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Chern classes of singular metrics on vector bundles

Banff International Research Station for Mathematical Innovation and Discovery

For holomorphic line bundles, it has turned out to be useful to not just consider smooth metrics, but also singular metrics which are not necessarily smooth, and which can degenerate. In relation to vanishing theorems and other properties of the line bundle, one considers plurisubharmonicity properties of the possibly singular metric which correspond to notions of positivity for the line bundle. In particular, having a positive singular metric means that the first Chern form associated to the metric is a closed positive \((1,1)\)-current. More recently, singular metrics on holomorphic vector bundles have been considered, Griffiths positivity of a singular metric on a vector bundle is defined in terms of plurisubharmonicity. For a vector bundle with a Griffiths positive singular metric, there is a naturally defined first Chern class which is a closed positive \((1,1)\)-current, but there are examples where the full curvature matrix is not of order 0. I will discuss joint work with Hossein Raufi, Jean Ruppenthal and Martin Sera, where we show that one can give a natural meaning to the k:th Chern form \(c_k(h)\) of a singular Griffiths positive metric h as a closed \((k,k)\)-current of order 0, as long as h is non-degenerate outside a subvariety of codimension at least \(k\). The proof builds on pluripotential theory, and in particular, one consider in the spirit of Bedford-Taylor products like \((dd^c \varphi)^q \wedge T\), where \(\varphi\) is plurisubharmonic and \(T\) is a closed positive \( (q,q) \)-current.

Mathematics; Several complex variables and analytic spaces; Potential theory; Global analysis

video/mp4

Author affiliation: University of Wuppertal

2017-02-08

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/59599

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/