Lagrangian Fibrations

Huybrechts, Daniel; Mauri, Mirko

doi:10.1007/s00032-022-00349-y

Cited by 5 publications

(7 citation statements)

References 61 publications

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“…On the other hand, we have Z ⊂ ∆ i (π M ) where the latter has codimension at least i by (16). Therefore Z is an i-dimensional irreducible component of ∆ i (π M ), and the irreducible component W is of the form R Z .…”

Section: Proposition 22mentioning

confidence: 97%

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Fourier-Mukai transforms and the decomposition theorem for integrable systems

Maulik¹,

Shen²,

Yin³

2023

Preprint

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We study the interplay between the Fourier-Mukai transform and the decomposition theorem for an integrable system π : M → B. Our main conjecture is that the Fourier-Mukai transform of sheaves of Kähler differentials, after restriction to the formal neighborhood of the zero section, are quantized by the Hodge modules arising in the decomposition theorem for π. For an integrable system, our formulation unifies the Fourier-Mukai calculation of the structure sheaf by Arinkin-Fedorov, the theorem of the higher direct images by Matsushita, and the "perverse = Hodge" identity by the second and the third authors.As evidence, we show that these Fourier-Mukai images are Cohen-Macaulay sheaves with middle-dimensional support on the relative Picard space, with support governed by the higher discriminants of the integrable system. We also prove the conjecture for smooth integrable systems and certain 2-dimensional families with nodal singular fibers. Finally, we sketch the proof when cuspidal fibers appear.

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Section: Proposition 22mentioning

confidence: 97%

“…The perverse-Hodge symmetry. The motivation for Conjecture 0.1 is an effort to understand and categorify the perverse-Hodge symmetry for Lagrangian fibrations [31]; see also [11,10,16,32].…”

mentioning

confidence: 99%

Fourier-Mukai transforms and the decomposition theorem for integrable systems

Maulik¹,

Shen²,

Yin³

2023

Preprint

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“…This means that B is a complex manifold of dimension n, and that π is a proper surjective holomorphic mapping whose smooth fibers are Lagrangian, in the sense that σ restricts to zero on every smooth fiber of π. For a very nice introduction to the general theory of Lagrangian fibrations, see the recent paper by Huybrechts and Mauri [HM22]. It is known that the smooth fibers are abelian varieties of dimension n. Moreover, according to a theorem by Matsushita [Mat00, Thm.…”

Section: Lagrangian Fibrationsmentioning

confidence: 99%

“…I thank Daniel Huybrechts for his detailed comments about a first draft of the paper. Finally, I am grateful to Daniel Huybrechts and Mirko Mauri for their survey paper [HM22] about Lagrangian fibrations, from which I learned a lot of the general theory.…”

Section: Acknowledgementsmentioning

confidence: 99%

Hodge theory and Lagrangian fibrations on holomorphic symplectic manifolds

Schnell¹

2023

Preprint

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“…There are also a number of partial results towards Conjecture 1 in general, see [25] for an excellent recent overview. It is known that B must be a Kähler space (see e.g.…”

Section: Introductionmentioning

confidence: 99%

Special Kähler geometry and holomorphic Lagrangian fibrations

Li,

Tosatti

2024

Comptes Rendus. Mathématique

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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.Résumé. 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.

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Lagrangian Fibrations

Abstract: We review the theory of Lagrangian fibrations of hyperkähler manifolds as initiated by Matsushita. We also discuss more recent work of Shen–Yin and Harder–Li–Shen–Yin. Occasionally, we give alternative arguments and complement the discussion by additional observations.

Cited by 5 publications

References 61 publications

Fourier-Mukai transforms and the decomposition theorem for integrable systems

Fourier-Mukai transforms and the decomposition theorem for integrable systems

Hodge theory and Lagrangian fibrations on holomorphic symplectic manifolds

Special Kähler geometry and holomorphic Lagrangian fibrations

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