Curvature of higher direct images
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 30 (2021) no. 1, pp. 171-201.

Given a holomorphic family f:𝒳S of compact complex manifolds and a relatively ample line bundle L𝒳, 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).

Étant donné une famille holomorphe f:𝒳S de variétés complexes compactes lisses et un fibré en droites L𝒳 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.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1670
Classification: 32L10, 32G05, 14DXX
Keywords: Curvature of higher direct image sheaves, Deformations of complex structures, Families, Fibrations
Naumann, Philipp 1

1 Philipp Naumann, Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany
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Naumann, Philipp. Curvature of higher direct images. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 30 (2021) no. 1, pp. 171-201. doi : 10.5802/afst.1670. http://www.numdam.org/articles/10.5802/afst.1670/

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