Summary: Hermitian­holomorphic (2)­Gerbes and tame symbols
Ettore Aldrovandi
Department of Mathematics
Florida State University
Tallahassee, FL 32306­4510, USA
aldrovandi@math.fsu.edu
Abstract
The tame symbol of two invertible holomorphic functions can be obtained by computing their cup
product in Deligne cohomology, and it is geometrically interpreted as a holomorphic line bundle with
connection. In a similar vein, certain higher tame symbols later considered by Brylinski and McLaughlin are
geometrically interpreted as holomorphic gerbes and 2­gerbes with abelian band and a suitable connective
structure.
In this paper we observe that the line bundle associated to the tame symbol of two invertible holomor­
phic 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 hermitian­holomorphic structure on a gerbe is also proven. Similar results are proved
for 2­gerbes as well.

Source: Aldrovandi, Ettore - Department of Mathematics, Florida State University

Collections: Mathematics