Singular Cohomology Groups Research Articles

Semi-algebraic chains on projective varieties and the Abel-Jacobi map for higher Chow cycles

We will show that the singular cohomology groups of a smooth quasi-projective complex variety relative to a normal crossing divisor can be described in terms of delta-admissible chains. Roughly speaking, a delta-admissible chain is a simplicial semi-algebraic chain meeting the faces properly. As an application, we show that the Abel-Jacobi map for higher Chow cycles can be described via delta-admissible chains. As an example, we will describe the Hodge realization of the polylog cycles constructed by Bloch-Kriz in terms of the Abel-Jacobi map.

The Mumford�Tate conjecture for products of abelian varieties

Let $X$ be a smooth projective variety over a finitely generated field $K$ of characteristic~$0$ and fix an embedding $K \subset \mathbb{C}$. The Mumford--Tate conjecture is a precise way of saying that certain extra structure on the $\ell$-adic \'etale cohomology groups of~$X$ (namely, a Galois representation) and certain extra structure on the singular cohomology groups of~$X$ (namely, a Hodge structure) convey the same information. The main result of this paper says that if $A_1$ and~$A_2$ are abelian varieties (or abelian motives) over~$K$, and the Mumford--Tate conjecture holds for both~$A_1$ and~$A_2$, then it holds for $A_1 \times A_2$. These results do not depend on the embedding $K \subset \CC$.

Real cohomology and the powers of the fundamental ideal in the Witt ring

Let A be a local ring in which 2 is invertible. It is known that the localization of the cohomology ring H et∗(A, ℤ∕2) with respect to the class (−1) ∈ H et1(A, ℤ∕2) is isomorphic to the ring C(sperA, ℤ∕2) of continuous ℤ∕2-valued functions on the real spectrum of A. Let In(A) denote the powers of the fundamental ideal in the Witt ring of symmetric bilinear forms over A. The starting point of this article is the “integral” version: the localization of the graded ring ⊕ n≥0In(A) with respect to the class 〈〈−1〉〉 := 〈1,1〉∈ I(A) is isomorphic to the ring C(sperA, ℤ) of continuous ℤ-valued functions on the real spectrum of A. This has interesting applications to schemes. For instance, for any algebraic variety X over the field of real numbers ℝ and any integer n strictly greater than the Krull dimension of X, we obtain a bijection between the Zariski cohomology groups HZar∗(X,ℐn) with coefficients in the sheaf ℐn associated to the n-th power of the fundamental ideal in the Witt ring W(X) and the singular cohomology groups Hsing∗(X(ℝ), ℤ).

Differential cohomology and locally covariant quantum field theory

We study differential cohomology on categories of globally hyperbolic Lorentzian manifolds. The Lorentzian metric allows us to define a natural transformation whose kernel generalizes Maxwell's equations and fits into a restriction of the fundamental exact sequences of differential cohomology. We consider smooth Pontryagin duals of differential cohomology groups, which are subgroups of the character groups. We prove that these groups fit into smooth duals of the fundamental exact sequences of differential cohomology and equip them with a natural presymplectic structure derived from a generalized Maxwell Lagrangian. The resulting presymplectic Abelian groups are quantized using the [Formula: see text]-functor, which yields a covariant functor from our categories of globally hyperbolic Lorentzian manifolds to the category of [Formula: see text]-algebras. We prove that this functor satisfies the causality and time-slice axioms of locally covariant quantum field theory, but that it violates the locality axiom. We show that this violation is precisely due to the fact that our functor has topological subfunctors describing the Pontryagin duals of certain singular cohomology groups. As a byproduct, we develop a Fréchet–Lie group structure on differential cohomology groups.

Holomorphic cohomology groups on compact Kähler complex spaces

Let ( X , O X ) be a compact (reduced) complex space, bimeromorphic to a Kähler manifold. The singular cohomology groups H q ( X , C ) carry a mixed Hodge structure. In particular they carry a weight filtration W − l H q ( X , C ) ( l = 0 , … , q ), and the graded quotients W − l H q ( X , C ) W − l − 1 H q ( X , C ) have a direct sum decomposition into holomorphic invariants as ⊕ r + s = q − l ( W − l H q ( X , C ) W − l − 1 H q ( X , C ) ) ( r , s ) . Here we investigate the relationships between the above invariants for r = 0 and the cohomology groups H q ( X , O ˜ X ) , where O ˜ X is the sheaf of weakly holomorphic functions on X. Moreover, according to the smooth case, we characterize the topological line bundles L on X such that the class of c 1 ( L ) in W 0 H 2 ( X , C ) W − 1 H 2 ( X , C ) has pure type ( 1 , 1 ) .

The Leray spectral sequence is motivic

Our goal is to prove that the Leray spectral sequence associated to a map of algebraic varieties is motivic in the following sense: If the singular cohomology groups of the category of quasiprojective varieties defined over a subfield of C can be canonically endowed with some additional structure, such as its mixed Hodge structure or its motive (in Nori's sense or as a product of realizations). Then the Leray spectral sequence for a projective map in this category is automatically compatible with this structure. As byproduct, we get a relatively cheap construction of a mixed Hodge structure on the cohomology of variation of Hodge structure of geometric origin.

Rational isomorphisms between K-theories and cohomology theories

The well known isomorphism relating the rational algebraic K-theory groups and the rational motivic cohomology groups of a smooth variety over a field of characteristic 0 is shown to be realized by a map (the “Segre map”) of infinite loop spaces. Moreover, the associated Chern character map on rational homotopy groups is shown to be a ring isomorphism. A technique is introduced that establishes a useful general criterion for a natural transformation of functors on quasi-projective complex varieties to induce a homotopy equivalence of semi-topological singular complexes. Since semi-topological K-theory and morphic cohomology can be formulated as the semi-topological singular complexes associated to algebraic K-theory and motivic cohomology, this criterion provides a rational isomorphism between the semi-topological K-theory groups and the morphic cohomology groups of a smooth complex variety. Consequences include a Riemann-Roch theorem for the Chern character on semi-topological K-theory and an interpretation of the “topological filtration” on singular cohomology groups in K-theoretic terms.

The equivariant Serre spectral sequence as an application of a spectral sequence of Spanier

E.H. Spanier (1992) has constructed, for a cohomology theory defined on a triangulated space and locally constant on each open simplex, a spectral sequence whose E 2-term consists of certain simplicial cohomology groups, converging to the cohomology of the space. In this paper we study a closed G-fibration ƒ: Y → X, where G is a finite group. We show that if the base- G-space X is equivariantly triangulated and Y is paracompact, then Spanier's spectral sequence yields an equivariant Serre spectral sequence for ƒ. The main point here is to identify the equivariant singular cohomology groups of X with appropriate simplicial cohomology groups of the orbit space X G .

One Remark on the Realizability of Singular Cohomology Groups

One Remark on the Realizability of Singular Cohomology Groups

Singular Cohomology Groups

Singular Cohomology Groups

On the Realizability of Singular Cohomology Groups

On the Realizability of Singular Cohomology Groups

On the realizability of singular cohomology groups

On the realizability of singular cohomology groups