Special Kähler geometry and holomorphic Lagrangian fibrations

Li, Yang; Tosatti, Valentino

Comptes Rendus. Mathématique, Complex algebraic geometry, in memory of Jean-Pierre Demailly, Tome 362 (2024), pp. 171-196

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.

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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