Deligne Cohomology Research Articles

Simplicial Abel-Jacobi maps and reciprocity laws

We describe an explicit morphism of complexes that induces the cycle-class maps from (simplicially described) higher Chow groups to rational Deligne cohomology. The reciprocity laws satisfied by the currents we introduce for this purpose are shown to provide a clarifying perspective on functional equations satisfied by complex-valued di- and trilogarithms.

Massey products in differential cohomology via stacks

We extend Massey products from cohomology to differential cohomology via stacks, organizing and generalizing existing constructions in Deligne cohomology. We study the properties and show how they are related to more classical Massey products in de Rham, singular, and Deligne cohomology. The setting and the algebraic machinery via stacks allow for computations and make the construction well-suited for applications. We illustrate with several examples from differential geometry and mathematical physics.

Exact Computations in Topological Abelian Chern-Simons and BF Theories

We introduce Deligne cohomology that classifies U1 fibre bundles over 3 manifolds endowed with connections. We show how the structure of Deligne cohomology classes provides a way to perform exact (nonperturbative) computations in U1 Chern-Simons theory (BF theory, resp.) at the level of functional integrals. The partition functions (and observables) of these theories are strongly related to topological invariants well known to the mathematicians.

Aeppli–Bott-Chern cohomology and Deligne cohomology from a viewpoint of Harvey–Lawson’s spark complex

By comparing Deligne complex and Aeppli–Bott-Chern complex, we construct a differential cohomology \(\widehat{H}^*(X, *, *)\) that plays the role of Harvey–Lawson spark group \(\widehat{H}^*(X, *)\), and a cohomology \(H^*_{\mathrm{ABC}}(X; \mathbb Z(*, *))\) that plays the role of Deligne cohomology \(H^*_{\mathcal {D}}(X; \mathbb Z(*))\) for every complex manifold X. They fit in the short exact sequence $$\begin{aligned} 0\rightarrow H^{k+1}_{\mathrm{ABC}}(X; \mathbb Z(p, q)) \rightarrow \widehat{H}^k(X, p, q) \overset{\delta _1}{\rightarrow } Z^{k+1}_I(X, p, q) \rightarrow 0 \end{aligned}$$ and \(\widehat{H}^{\bullet }(X, \bullet , \bullet )\) possess ring structure and refined Chern classes, acted by the complex conjugation, and if some primitive cohomology groups of X vanish, there is a Lefschetz isomorphism. Furthermore, the ring structure of \(H^{\bullet }_{\mathrm{ABC}}(X; \mathbb Z(\bullet , \bullet ))\) inherited from \(\widehat{H}^{\bullet }(X, \bullet , \bullet )\) is compatible with the one of the analytic Deligne cohomology \(H^{\bullet }(X; \mathbb Z(\bullet ))\). We compute \(\widehat{H}^*(X, *, *)\) for X the Iwasawa manifold and its small deformations and get a refinement of the classification given by Nakamura.

A new construction of cyclic homology

Based on the ideas of Cuntz and Quillen, we give a simple construction of cyclic homology of unital algebras in terms of the noncommutative de Rham complex and a certain differential similar to the equivariant de Rham differential. We describe the Connes exact sequence in this setting. We define equivariant Deligne cohomology and construct, for each n > 0, a natural map from cyclic homology of an algebra to the GL_n-equivariant Deligne cohomology of the variety of n-dimensional representations of that algebra. The bridge between cyclic homology and equivariant Deligne cohomology is provided by extended cyclic homology, which we define and compute here, based on the extended noncommutative de Rham complex introduced previously by the authors.

Exponential complexes, period morphisms, and characteristic classes

We introduce exponential complexes of sheaves on manifolds. They are resolutions of the (Tate twisted) constant sheaves of the rational numbers, generalising the short exact exponential sequence. There are canonical maps from the exponential complexes to the de Rham complex. Using this, we introduce new complexes calculating the rational Deligne cohomology. We call them exponential Deligne complexes. Their advantage is that, at least at the generic point of a complex variety, one can define Beilinson's regulator map to the rational Deligne cohomology on the level of complexes. Namely, we define period morphisms. We use them to produce homomorphisms from motivic complexes to the exponential Deligne complexes at the generic point. Combining this with the construction of Chern classes with coefficients in the bigrassmannian complexes, we get, for the weights up to four, local explicit formulas for the Chern classes in the rational Deligne cohomology via polylogarithms.

Flat pairing and generalized Cheeger–Simons characters

Let \(h^{\bullet }\) be a multiplicative cohomology theory, \(h_{\bullet }\) its dual homology theory and \(\hat{h}^{\bullet }\) a differential refinement. We first construct the natural pairing between \(h_{\bullet }\) and the flat part of \(\hat{h}^{\bullet }\), generalizing the holonomy of a flat Deligne cohomology class. Then, in order to generalize the holonomy of any Deligne cohomology class, we define the generalized Cheeger–Simons characters. The latter are functions from suitably defined differential cycles to the cohomology ring of the point, such that the value on a trivial cycle only depends on the curvature.

Arakelov motivic cohomology II

We show that the constructions done in part I generalize their classical counterparts: firstly, the classical Beilinson regulator is induced by the abstract Chern class map from BGL \operatorname {BGL} to the Deligne cohomology spectrum. Secondly, Arakelov motivic cohomology is a generalization of arithmetic K K -theory and arithmetic Chow groups. For example, this implies a decomposition of higher arithmetic K K -groups in its Adams eigenspaces. Finally, we give a conceptual explanation of the height pairing: it is the natural pairing of motivic homology and Arakelov motivic cohomology.

Decomposition matrices for low-rank unitary groups

We study the decomposition matrices of the unipotent ℓ-blocks of finite special unitary groups SU n ( q ) for unitary primes ℓ larger than n. Up to few unknown entries, we give a complete solution for n = 2 , ... , 10 . We also prove a general result for two-column partitions when ℓ divides q + 1 . This is achieved using projective modules coming from the ℓ-adic cohomology of Deligne–Lusztig varieties.

Semistrict higher gauge theory

We develop semistrict higher gauge theory from first principles. In particular, we describe the differential Deligne cohomology underlying semistrict principal 2-bundles with connective structures. Principal 2-bundles are obtained in terms of weak 2-functors from the Cech groupoid to weak Lie 2-groups. As is demonstrated, some of these Lie 2-groups can be differentiated to semistrict Lie 2-algebras by a method due to Severa. We further derive the full description of connective structures on semistrict principal 2-bundles including the non-linear gauge transformations. As an application, we use a twistor construction to derive superconformal constraint equations in six dimensions for a non-Abelian N=(2,0) tensor multiplet taking values in a semistrict Lie 2-algebra.

On a canonical class of Green currents for the unit sections of abelian schemes

We show that on any abelian scheme over a complex quasi-projective smooth variety, there is a Green current for the zero-section, which is axiomatically determined up to \partial and \bar\partial -exact differential forms. On an elliptic curve, this current specialises to a Siegel function. We prove generalisations of classical properties of Siegel functions, like distribution relations and reciprocity laws. Furthermore, as an application of a refined version of the arithmetic Riemann-Roch theorem, we show that the above current, when restricted to a torsion section, is the realisation in analytic Deligne cohomology of an element of the (Quillen) K_1 group of the base, the corresponding denominator being given by the denominator of a Bernoulli number. This generalises the second Kronecker limit formula and the denominator 12 computed by Kubert, Lang and Robert in the case of Siegel units. Finally, we prove an analog in Arakelov theory of a Chern class formula of Bloch and Beauville, where the canonical current plays a key role.

Bigraded invariants for real curves

For a proper smooth real algebraic curve we compute the ring structure of both its ordinary bigraded Gal.C=R/‐equivariant cohomology [6] and its integral Deligne cohomology for real varieties [12]. These rings reflect both the equivariant topology and the real algebraic structure of and they are recipients of natural transformations from motivic cohomology. We conjecture that they completely detect the motivic torsion classes. 55N91; 14P25

Hodge filtered complex bordism

We construct Hodge filtered cohomology groups for complex manifolds that combine the topological information of generalized cohomology theories with geometric data of Hodge filtered holomorphic forms. This theory provides a natural generalization of Deligne cohomology. For smooth complex algebraic varieties, we show that the theory satisfies a projective bundle formula and $\A^1$-homotopy invariance. Moreover, we obtain transfer maps along projective morphisms.

Relative differential cohomology

Let h be a rationally even cohomology theory and h^ the natural differential refinement, as defined by Hopkins and Singer. We consider the possible definitions of the relative differential cohomology groups, generalizing the analogous picture for the Deligne cohomology, and we show the corresponding long exact sequence in each case.

Projective modules in the intersection cohomology of Deligne–Lusztig varieties

We formulate a strong positivity conjecture on characters afforded by the Alvis–Curtis dual of the intersection cohomology of Deligne–Lusztig varieties. This conjecture provides a powerful tool to determine decomposition numbers of unipotent ℓ-blocks of finite reductive groups.

Relative Deligne cohomology and Cheeger-Simons characters

In the paper [1] (arXiv:math/0408333) the authors discuss two possible definitions of the relative Cheeger-Simons characters, the second one fitting into a long exact sequence. Here we relate that picture to the one of the relative Deligne cohomology groups, defined via the mapping cone: we show that there are three meaningful relative groups, and we analyze the corresponding definition of relative Cheeger-Simons characters in each case. We then extract another definition, corresponding to the one fitting into a long exact sequence in [1]. Finally we show how the explicit formulas for the holonomy, the transgression maps and the integration can be extended to the relative case.

Cohomology of Deligne–Lusztig varieties for short-length regular elements in exceptional groups

We determine the cohomology of Deligne–Lusztig varieties associated with some short-length regular elements for split groups of type F4 and En. As a byproduct, we obtain conjectural Brauer trees for the principal Φ14-block of E7 and the principal Φ24-block of E8.

A note on decomposition numbers for groups of Lie type of small rank

Using virtual projective characters coming from the ℓ-adic cohomology of Deligne–Lusztig varieties, we determine new decomposition numbers for principal blocks of some groups of Lie type of small rank, including G2(q) and Steinbergʼs triality groups D43(q).

GEOMETRY OF SPIN ANDSPINcSTRUCTURES IN THE M-THEORY PARTITION FUNCTION

We study the effects of having multiple Spin structures on the partition function of the spacetime fields in M-theory. This leads to a potential anomaly which appears in the eta invariants upon variation of the Spin structure. The main sources of such spaces are manifolds with nontrivial fundamental group, which are also important in realistic models. We extend the discussion to the Spinccase and find the phase of the partition function, and revisit the quantization condition for the C-field in this case. In type IIA string theory in 10 dimensions, the (mod 2) index of the Dirac operator is the obstruction to having a well-defined partition function. We geometrically characterize manifolds with and without such an anomaly and extend to the case of nontrivial fundamental group. The lift to KO-theory gives the α-invariant, which in general depends on the Spin structure. This reveals many interesting connections to positive scalar curvature manifolds and constructions related to the Gromov–Lawson–Rosenberg conjecture. In the 12-dimensional theory bounding M-theory, we study similar geometric questions, including choices of metrics and obtaining elements of K-theory in 10 dimensions by pushforward in K-theory on the disk fiber. We interpret the latter in terms of the families index theorem for Dirac operators on the M-theory circle and disk. This involves superconnections, eta forms, and infinite-dimensional bundles, and gives elements in Deligne cohomology in lower dimensions. We illustrate our discussion with many examples throughout.

SL2(ℂ)-character variety of a hyperbolic link and regulator

In this paper, we study the SL2(C) character variety of a hyperbolic link in S 3 . We analyze a special smooth projective variety Y h arising from some 1-dimensional irreducible slices on the character variety. We prove that a natural symbol obtained from these 1-dimensional slices is a torsion in K2(C(Y h )). By using the regulator map from K2 to the corresponding Deligne cohomology, we get some variation formulae on some Zariski open subset of Y h . From this we give some discussions on a possible parametrized volume conjecture for both hyperbolic links and knots.