Lagrangian fibrations on hyperkähler manifolds - On a question of Beauville

Greb, Daniel; Lehn, Christian; Rollenske, Sönke

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 46 (2013) no. 3, pp. 375-403.

Soit $X$ une variété hyperkählérienne compacte contenant un tore complexe $L$ en tant que sous-variété lagrangienne. A. Beauville a posé la question suivante : la variété $X$ admet-elle une fibration lagrangienne de fibre $L$ ? Nous démontrons que c’est le cas si $X$ n’est pas projective. Si $X$ est projective nous montrons l’existence d’une fibration lagrangienne presque holomorphe de fibre $L$ sous des hypothèses plus restrictives sur la paire $(X,L)$. Ces hypothèses peuvent se formuler de deux manières : en termes topologiques ou grâce à la théorie des déformations de $(X,L)$. Par ailleurs, nous démontrons que pour une telle fibration lagrangienne presque holomorphe il y a toujours un bon modèle minimal lisse, c’est-à-dire une variété hyperkählérienne birationelle à $X$ sur laquelle la fibration est holomorphe.

Let $X$ be a compact hyperkähler manifold containing a complex torus $L$ as a Lagrangian subvariety. Beauville posed the question whether $X$ admits a Lagrangian fibration with fibre $L$. We show that this is indeed the case if $X$ is not projective. If $X$ is projective we find an almost holomorphic Lagrangian fibration with fibre $L$ under additional assumptions on the pair $(X,L)$, which can be formulated in topological or deformation-theoretic terms. Moreover, we show that for any such almost holomorphic Lagrangian fibration there exists a smooth good minimal model, i.e., a hyperkähler manifold birational to $X$ on which the fibration is holomorphic.

Classification : 53C26, 14D06, 14E30, 32G10, 32G05
Keywords: hyperkähler manifold, lagrangian fibration
Mot clés : variété hyperkählérienne, fibration lagrangienne

[1] E. Amerik & F. Campana, Fibrations méromorphes sur certaines variétés à fibré canonique trivial, Pure Appl. Math. Q. 4 (2008), 509-545. | MR

[2] D. Barlet, Espace analytique réduit des cycles analytiques complexes compacts d'un espace analytique complexe de dimension finie, in Fonctions de plusieurs variables complexes, II (Sém. François Norguet, 1974-1975), Lecture Notes in Math. 482, Springer, 1975, 1-158. | MR

[3] D. Barlet, How to use the cycle space in complex geometry, in Several complex variables (Berkeley, CA, 1995-1996), Math. Sci. Res. Inst. Publ. 37, Cambridge Univ. Press, 1999, 25-42. | MR

[4] T. Bauer, F. Campana, T. Eckl, S. Kebekus, T. Peternell, S. Rams, T. Szemberg & L. Wotzlaw, A reduction map for nef line bundles, in Complex geometry (Göttingen, 2000), Springer, 2002, 27-36. | MR

[5] A. Beauville, Variétés kählériennes dont la première classe de Chern est nulle, J. Differential Geom. 18 (1983), 755-782. | MR

[6] A. Beauville, Symplectic singularities, Invent. Math. 139 (2000), 541-549. | MR

[7] A. Beauville, Holomorphic symplectic geometry: a problem list, in Complex and differential geometry, Springer Proc. Math. 8, Springer, 2011, 49-63. | MR

[8] C. Birkar, P. Cascini, C. D. Hacon & J. Mckernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), 405-468. | MR

[9] F. Campana, Isotrivialité de certaines familles kählériennes de variétés non projectives, Math. Z. 252 (2006), 147-156. | MR

[10] F. Campana, K. Oguiso & T. Peternell, Non-algebraic hyperkähler manifolds, J. Differential Geom. 85 (2010), 397-424. | MR

[11] P. Cascini & V. Lazić, New outlook on the Minimal Model program I, preprint arXiv:1009.3188, to appear in Duke Math J. | MR

[12] J.-P. Demailly, Multiplier ideal sheaves and analytic methods in algebraic geometry, in School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000), ICTP Lect. Notes 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001, 1-148. | MR

[13] G. Dloussky & A. Teleman, Infinite bubbling in non-Kählerian geometry, Math. Ann. 353 (2012), 1283-1314. | MR

[14] R. Donagi & E. Markman, Spectral covers, algebraically completely integrable, Hamiltonian systems, and moduli of bundles, in Integrable systems and quantum groups (Montecatini Terme, 1993), Lecture Notes in Math. 1620, Springer, 1996, 1-119. | MR

[15] A. Douady, Le problème des modules pour les sous-espaces analytiques compacts d'un espace analytique donné, Ann. Inst. Fourier (Grenoble) 16 (1966), 1-95. | MR

[16] A. Fujiki, Closedness of the Douady spaces of compact Kähler spaces, Publ. Res. Inst. Math. Sci. 14 (1978/79), 1-52. | MR

[17] A. Fujiki, On the Douady space of a compact complex space in the category $𝒞$, Nagoya Math. J. 85 (1982), 189-211. | MR

[18] H. Grauert, T. Peternell & R. Remmert (éds.), Several complex variables. VII, Encyclopaedia of Math. Sciences 74, Springer, 1994. | MR

[19] C. D. Hacon & C. Xu, Existence of log canonical closures, preprint arXiv:1105.1169. | MR

[20] D. Huybrechts, Compact hyperkähler manifolds: basic results, Invent. Math. 135 (1999), 63-113. | MR

[21] D. Huybrechts, Compact hyperkähler manifolds, in Calabi-Yau manifolds and related geometries (Nordfjordeid, 2001), Universitext, Springer, 2003, 161-225. | MR

[22] J.-M. Hwang, Geometry of minimal rational curves on Fano manifolds, in School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000), ICTP Lect. Notes 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001, 335-393. | MR

[23] J.-M. Hwang & R. M. Weiss, Webs of Lagrangian tori in projective symplectic manifolds, preprint arXiv:1201.2369, to appear in Inv. math. | MR

[24] Y. Kawamata, Flops connect minimal models, Publ. Res. Inst. Math. Sci. 44 (2008), 419-423. | MR

[25] K. Kodaira & D. C. Spencer, On deformations of complex analytic structures. III. Stability theorems for complex structures, Ann. of Math. 71 (1960), 43-76. | MR

[26] J. Kollár & S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics 134, Cambridge Univ. Press, 1998. | MR

[27] C.-J. Lai, Varieties fibered by good minimal models, Math. Ann. 350 (2011), 533-547. | MR

[28] R. Lazarsfeld, Positivity in algebraic geometry. II, Ergebnisse Math. Grenzg. 49, Springer, 2004. | MR

[29] C. Lehn, Symplectic Lagrangian fibrations, Dissertation, Johannes Gutenberg Universität, Mainz, 2011.

[30] D. I. Lieberman, Compactness of the Chow scheme: applications to automorphisms and deformations of Kähler manifolds, in Fonctions de plusieurs variables complexes, III (Sém. François Norguet, 1975-1977), Lecture Notes in Math. 670, Springer, 1978, 140-186. | MR

[31] D. Matsushita, On fibre space structures of a projective irreducible symplectic manifold, Topology 38 (1999), 79-83. | MR

[32] D. Matsushita, Equidimensionality of Lagrangian fibrations on holomorphic symplectic manifolds, Math. Res. Lett. 7 (2000), 389-391. | MR

[33] D. Matsushita, Addendum to [31], Topology 40 (2001), 431-432. | MR

[34] D. Matsushita, Holomorphic symplectic manifolds and Lagrangian fibrations, Acta Appl. Math. 75 (2003), 117-123. | MR

[35] D. Matsushita, On nef reductions of projective irreducible symplectic manifolds, Math. Z. 258 (2008), 267-270. | MR

[36] D. Matsushita, On almost holomorphic Lagrangian fibrations, preprint arXiv:1209.1194. | MR

[37] D. Matsushita, On deformations of Lagrangian fibrations, preprint arXiv:0903.2098.

[38] B. G. Moishezon, On $n$-dimensional compact complex varieties with $n$ algebraically independent meromorphic functions. I-III, Am. Math. Soc. Transl. 63 (1967), 51-177.

[39] Y. Namikawa, On deformations of $\mathbb{Q}$-factorial symplectic varieties, J. reine angew. Math. 599 (2006), 97-110. | MR

[40] Z. Ran, Lifting of cohomology and unobstructedness of certain holomorphic maps, Bull. Amer. Math. Soc. (N.S.) 26 (1992), 113-117. | MR

[41] J. Sawon, Abelian fibred holomorphic symplectic manifolds, Turkish J. Math. 27 (2003), 197-230. | MR

[42] M. Verbitsky, HyperKähler SYZ conjecture and semipositive line bundles, Geom. Funct. Anal. 19 (2010), 1481-1493. | MR

[43] C. Voisin, Sur la stabilité des sous-variétés lagrangiennes des variétés symplectiques holomorphes, in Complex projective geometry (Trieste, 1989/Bergen, 1989), London Math. Soc. Lecture Note Ser. 179, Cambridge Univ. Press, 1992, 294-303. | MR

[44] J. Wierzba, Birational geometry of symplectic 4-folds, unpublished manuscript, currently available at http://www.mimuw.edu.pl/~jarekw/postscript/bir4fd.ps, 2002.