Summary: Hermitian-holomorphic (2)-Gerbes and tame symbols
Department of Mathematics
Florida State University
Tallahassee, FL 32306-4510, USA
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 2-gerbes with abelian band and a suitable connective
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 hermitian-holomorphic structure on a gerbe is also proven. Similar results are proved
for 2-gerbes as well.