Holmstrom Deligne-Beilinson cohomology

Deligne-Beilinson cohomology

Discussion with Scholl, Nov 2007

Deligne cohomology works nive when $X$ is proper, but otherwise, need D-B cohomology in order to get good theory. I think D-B cohomology can be identified with absolute Hodge cohomology? see notes from discussion.

---

Deligne-Beilinson cohomology

Chapter 10 in the book of Burgos Gil

A chapter in the Beilinson volume: Esnault, Viehweg

---

Deligne-Beilinson cohomology

There is a product structure, described by Esnault-Viehweg.

---

Deligne-Beilinson cohomology

Real or complex algebraic varieties.

MR1156507 (93a:14022) Levine, Marc(1-NORE) Deligne-Be\u\i linson cohomology for singular varieties. Algebraic $K$-theory, commutative algebra, and algebraic geometry (Santa Margherita Ligure, 1989), 113–146, Contemp. Math., 126, Amer. Math. Soc., Providence, RI, 1992.

---

Deligne-Beilinson cohomology

Chern class from K-theory to Deligne-Beilinson cohomology: see http://www.math.uiuc.edu/K-theory/0065.

Following Burgos-Gil: D-B cohomology can also be defined as the the hypercohomology of a certain complex of sheaves in the Zariski topology. Therefore, one can apply Gillet’s construction of characteristic classes for higher K-theory. Beilinson’s regulator map is the Chern character map between higher K-theory and real D-B cohomology.

Here is a down-to-earth (ad hoc) definition in the special case of a number field. First we define a map

and then we define, for a number field $k$ with a ring of integers $O$, the regulator map as

by composing taking the product of all complex embeddings $\Sigma$ (taking invariance under complex conjugation?) and composing with $r_{Be, \mathbb{C}}$. Along the way we use various kinds of cohomology of $GL_n(\mathbb{C})$ a lot.

One can compare this with Borel’s regulator, and with the right normalisation, Borel’s regulator is twice Beilinson’s.

---

MR1736876 (2001c:14042) Dupont, Johan(DK-ARHS-MI); Hain, Richard(1-DUKE); Zucker, Steven(1-JHOP) Regulators and characteristic classes of flat bundles. (English summary)

---

Deligne-Beilinson cohomology

Burgos: Arithmetic Chow rings and Deligne-Beilinson cohomology, 1997.

MR1014822 (90j:14025) Esnault, Hélène(D-MPI) On the Loday symbol in the Deligne-Be\u\i linson cohomology. $K$-Theory 3 (1989), no. 1, 1–28.

---

Deligne-Beilinson cohomology

MathSciNet

Google Scholar

Google

arXiv: Experimental full text search

arXiv: Abstract search

---

Deligne-Beilinson cohomology

AG (Algebraic geometry)

---

Deligne-Beilinson cohomology

Mixed, Arithmetic

---

Deligne-Beilinson cohomology

Brief notes from Esnault-Viehweg

(see link under Online references below)

Deligne cohomology is defined for a complex manifold. Deligne-Beilinson cohomology is defined for a quasi-projective complex manifold, and presumably coincides with Deligne cohomology in the compact/proper case.

We can extend the definition of D-B cohomology to simplicial schemes over $\mathbb{C}$ (separated, of finite type).

There is a cycle class map from the Chow ring to the D-B cohomology ring, which is related to the Abel-Jacobi map.

We can define Chern classes of vector bundles in D-B cohomology.

Beilinson describes D-B cohomology as an extension of Hodge structures.

Brief notes from Burgos Gil, chapter 10.

One considers a proper smooth algebraic varity $\bar{X}$ over $\mathbb{C}$, with a normal crossings divisor D. Put $X = \bar{X} - D$. For a subgroup $\Lambda$ of $\mathbb{C}$ one defines a complex of sheaves, in the analytic topology, $\Lambda(p)_\mathcal{D}$ (or $\Lambda(p)_{D}$). The hypercohomology of this sheaf is the Deligne-Beilinson cohomology of $X$. Notation: $H^m_\mathcal{D} (X, \Lambda(p) )$.

Alternative definition in terms of certain complexes. Some Hodge theory; if $X$ is projective, then $H^{2p}_\mathcal{D} (X, \mathbb{Z}(p) )$ is en extension of $H^{p,p}(X, \mathbb{Z}(p) )$ by the intermediate Jacobian $J_p(X)$.

Representation of real D-B cohomology in terms of smooth differential forms.

Definition of D-B cohomology for real varieties.

Remark: “In general it is not clear how to define an integral/rational structure on D-B cohomology. However, when $X$ is defined over a number field one can do something for certain values of $m$ and $p$.

Notes from Feliu section 1.4

Description of the truncated Deligne complex (the square of (p,q)-forms with the extra diagonal arrow on top) which computes the Deligne cohomology $H^n_D(X, \mathbb{R}(p) )$ of an (open) complex algebraic manifold, in degrees $n \leq 2p$. Also description of the product structure in terms of this complex. The only reference is to Burgos Gil: Arithmetic Chow rings and Deligne-Beilinson cohomology. Description of the complex of currents

Functoriality: Contravariant functoriality by pullback of differential forms (with log singularities). Covariant functoriality for proper morphisms $f$ of equidimensional cplx algebraic manifolds (index shifting).

Cohomology with supports. Cycle map from Chow groups as expected.

Chern classes: explicit descriptions of $c_1: Pic(X) \to H^2(X, \mathbb{R}(1) )$ and $c_1: \Gamma(X, \mathbf{G}_m ) \to H^1(X, \mathbb{R}(1) )$.

For real varieties (i.e. a complex algebraic manifold with an antilinear involution), can compute real D-B cohomology by taking fixed part either of the ordinary D-B cohomology or on the Deligne complex. She actually says that the real D-B cohomology can be computed as the cohomology (not hypercohomology) of the real Deligne complex, but this must be wrong?

---

Deligne-Beilinson cohomology

See also: Deligne cohomology, Absolute Hodge cohomology, Deligne homology, Bloch-Ogus cohomology

---

Deligne-Beilinson cohomology

Soule http://www.ams.org/mathscinet-getitem?mr=991985 contains a review of D-B cohomology.

Zucker-Brylinski review some material.

---

Deligne-Beilinson cohomology

Dimension of the cohom gps “determined by the Hodge structure of X tensored with the reals”, it said somewhere on the internet I think. How does this work??

nLab page on Deligne-Beilinson cohomology