Hermitian-holomorphic (2)-gerbes and tame symbols

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 2-gerbes 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 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.

2-Gerbes bound by complexes of gr-stacks, and cohomology

“…The following theorem generalizes previous results on connective and Hermitian structures on 2-gerbes, see [5][6][7]. Proof.…”

“…More recently, we have similarly introduced the concept of Hermitian structure on an abelian gerbe bound by O X , by modeling it on the corresponding familiar notion of invertible sheaf equipped with a fiber Hermitian metric [7]. In simplified terms, this structure is also of the type introduced above, namely we find that in this case it can be conveniently encoded in the structure of gerbe bound by the complex O X log|·| −−→ E 0 X , where the latter denotes the sheaf of smooth real functions on X .…”

“…However, a more interesting group H 4 D (X, 2) does appear in the following way: in [7] we introduced a complex, denoted Γ (2) • (defined in Section 7.1), and we (informally) argued that the hypercohomology group H 4 (X, Γ (2) • ) classifies 2-gerbes equipped with both a connective structureà la 2 The absolute cohomology groups in that range are zero. 3 There is of course an interest in knowing that, say, H 3 D (X, Z(1)) classifies abelian gerbes bound by O X , however the nice connection with regulators, etc.…”

“…(1) • X provides a characterization of the canonical connection associated with a Hermitian line bundle [10,7]. We will also need a leaner version of the complex (1.2.7) introduced in [11], namely:…”

“…A slight modification of the notion of connective structure recalled in Section 2.2 is to consider the length 2 complex [7]:…”