Modifications of torsion-free coherent analytic sheaves

Ruppenthal, Jean; Sera, Martin

Annales de l'Institut Fourier, Tome 67 (2017) no. 1, pp. 237-265.

Nous étudions la transformation de faisceaux analytiques cohérents à travers des modifications propres. Plus précisément, nous étudions les images directes de faisceaux images-inverses, et les préimages sans torsion de faisceaux images directes. Sous certaines conditions, il est démontré que les faisceaux cohérents sans torsion peuvent être réalisés comme images directes de faisceaux localement libres à travers des modifications. Ainsi, il est possible d’étudier les faisceaux cohérents modulo torsion en ramenant le problème à l’étude de fibrés vectoriels sur des variétés. Nous appliquons ceci aux faisceaux d’idéaux réduits et au faisceau canonique de $n$-formes holomorphes de Grauert–Riemenschneider.

We study the transformation of torsion-free coherent analytic sheaves under proper modifications. More precisely, we study direct images of inverse image sheaves, and torsion-free preimages of direct image sheaves. Under some conditions, it is shown that torsion-free coherent sheaves can be realized as the direct image of locally free sheaves under modifications. Thus, it is possible to study coherent sheaves modulo torsion by reducing the problem to study vector bundles on manifolds. We apply this to reduced ideal sheaves and to the Grauert–Riemenschneider canonical sheaf of holomorphic $n$-forms.

Classification : 32C35, 32S45, 32L05, 32L10
Keywords: modifications, coherent analytic sheaves, torsion-free preimage
Mot clés : modifications, faisceaux analytiques cohérents, préimages sans torsion

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