Abstract. We study singular hermitian metrics on holomorphic vector bundles, following Berndtsson-Pȃun. Previous work by Raufi has shown that for such metrics, it is in general not possible to define the curvature as a current with measure coefficients. In this paper we show that despite this, under appropriate codimension restrictions on the singular set of the metric, it is still possible to define Chern forms as closed currents of order 0 with locally finite mass, which represent the Chern classes of the vector bundle.
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.
The problem of optimization of rigid bodies moving in the deformable media is considered. The shape of the axisymmetric impactors has been taken as an unknown design variable. The total resistance force, the mass of material and the volume are taken as components of the minimized vector-functional. Formulated multiobjective minimization problem for vector-functional is investigated analytically. As an example the Pareto-optimal set of optimal shape and Pareto-front are found for the rigid thin-walled axisymmetric shells having minimal total resistence force and the mass of the shell material
We consider mixed Monge–Ampère products of quasiplurisubharmonic functions with analytic singularities, and show that such products may be regularized as explicit one‐parameter limits of mixed Monge–Ampère products of smooth functions, generalizing results of Andersson, Błocki and the last author in the case of non‐mixed Monge–Ampère products. Connections to the theory of residue currents, going back to Coleff–Herrera, Passare and others, play an important role in the proof. As a consequence we get an approximation of Chern and Segre currents of certain singular hermitian metrics on vector bundles by smooth forms in the corresponding Chern and Segre classes.
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