A Hodge-theoretic proof of Hwang’s theorem on base manifolds of Lagrangian fibrations
[Une preuve de nature Hodge théorique d’un théorème de Hwang sur les variétés de base des fibrations lagrangiennes]
Bakker, Benjamin1 ; Schnell, Christian2
1Dept. of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago (USA)
2Department of Mathematics, Stony Brook University, Stony Brook, NY (USA)
Annales de l'Institut Fourier, Online first, 40 p.

We give a Hodge-theoretic proof of Hwang’s theorem, which says that if the base of a Lagrangian fibration of an irreducible holomorphic symplectic manifold is smooth, it must be projective space.

On donne une preuve nouvelle d’un théorème de Hwang, qui dit que si la base d’une fibration lagrangienne sur une variété symplectique holomorphe irréductible est lisse, alors il s’agit de l’espace projectif.

Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : 10.5802/aif.3786
Classification : 14C30, 14J42
Keywords: hyperkähler manifold, holomorphic symplectic manifold, Lagrangian fibration, Hwang’s theorem, variation of Hodge structure, Hodge module, rational curves
Mots-clés : variété hyperkählérienne, variété symplectique holomorphe, fibration lagrangienne, théorème de Hwang, variation de structure de Hodge, module de Hodge, courbes rationnelles

Bakker, Benjamin  1   ; Schnell, Christian  2

1 Dept. of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago (USA)
2 Department of Mathematics, Stony Brook University, Stony Brook, NY (USA) 
@unpublished{AIF_0__0_0_A75_0,
     author = {Bakker, Benjamin and Schnell, Christian},
     title = {A {Hodge-theoretic} proof of {Hwang{\textquoteright}s} theorem on base manifolds of {Lagrangian} fibrations},
     journal = {Annales de l'Institut Fourier},
     year = {2026},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     doi = {10.5802/aif.3786},
     language = {en},
     note = {Online first},
}
TY  - UNPB
AU  - Bakker, Benjamin
AU  - Schnell, Christian
TI  - A Hodge-theoretic proof of Hwang’s theorem on base manifolds of Lagrangian fibrations
JO  - Annales de l'Institut Fourier
PY  - 2026
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3786
LA  - en
ID  - AIF_0__0_0_A75_0
ER  - 
%0 Unpublished Work
%A Bakker, Benjamin
%A Schnell, Christian
%T A Hodge-theoretic proof of Hwang’s theorem on base manifolds of Lagrangian fibrations
%J Annales de l'Institut Fourier
%D 2026
%V 0
%N 0
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3786
%G en
%F AIF_0__0_0_A75_0
Bakker, Benjamin; Schnell, Christian. A Hodge-theoretic proof of Hwang’s theorem on base manifolds of Lagrangian fibrations. Annales de l'Institut Fourier, Online first, 40 p.

[1] Araujo, Carolina Rational curves of minimal degree and characterizations of projective spaces, Math. Ann., Volume 335 (2006) no. 4, pp. 937-951 | DOI | MR | Zbl

[2] Bakker, Ben A short proof of a conjecture by Matsushita (2022) | arXiv | Zbl

[3] Campana, Frédéric Local projectivity of Lagrangian fibrations on hyperkähler manifolds, Manuscr. Math., Volume 164 (2021) no. 3-4, pp. 589-591 | MR | Zbl | DOI

[4] Cattani, Eduardo; Kaplan, Aroldo Degenerating variations of Hodge structure, Actes du Colloque de Théorie de Hodge (Luminy, 1987) (Barlet, D.; Esnault, H.; Elzein, F.; Verdier, J. L.; Viehweg, E., eds.) (Astérisque), Société Mathématique de France, 1989 no. 179-180, pp. 67-96 | Numdam | MR | Zbl

[5] Cho, Koji; Miyaoka, Yoichi; Shepherd-Barron, Nicholas I. Characterizations of projective space and applications to complex symplectic manifolds, Higher dimensional birational geometry (Kyoto, 1997) (Advanced Studies in Pure Mathematics), Volume 35, Mathematical Society of Japan, 2002, pp. 1-88 | Zbl | DOI | MR

[6] Donagi, Ron; Markman, Eyal Cubics, integrable systems, and Calabi-Yau threefolds, Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993) (Israel Mathematical Conference Proceedings), Volume 9, Bar-Ilan University (1996), pp. 199-221 | Zbl | MR

[7] Freed, Daniel S. Special Kähler manifolds, Commun. Math. Phys., Volume 203 (1999) no. 1, pp. 31-52 | Zbl | DOI | MR

[8] Fujita, Takao On Kähler fiber spaces over curves, J. Math. Soc. Japan, Volume 30 (1978) no. 4, pp. 779-794 | Zbl | DOI | MR

[9] Hwang, Jun-Muk Deformation of holomorphic maps onto Fano manifolds of second and fourth Betti numbers 1, Ann. Inst. Fourier, Volume 57 (2007) no. 3, pp. 815-823 | DOI | MR | Numdam | Zbl

[10] Hwang, Jun-Muk Base manifolds for fibrations of projective irreducible symplectic manifolds, Invent. Math., Volume 174 (2008) no. 3, pp. 625-644 | DOI | MR | Zbl

[11] Hwang, Jun-Muk; Oguiso, Keiji Characteristic foliation on the discriminant hypersurface of a holomorphic Lagrangian fibration, Am. J. Math., Volume 131 (2009) no. 4, pp. 981-1007 | DOI | MR | Zbl

[12] Kebekus, Stefan Characterizing the projective space after Cho, Miyaoka and Shepherd-Barron, Complex geometry (Göttingen, 2000), Springer, 2002, pp. 147-155 | MR | DOI | Zbl

[13] Kollár, János Higher direct images of dualizing sheaves. I, Ann. Math. (2), Volume 123 (1986) no. 1, pp. 11-42 | DOI | MR | Zbl

[14] Kollár, János Higher direct images of dualizing sheaves. II, Ann. Math. (2), Volume 124 (1986) no. 1, pp. 171-202 | DOI | MR | Zbl

[15] Kollár, János Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 32, Springer, 1996 | DOI | MR

[16] Li, Yang; Tosatti, Valentino Special Kähler geometry and holomorphic Lagrangian fibrations (2023) | arXiv

[17] Matsushita, Daisuke On fibre space structures of a projective irreducible symplectic manifold, Topology, Volume 38 (1999) no. 1, pp. 79-83 | DOI | MR

[18] Matsushita, Daisuke On fibre space structures of a projective irreducible symplectic manifold, Topology, Volume 38 (1999) no. 1, pp. 79-83 | DOI | MR

[19] Matsushita, Daisuke Higher direct images of dualizing sheaves of Lagrangian fibrations, Am. J. Math., Volume 127 (2005) no. 2, pp. 243-259 | MR | DOI | Zbl

[20] Nagata, Masayoshi On the purity of branch loci in regular local rings, Ill. J. Math., Volume 3 (1959), pp. 328-333 | MR | Zbl

[21] Peters, Chris A criterion for flatness of Hodge bundles over curves and geometric applications, Math. Ann., Volume 268 (1984) no. 1, pp. 1-19 | DOI | MR | Zbl

[22] Sabbah, Claude; Schnell, Christian Degenerating complex variations of Hodge structure in dimension one (2022) | arXiv | Zbl

[23] Saito, Morihiko Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci., Volume 24 (1988) no. 6, pp. 849-995 | DOI | MR | Zbl

[24] Saito, Morihiko Mixed Hodge modules, Publ. Res. Inst. Math. Sci., Volume 26 (1990) no. 2, pp. 221-333 | DOI | MR | Zbl

[25] Saito, Morihiko On Kollár’s conjecture, Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989) (Proceedings of Symposia in Pure Mathematics), Volume 52(2), American Mathematical Society, 1991, pp. 509-517 | DOI | MR | Zbl

[26] Schmid, Wilfried Variation of Hodge structure: the singularities of the period mapping, Invent. Math., Volume 22 (1973), pp. 211-319 | DOI | MR | Zbl

[27] Schnell, Christian Complex analytic Néron models for arbitrary families of intermediate Jacobians, Invent. Math., Volume 188 (2012) no. 1, pp. 1-81 | DOI | MR | Zbl

[28] Schnell, Christian Hodge theory and Lagrangian fibrations on holomorphic symplectic manifolds (2023) | arXiv

[29] Schnell, Christian; Yang, Ruijie Higher multiplier ideals, J. Reine Angew. Math. (2026) | DOI | Zbl | MR

[30] The Stacks Project Authors Stacks Project (http://stacks.math.columbia.edu)

[31] Voisin, Claire Torsion points of sections of Lagrangian torus fibrations and the Chow ring of hyper-Kähler manifolds, Geometry of moduli (Abel Symposia), Volume 14, Springer, 2018, pp. 295-326 | DOI | MR | Zbl

Cité par Sources :