Special Kähler geometry and holomorphic Lagrangian fibrations

Yang Li; Valentino Tosatti

doi:10.5802/crmath.629

Article de recherche - Géométrie algébrique

Special Kähler geometry and holomorphic Lagrangian fibrations
[Géométrie kählérienne spéciale et fibrations lagrangiennes holomorphes]

Yang Li ¹ ; Valentino Tosatti ²

¹ Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
² Courant Institute of Mathematical Sciences, New York University, 251 Mercer St, New York, NY 10012, USA

Comptes Rendus. Mathématique, Volume 362 (2024) no. S1, pp. 171-196.

Résumé
Abstract

Étant donné une fibration lagrangienne holomorphe d’une variété hyperkählérienne compacte, nous utilisons la géométrie différentielle de la métrique kählérienne spéciale qui existe sur la base au dehors du lieu discriminant, et montrons que l’image réciproque du fibré tangent de la base par le morphisme d’évaluation d’une famille de courbes rationnelles minimales admet une décomposition parallèle. La décomposition n’est pas triviale lorsque la base n’est pas un espace projectif demi-dimensionnel. En combinant cela avec des résultats de Voisin, Hwang et Bakker–Schnell, nous en déduisons que la base doit être un espace projectif, résultat prouvé pour la première fois par Hwang.

Given a holomorphic Lagrangian fibration of a compact hyperkähler manifold, we use the differential geometry of the special Kähler metric that exists on the base away from the discriminant locus, and show that the pullback of the tangent bundle of the base to the total space of a family of minimal rational curves admits a parallel splitting. The splitting is nontrivial when the base is not half-dimensional projective space. Combining this with results of Voisin, Hwang and Bakker–Schnell, we deduce that the base must be projective space, a result first proved by Hwang.

Reçu le : 2023-08-20
Accepté le : 2024-03-02
Publié le : 2024-06-06

DOI : 10.5802/crmath.629

Affiliations des auteurs :

Yang Li ¹ ; Valentino Tosatti ²

¹ Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
² Courant Institute of Mathematical Sciences, New York University, 251 Mercer St, New York, NY 10012, USA

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Special {K\"ahler} geometry and holomorphic {Lagrangian} fibrations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {171--196},
     publisher = {Acad\'emie des sciences, Paris},
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     doi = {10.5802/crmath.629},
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Yang Li; Valentino Tosatti. Special Kähler geometry and holomorphic Lagrangian fibrations. Comptes Rendus. Mathématique, Volume 362 (2024) no. S1, pp. 171-196. doi : 10.5802/crmath.629. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.629/

Bibliographie
Cité par

[1] Benjamin Bakker A short proof of a conjecture of Matsushita (2022) (https://arxiv.org/abs/2209.00604)

[2] Daniel Barlet Développement asymptotique des fonctions obtenues par intégration sur les fibres, Invent. Math., Volume 68 (1982) no. 1, pp. 129-174 | DOI | Zbl

[3] Arnaud Beauville Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differ. Geom., Volume 18 (1983) no. 4, pp. 755-782 | Zbl

[4] Sébastien Boucksom; Mattias Jonsson Tropical and non-Archimedean limits of degenerating families of volume forms, J. Éc. Polytech., Math., Volume 4 (2017), pp. 87-139 | DOI | Numdam | MR | Zbl

[5] Fedor Bogomolov; Nikon Kurnosov Lagrangian fibrations for IHS fourfolds (2018) (https://arxiv.org/abs/1810.11011)

[6] Arend Bayer; Emanuele Macrì MMP for moduli of sheaves on K3’s via wall-crossing: nef and movable cones, Lagrangian fibrations, Invent. Math., Volume 198 (2014) no. 3, pp. 505-590 | DOI | MR | Zbl

[7] Benjamin Bakker; Christian Schnell A Hodge-theoretic proof of Hwang’s theorem on base manifolds of Lagrangian fibrations (2023) (https://arxiv.org/abs/2311.08977)

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

[9] Martin Callies; Andriy Haydys Local models of isolated singularities for affine special Kähler structures in dimension two, Int. Math. Res. Not., Volume 2020 (2020) no. 17, pp. 5215-5235 | DOI | Zbl

[10] Koji Cho; Yoichi Miyaoka; Nicholas I. Shepherd-Barron 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 | MR | Zbl

[11] Jean-Pierre Demailly Complex analytic and differential geometry (Online book, https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf)

[12] Ron Donagi; Edward Witten Supersymmetric Yang-Mills theory and integrable systems, Nucl. Phys., B, Volume 460 (1996) no. 2, pp. 299-334 | DOI | MR | Zbl

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

[14] Daniel Greb; Stefan Kebekus; Thomas Peternell Reflexive differential forms on singular spaces. Geometry and cohomology, J. Reine Angew. Math., Volume 697 (2014), pp. 57-89 | DOI | MR | Zbl

[15] Daniel Greb; Christian Lehn Base manifolds for Lagrangian fibrations on hyperkähler manifolds, Int. Math. Res. Not., Volume 2014 no. 19, pp. 5483-5487 | DOI | Zbl

[16] Hans Grauert; Reinhold Remmert Plurisubharmonische Funktionen in komplexen Räumen, Math. Z., Volume 65 (1956), pp. 175-194 | DOI | Zbl

[17] Mark Gross; Valentino Tosatti; Yuguang Zhang Collapsing of abelian fibered Calabi–Yau manifolds, Duke Math. J., Volume 162 (2013) no. 3, pp. 517-551 | MR | Zbl

[18] Mark Gross; Valentino Tosatti; Yuguang Zhang Gromov–Hausdorff collapsing of Calabi–Yau manifolds, Commun. Anal. Geom., Volume 24 (2016) no. 1, pp. 93-113 | DOI | MR | Zbl

[19] Mark Gross; Valentino Tosatti; Yuguang Zhang Geometry of twisted Kähler–Einstein metrics and collapsing, Commun. Math. Phys., Volume 380 (2020) no. 3, pp. 1401-1438 | DOI | Zbl

[20] Robin Hartshorne Algebraic geometry, Graduate Texts in Mathematics, 52, Springer, 1977 | DOI

[21] Andriy Haydys Isolated singularities of affine special Kähler metrics in two dimensions, Commun. Math. Phys., Volume 340 (2015) no. 3, pp. 1231-1237 | DOI | MR | Zbl

[22] Claus Hertling $t t^{*}$ geometry, Frobenius manifolds, their connections, and the construction for singularities, J. Reine Angew. Math., Volume 555 (2003), pp. 77-161 | MR | Zbl

[23] D. Huybrechts; M. Mauri Lagrangian fibrations, Milan J. Math., Volume 90 (2022) no. 2, pp. 459-483 | DOI | MR | Zbl

[24] Jun-Muk Hwang; Keiji Oguiso 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

[25] Hans-Joachim Hein; Valentino Tosatti Remarks on the collapsing of torus fibered Calabi-Yau manifolds, Bull. Lond. Math. Soc., Volume 47 (2015) no. 6, pp. 1021-1027 | MR | Zbl

[26] Jun-Muk Hwang 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 | Numdam | MR | Zbl

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

[28] Jun-Muk Hwang Mori geometry meets Cartan geometry: varieties of minimal rational tangents, Proceedings of the International Congress of Mathematicians–Seoul 2014. Vol. 1. Plenary lectures and ceremonies, KM Kyung Moon Sa, 2014, pp. 369-394 | Zbl

[29] Daniel Huybrechts; Chenyang Xu Lagrangian fibrations of hyperkähler fourfolds, J. Inst. Math. Jussieu, Volume 21 (2022) no. 3, pp. 921-932 | DOI | Zbl

[30] Dano Kim Canonical bundle formula and degenerating families of volume forms (2019) (https://arxiv.org/abs/1910.06917)

[31] Seán Keel; James McKernan Rational curves on quasi-projective surfaces, Mem. Am. Math. Soc., Volume 669 (1999), p. viii+153 | MR | Zbl

[32] Shoshichi Kobayashi Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, 15, Princeton University Press, 1987, xi+304 pages

[33] János Kollár Cone theorems and bug-eyed covers, J. Algebr. Geom., Volume 1 (1992) no. 2, pp. 293-323 | MR | Zbl

[34] János Kollár Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 32, Springer, 1995, viii+320 pages | Zbl

[35] Zhiqin Lu A note on special Kähler manifolds, Math. Ann., Volume 313 (1999) no. 4, pp. 711-713 | MR | Zbl

[36] Eyal Markman Lagrangian fibrations of holomorphic-symplectic varieties of $K 3^{[n]}$ -type, Algebraic and complex geometry (Springer Monographs in Mathematics), Volume 71, Springer, 2014, pp. 241-283 | DOI | MR | Zbl

[37] Daisuke Matsushita Holomorphic symplectic manifolds and Lagrangian fibrations, Acta Appl. Math., Volume 75 (2003) no. 1-3, pp. 117-123 | DOI | MR

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

[39] Daisuke Matsushita On deformations of Lagrangian fibrations, $K 3$ Surfaces and their moduli (Progress in Mathematics), Volume 315, Birkhäuser, 2016, pp. 237-243 | DOI | MR | Zbl

[40] Daisuke Matsushita On isotropic divisors on irreducible symplectic manifolds, Higher dimensional algebraic geometry–in honour of Professor Yujiro Kawamata’s sixtieth birthday (Advanced Studies in Pure Mathematics), Volume 74, Mathematical Society of Japan, 2017, pp. 291-312 | DOI | MR | Zbl

[41] Daisuke Matsushita On fibre space structures of a projective irreducible symplectic manifold, Topology, Volume 38 (1999) no. 1, pp. 79-83 Addendum in ibid. 40 (2001), no. 2, pp. 431–432 | DOI | MR | Zbl

[42] Yoichi Miyaoka; Shigefumi Mori A numerical criterion for uniruledness, Ann. Math., Volume 124 (1986) no. 1, pp. 65-69 | DOI | MR | Zbl

[43] Ngaiming Mok The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature, J. Differ. Geom., Volume 27 (1988) no. 2, pp. 179-214 | Zbl

[44] Shigefumi Mori Projective manifolds with ample tangent bundles, Ann. Math., Volume 110 (1979) no. 3, pp. 593-606 | DOI | MR | Zbl

[45] Y. Nagai Dual fibration of a projective Lagrangian fibration (2005) (Unpublished preprint)

[46] Yoshinori Namikawa Projectivity criterion of Moishezon spaces and density of projective symplectic varieties, Int. J. Math., Volume 13 (2002) no. 2, pp. 125-135 | DOI | MR | Zbl

[47] Takeo Ohsawa Analysis of several complex variables, Iwanami Series in Modern Mathematics. Translations of Mathematical Monographs, 211, American Mathematical Society, 2002, xvii+121 pages | DOI | Zbl

[48] Wenhao Ou Lagrangian fibrations on symplectic fourfolds, J. Reine Angew. Math., Volume 746 (2019), pp. 117-147 | MR | Zbl

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

[50] Philip Sieder Varieties with ample tangent sheaves, Manuscr. Math., Volume 157 (2018) no. 1-2, pp. 257-261 | DOI | MR | Zbl

[51] Jian Song; Gang Tian Canonical measures and Kähler–Ricci flow, Flow, Volume 25 (2012) no. 2, pp. 303-353 | Zbl

[52] Junliang Shen; Qizheng Yin Topology of Lagrangian fibrations and Hodge theory of hyper-Kähler manifolds. With Appendix B by Claire Voisin, Duke Math. J., Volume 171 (2022) no. 1, pp. 209-241 | Zbl

[53] Yum-Tong Siu; Shing-Tung Yau Compact Kähler manifolds of positive bisectional curvature, Invent. Math., Volume 59 (1980) no. 2, pp. 189-204 | Zbl

[54] Shigeharu Takayama Simple connectedness of weak Fano varieties, J. Algebr. Geom., Volume 9 (2000) no. 2, pp. 403-407 | MR | Zbl

[55] Shigeharu Takayama Asymptotic expansions of fiber integrals over higher-dimensional bases, J. Reine Angew. Math., Volume 773 (2021), pp. 67-128 | DOI | MR | Zbl

[56] Valentino Tosatti Adiabatic limits of Ricci-flat Kähler metrics, J. Differ. Geom., Volume 84 (2010) no. 2, pp. 427-453 | Zbl

[57] Valentino Tosatti; Yuguang Zhang Finite time collapsing of the Kähler-Ricci flow on threefolds, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 18 (2018) no. 1, pp. 105-118 | Zbl

[58] Valentino Tosatti; Yuguang Zhang Collapsing hyperkähler manifolds, Ann. Sci. Éc. Norm. Supér., Volume 52 (2020) no. 3, pp. 751-786 | DOI | Zbl

[59] Bert van Geemen; Claire Voisin On a conjecture of Matsushita, Int. Math. Res. Not., Volume 2016 (2016) no. 10, pp. 3111-3123 | MR | Zbl

[60] Claire Voisin 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 | Zbl

[61] Chenyang Xu Strong rational connectedness of surfaces, J. Reine Angew. Math., Volume 665 (2012), pp. 189-205 | MR | Zbl

[62] Ken-Ichi Yoshikawa On the boundary behavior of the curvature of $L^{2}$ -metrics (2010) (https://arxiv.org/abs/1007.2836)

[63] Kōta Yoshioka Bridgeland’s stability and the positive cone of the moduli spaces of stable objects on an abelian surface, Development of moduli theory–Kyoto 2013 (Advanced Studies in Pure Mathematics), Volume 69, Mathematical Society of Japan, 2013, pp. 473-537 | Zbl

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