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arXiv · math/0310027

Hermitian-holomorphic (2)-Gerbes and tame symbols

Ettore Aldrovandi
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Abstract

We observe that the line bundle associated to the tame symbol of two invertible holomorphic functions also carries a fairly canonical hermitian metric, hence it represents a class in a Hermitian holomorphic Deligne cohomology group. We put forward an alternative definition of hermitian holomorphic structure on a gerbe which is closer to the familiar one for line bundles and does not rely on an explicit ``reduction of the structure group.'' Analogously to the case of holomorphic line bundles, a uniqueness property for the connective structure compatible with the hermitian-holomorphic structure on a gerbe is also proven. Similar results are proved for 2-gerbes as well. We then show the hermitian structures so defined propagate to a class of higher tame symbols previously considered by Brylinski and McLaughlin, which are thus found to carry corresponding hermitian-holomorphic structures. Therefore we obtain an alternative characterization for certain higher Hermitian holomorphic Deligne cohomology groups.

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Ettore Aldrovandi. 2004-10-13. Hermitian-holomorphic (2)-Gerbes and tame symbols. https://arxiv.org/abs/math/0310027

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