Soc., Providence, RI, 2010, pp. 495–524. MR 2743822 (2012d:14015)
- János Kollár, Karen E. Smith, and Alessio Corti, Rational and nearly rational varieties, Cambridge Studies in Advanced Mathematics, vol. 92, Cambridge University Press, Cambridge, 2004. MR 2062787 (2005i:14063)
- Teruhisa Matsusaka and David Mumford, Two fundamental theorems on deformations of polarized varieties, Amer. J. Math. 86 (1964), 668–684. MR 0171778 (30 \#2005)
- Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273 (88h:13001)
- Daisuke Matsushita, On fibre space structures of a projective irreducible symplectic manifold, Topology 38 (1999), no. 1, 79–83. MR 1644091 (99f:14054), DOI https://doi.org/10.1016/S0040-9383%2898%2900003-2
- Daisuke Matsushita, On singular fibres of Lagrangian fibrations over holomorphic symplectic manifolds, Math. Ann. 321 (2001), no. 4, 755–773. MR 1872528 (2002i:32017), DOI https://doi.org/10.1007/s002080100251
- Daisuke Matsushita, Higher direct images of dualizing sheaves of Lagrangian fibrations, Amer. J. Math. 127 (2005), no. 2, 243–259. MR 2130616 (2006b:14075)
- Daisuke Matsushita, A canonical bundle formula for projective Lagrangian fibrations, preprint arXiv:0701.0122.
- David Mumford, Abelian varieties, with appendices by C. P. Ramanujam and Yuri Manin, corrected reprint of the second (1974) edition. Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, published for the Tata Institute of Fundamental Research, Bombay, by Hindustan Book Agency, New Delhi, 2008. MR 2514037 (2010e:14040)
- Yukihiko Namikawa, Toroidal compactification of Siegel spaces, Lecture Notes in Mathematics, vol. 812, Springer, Berlin, 1980. MR 584625 (82a:32034)
- Kieran G. O’Grady, Desingularized moduli spaces of sheaves on a $K3$, J. Reine Angew. Math. 512 (1999), 49–117. MR 1703077 (2000f:14066), DOI https://doi.org/10.1515/crll.1999.056
- Keiji Oguiso, Shioda-Tate formula for an abelian fibered variety and applications, J. Korean Math. Soc. 46 (2009), no. 2, 237–248. MR 2494474 (2009m:14011), DOI https://doi.org/10.4134/JKMS.2009.46.2.237
- Antonio Rapagnetta, On the Beauville form of the known irreducible symplectic varieties, Math. Ann. 340 (2008), no. 1, 77–95. MR 2349768 (2008i:14022), DOI https://doi.org/10.1007/s00208-007-0139-6
- Justin Sawon, Derived equivalence of holomorphic symplectic manifolds, Algebraic structures and moduli spaces, CRM Proc. Lecture Notes, vol. 38, Amer. Math. Soc., Providence, RI, 2004, pp. 193–211. MR 2096146 (2005m:14075)
- Justin Sawon, On the discriminant locus of a Lagrangian fibration, Math. Ann. 341 (2008), no. 1, 201–221. MR 2377475 (2008j:32021), DOI https://doi.org/10.1007/s00208-007-0189-9
- Justin Sawon, Twisted Fourier-Mukai transforms for holomorphic symplectic four-folds, Adv. Math. 218 (2008), no. 3, 828–864. MR 2414323 (2009g:14046), DOI https://doi.org/10.1016/j.aim.2008.01.013
- Justin Sawon, Deformations of holomorphic Lagrangian fibrations, Proc. Amer. Math. Soc. 137 (2009), no. 1, 279–285. MR 2439451 (2009j:32013), DOI https://doi.org/10.1090/S0002-9939-08-09473-2
- Justin Sawon, On Lagrangian fibrations by Jacobians I, J. Reine Angew. Math. 701 (2015), 127–151. MR 3331728, DOI https://doi.org/10.1515/crelle-2013-0023
- Nicholas I. Shepherd-Barron, Perfect forms and the moduli space of abelian varieties, Invent. Math. 163 (2006), no. 1, 25–45. MR 2208417 (2007e:14070), DOI https://doi.org/10.1007/s00222-005-0453-0
Additional Information
Justin Sawon
Affiliation:
Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599-3250
MR Author ID:
653333
Email:
sawon@email.unc.edu
Received by editor(s):
February 17, 2013
Received by editor(s) in revised form:
May 20, 2014, and November 13, 2014
Published electronically:
August 7, 2015
Article copyright:
© Copyright 2015
University Press, Inc.