Geometry of Deligne cohomology

Abstract

The aim of this paper is to give a geometric interpretation of holomorphic and smooth Deligne cohomology. Before stating the main results we recall the definition and basic properties of Deligne cohomology.Let X be a smooth complex projective variety and let Ω r X be the sheaf of germs of holomorphic r-forms on X. The qth Deligne complex of X is the complex of sheaveswhere Z(q) = (2π √ −1) q Z ⊂ C, and Z(q) X is the constant sheaf on X associated with the group Z(q). The hypercohomology H * (X, Z(q) D ) of the complex Z(q) D is called the Deligne cohomology of X. Our basic reference for Deligne cohomology is [EV].One of the key properties of Deligne cohomology is that for every p ≥ 1 the groupof the group H p,p Z (X) of integral (p, p)-classes of X by the the pth intermediate Jacobian J p (X) of Griffiths. For p = 1 the group H 2 (M, Z(1) D ) is isomorphic to the first cohomology group H 1 (O * X ) of the sheaf O * X of germs of non-vanishing holomorphic functions on X, and the sequence (1) reduces in this case to the classical short exact sequenceIt is well known that the group H 1 (X, O * X ) is isomorphic to the group CH 1 (X) of divisors of X modulo rational equivalence, the Jacobian J(X) is isomorphic to the group CH 1 hom (X) of rational equivalence classes of homologous to 0 divisors of X, and H 1,1 Z (X) is the image of the cycle map CH 1 (X) → H 2 (X; Z). One would like to have a similar cohomological description of the groups CH p (X) of codimension p cycles of X modulo rational equivalence, for p > 1, together with an analogous to (2) short exact sequence completely describing the image H 2p alg (X; Z) and the kernel CH PAWE L GAJERDeligne cohomology can be thought as a step in this direction. Indeed, the cycle homomorphism (3) lifts to a homomorphismis the Abel-Jacobi homomorphism. Another important property of Deligne cohomology is the existence of characteristic classes, called "regulators",from the algebraic K-groups of X into the Deligne cohomology of X, which generalize the classical Chern classes of holomorphic vector bundles. Several important conjectures of arithmetic algebraic geometry are formulated in terms of these regu-The second degree Deligne cohomology groups H 2 (X; Z(q) D ) have the following geometric interpretations.• H 2 (X; Z(0) D ) is the ordinary second cohomology group H 2 (X; Z) of X that can be identified with the group of smooth principal C * -bundles over X.• H 2 (X; Z(1) D ) is isomorphic to the group H 1 (O * X ) of isomorphism classes of holomorphic principal C * -bundles over X.• H 2 (X; Z(2) D ) is isomorphic to the group of isomorphism classes of holomorphic principal C * -bundles over X with holomorphic connections.• For every q > 2 the group H 2 (X; Z(q) D ) is isomorphic to the group of isomorphism classes of holomorphic principal C * -bundles with flat connections over X.Thanks to the above description of the groups H 2 (X; Z(q) D ) the geometric structure of regulators, cycle homomorphisms, and Abel-Jacobi homomorphisms has been completely understood in the c...

WZW Orientifolds and Finite Group Cohomology

“…) is the one of the isomorphism classes of flat line bundles on M and H 2 (M, D (2)) is the third real Deligne cohomology group [6,10]. The local data of a gerbe c = (B i , A ij , g ijk ) satisfy the cocycle condition 2 D 2 c = 0 and equivalent local data differ by a coboundary D 1 β with β = (Π i , χ ij ) so that the elements of the hypercohomology group H 2 (M, D(2)) are in a one-to-one correspondence with stable isomorphism classes of gerbes.…”

“…We shall call Γ the orbifold group. In a natural way, we may lift its action to the Abelian groups A n of (2.6)-(2.9) by defining 10) etc. This turns the complex K(D(2)) of (2.3) induced from the sheaf complex (2.5) into one of Γ-modules.…”

Bundle Gerbes for Chern-Simons and Wess-Zumino-Witten Theories

“…In [26] Theorem H , it is shown that for a smooth manifold M , the group H 4 (M, Z) is isomorphic to the group of isomorphism classes of smooth principal B 2 U (1)-bundles over M .…”

Landweber exact formal group laws and smooth cohomology theories

“…For cohomology theories based on geometric or analytic cycles there are often alternative models. This applies in particular to ordinary cohomology whose smooth extension has various different realizations (see Cheeger and Simons [7], Gajer [12], Brylinski [2], Dupont and Ljungmann [8], Hopkins and Singer [14] and Bunke, Kreck and Schick [3]). The papers by Simons and Sullivan [21] or Bunke and Schick [5] show that all these realizations are isomorphic.…”