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LAGRANGIAN FIBRATIONS OF HYPERKÄHLER FOURFOLDS

Published online by Cambridge University Press:  20 July 2020

Daniel Huybrechts Open the ORCID record for Daniel Huybrechts [Opens in a new window]  and
Chenyang Xu
Daniel Huybrechts
Affiliation:
Mathematisches Institut and Hausdorff Center for Mathematics, Universität Bonn, Endenicher Allee 60, 53115Bonn, Germany ([email protected])
Chenyang Xu
Affiliation:
Mathematics Department, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA ([email protected]) BICMR, Beijing, China ([email protected])
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Abstract

The base surface $B$ of a Lagrangian fibration of a projective, irreducible symplectic fourfold $X$ is shown to be isomorphic to $\mathbb{P}^{2}$.

Keywords

Lagrangian fibrationsirreducible symplectic varietieshyperkähler manifolds
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 3 , May 2022 , pp. 921 - 932
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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Footnotes

DH is supported by the SFB/TR 45 ‘Periods, Moduli Spaces and Arithmetic of Algebraic Varieties’ of the DFG (German Research Foundation). CX is partially supported by a Chern Professorship of the MSRI (NSF No. DMS-1440140) and by the National Science Fund for Distinguished Young Scholars (NSFC 11425101) ‘Algebraic Geometry’.

References

Alexeev, V., Log canonical singularities and complete moduli of stable pairs, preprint, 1996, arXiv:alg-geom/9608013.Google Scholar
Alexeev, V., Complete moduli in the presence of semiabelian group action, Ann. of Math. 155 (2002), 611–708.CrossRefGoogle Scholar
Alexeev, V. and Nakamura, I., On Mumford’s construction of degenerating abelian varieties, Tohoku Math. J. 51 (1999), 399–420.CrossRefGoogle Scholar
Berkovich, V., Spectral theory and analytic geometry over non-Archimedean fields, in Math. Surv. Mon. (American Mathematical Society, Providence, 1990).Google Scholar
Bogomolov, F. and Kurnosov, N., Lagrangian fibrations for IHS fourfolds, preprint, 2018, arXiv:1810.11011.Google Scholar
Brieskorn, E., Rationale Singularitäten komplexer Flächen, Invent. Math. 4 (1968), 336–358.CrossRefGoogle Scholar
Casalaina-Martin, S., A tour of stable reduction with applications, in A Celebration of Algebraic Geometry, Clay Math. Proc., Volume 18, pp. 65–117 (American Mathematical Society, Providence, 2013).Google Scholar
de Fernex, T., Kollár, J. and Xu, C., The dual complex of singularities, in Higher Dimensional Algebraic Geometry–in Honour of Professor Yujiro Kawamata’s Sixtieth Birthday, Adv. Stud. Pure Math., Volume 74, pp. 103–129 (Math. Soc. Japan, Tokyo, 2017).CrossRefGoogle Scholar
Edmonds, A., Transformation Groups and Low-Dimensional Manifolds, Contemp. Math., Volume 36, pp. 339–366 (American Mathematical Society, 1985).Google Scholar
Faltings, G. and Chai, C.-L., Degeneration of Abelian Varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3, Folge/A Series of Modern Surveys in Mathematics, Volume 22, (Springer, 1990).CrossRefGoogle Scholar
Frantzen, K., K3-surfaces with special symmetry, PhD thesis, Bochum (2008), http://webdoc.sub.gwdg.de/ebook/dissts/Bochum/Frantzen2008.pdf.Google Scholar
Fujiki, A., Finite automorphism groups of complex tori of dimension two, Publ. RIMS, Kyoto Univ. 24 (1988), 1–97.CrossRefGoogle Scholar
Ghys, E., Groups acting on the circle, Enseign. Math. 47(2) (2001), 329–407.Google Scholar
Friedman, R. and Morrison, D., The birational geometry of degenerations, Birkhäuser Prog. Math. 29 (1983).Google Scholar
Grothendieck, A., Groupes de Monodromie en Géométrie Algébrique I, SGA 7 I LNM, Volume 288, (Springer, 1972).Google Scholar
Halle, L. and Nicaise, J., Motivic zeta functions of degenerating Calabi–Yau varieties, Math. Ann. 370 (2018), 1277–1320.CrossRefGoogle Scholar
Hogadi, A. and Xu, C., Degenerations of rationally connected varieties, Trans. Amer. Math. Soc. 361 (2009), 3931–3949.CrossRefGoogle Scholar
Huybrechts, D., Compact hyperkähler manifolds, in Calabi–Yau Manifolds and Related Geometries (Nordfjordeid, 2001), Universitext, pp. 161–225 (Springer, Berlin, 2003).CrossRefGoogle Scholar
Hwang, J.-M., Base manifolds for fibrations of projective irreducible symplectic manifolds, Invent. Math. 174 (2008), 625–644.CrossRefGoogle Scholar
Hwang, J.-M. and Oguiso, K., Multiple fibers of holomorphic Lagrangian fibrations, Commun. Contemp. Math. 13 (2011), 309–329.CrossRefGoogle Scholar
Katsura, T., Generalized Kummer surfaces and their unirationality in characteristic p , J. Fac. Sci. Univ. Tokyo, Math. 34 (1987), 1–41.Google Scholar
Kollár, J., Singularities of the Minimal Model Program, Cambridge Tracts in Mathematics, Volume 200 (2013).CrossRefGoogle Scholar
Kollár, J., Nicaise, J. and Xu, C., Semi-stable extensions over 1-dimensional bases, Acta Math. Sinica 34 (2018), 103–113.CrossRefGoogle Scholar
Kovács, S. and Patakfalvi, Z., Projectivity of the moduli space of stable log-varieties and subadditivity of log-Kodaira dimension, J. Amer. Math. Soc. 30 (2017), 959–1021.CrossRefGoogle Scholar
Lipman, J., Rational singularities with applications to algebraic surfaces and unique factorization, Publ. Math. Inst. Hautes Études Sci. 36 (1969), 195–279.CrossRefGoogle Scholar
Liu, Y. and Xu, C., K-stability of cubic threefolds, Duke Math. J. 168(11) (2019), 2029–2073.CrossRefGoogle Scholar
Markman, E., Lagrangian fibrations of holomorphic-symplectic varieties of K3[n] -type, in Algebraic and Complex Geometry, Springer Proc. Math. Stat., Volume 71, pp. 241–283. (2014).Google Scholar
Markushevich, D., Completely integrable projective symplectic 4-dimensional varieties, Izv. Math. 59 (1995), 159–187.CrossRefGoogle Scholar
Matsuki, K., Introduction to the Mori Program, Universitext (Springer, New York, 2002).CrossRefGoogle Scholar
Matsushita, D., On fibre space structures of a projective irreducible symplectic manifold, Topology 38 (1999), 79–83.CrossRefGoogle Scholar
Matsushita, D., Addendum: ‘On fibre space structures of a projective irreducible symplectic manifold’ [Topology 38 (1999)], Topology 40 (2001), 431–432.CrossRefGoogle Scholar
Matsushita, D., Equidimensionality of Lagrangian fibrations on holomorphic symplectic manifolds, Math. Res. Lett. 7 (2000), 389–391.CrossRefGoogle Scholar
Mori, S., Flip theorem and the existence of minimal models for 3-folds, J. Amer. Math. Soc. 1 (1988), 117–254.CrossRefGoogle Scholar
Mumford, D., The topology of normal singularities of an algebraic surface and a criterion for simplicity, Publ. Math. Inst. Hautes Études Sci. 9 (1961), 5–22.CrossRefGoogle Scholar
Mustaţă, M. and Nicaise, J., Weight functions on non-Archimedean analytic spaces and the Kontsevich–Soibelman skeleton, Algebr. Geom. 2 (2015), 365–404.CrossRefGoogle Scholar
Nicaise, J., Berkovich skeleta and birational geometry, in Nonarchimedean and Tropical Geometry, Simons Symposia (ed. Baker, M. and Payne, S.), pp. 173–194 (Springer, Cham, 2016).CrossRefGoogle Scholar
Nicaise, J. and Xu, C., The essential skeleton of a degeneration of algebraic varieties, Am. J. Math. 138 (2016), 1645–1667.CrossRefGoogle Scholar
Ou, W., Lagrangian fibration on symplectic fourfolds, Crelle J. Reine Angew. Math. 746 (2019), 117–147.CrossRefGoogle Scholar