Ettore ALDROVANDI
(Florida State University)
Hermitian-holomorphic 2-gerbes and tame symbols
We give a definition for a hermitian structure on a 2-gerbe with
abelian band on a complex analytic manifold or algebraic
variety. We also have the associated notion of type $(1,0)$
connective structure, with is the analog of the Griffiths or
canonical connection for a line bundle with hermitian fiber
metric.
We show that hermitian 2-gerbes are classified in a suitable
sense by Hermitian-holomorphic Deligne cohomology groups. Such
groups refine the ordinary Deligne cohomology groups and have
lifts of the Tame Symbol maps which, just as in $K$-theory and
ordinary Deligne cohomology, are expressed as cup-products.
One example is provided by the cup-product of two line bundles
$L$ and $M$ with hermitian fiber metric: for an algebraic curve
(or more generally for a family of curves) it agrees with the
Deligne symbol $$ for the determinant of cohomology. When
both line bundles coincide with the tangent bundle equipped with
a conformal metric, the symbol can be shown to agree with the
Liouville action functional.
Reference:
-
E. Aldrovandi, Hermitian-holomorphic (2)-Gerbes and tame symbols,
arXiv:math.CT/0310027.