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Curvature of higher direct images
Philipp Naumann1
1 Philipp Naumann, Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 30 (2021) no. 1, pp. 171-201.

Étant donné une famille holomorphe f:𝒳→S de variétés complexes compactes lisses et un fibré en droites L→𝒳 relativement ample, les faisceaux images directes R n-p f * Ω 𝒳/S p (L) possèdent des métriques hermitiennes induites. Nous donnons une formule explicite pour le tenseur de courbure de ces images directes. Ceci généralise un résultat de Schumacher dans [11], où il a calculé la courbure de R n-p f * Ω 𝒳/S p (K 𝒳/S ⊗m ) pour une famille de variétés canoniquement polarisées. Dans le cas p=n, la formule coïncide avec la formule de Berndtsson obtenue dans [3]. Donc si L est globalement ample, nous prouvons à nouveau son résultat sur la positivité de f * (K 𝒳/S ⊗L) dans le sens de Nakano.

Given a holomorphic family f:𝒳→S of compact complex manifolds and a relatively ample line bundle L→𝒳, the higher direct images R n-p f * Ω 𝒳/S p (L) carry induced hermitian metrics. We give an explicit formula for the curvature tensor of these direct images. This generalizes a result of Schumacher in [11], where he computed the curvature of R n-p f * Ω 𝒳/S p (K 𝒳/S ⊗m ) for a family of canonically polarized manifolds. For p=n, the formula coincides with a formula of Berndtsson obtained in [3]. Thus, when L is globally ample, we reprove his result on the Nakano positivity of f * (K 𝒳/S ⊗L).

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/afst.1670
Classification : 32L10, 32G05, 14DXX
Mots clés : Curvature of higher direct image sheaves, Deformations of complex structures, Families, Fibrations
Philipp Naumann 1

1 Philipp Naumann, Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Philipp Naumann. Curvature of higher direct images. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 30 (2021) no. 1, pp. 171-201. doi : 10.5802/afst.1670. https://afst.centre-mersenne.org/articles/10.5802/afst.1670/

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