The formal de Rham complex

We propose a formal construction generalizing the classic de Rham complex to a wide class of models in mathematical physics and analysis. The presentation is divided into a sequence of definitions and elementary, easily verified statements; proofs are therefore given only in the key case. Linear operations are everywhere performed over a fixed number field $\mathbb{F} = \mathbb{R},\mathbb{C}$ . All linear spaces, algebras, and modules, although not stipulated explicitly, are by definition or by construction endowed with natural locally convex topologies, and their morphisms are continuous.     

Decomposition of the de Rham complex

We give a reformulation of Deligne and Illusie's characteristicp proof of the degeneration of the “Hodge-to-de Rham” spectral sequence, which replaces patching in the derived category by an explicit quasi-isomorphism.     

Generalized Calabi-Yau manifolds and the chiral de Rham complex

We show that the chiral de Rham complex of a generalized Calabi-Yau manifold carries N=2 supersymmetry. We discuss the corresponding topological twist for this N=2 algebra. We interpret this as an algebroid version of the super-Sugawara or Kac-Todorov construction.     

On approximations of the de Rham complex and their cohomology

For a commutative algebra R, its de Rham cohomology is an important invariant of R. In the paper, an infinite chain of de Rham-like complexes is introduced where the first member of the chain is the de Rham complex. The complexes are called approximations of the de Rham complex. Their cohomologies are found for polynomial rings and algebras of power series over a field of characteristic zero.     

The local index density of the perturbed de Rham complex

Czechoslovak Mathematical Journal - A perturbation of the de Rham complex was introduced by Witten for an exact 1-form Θ and later extended by Novikov for a closed 1-form on a Riemannian...     

The de Rham complex of a quasihomogeneous complete intersection

Functional Analysis and Its Applications -     

The de Rham homotopy theory of complex algebraic varieties II

We show that the local system of homotopy groups, associated with a topologically locally trivial family of smooth pointed varieties, underlies a good variation of mixed Hodge structure. In particular we show that there is a limit mixed Hodge structure on homotopy associated with a degeneration of such varieties.Supported in part by the National Science Foundation grant DMS-8401175.     

de Rham transforms for subdivision schemes

For any subdivision scheme, we define its de Rham transform, which generalizes the de Rham and Chaikin corner cutting. The main property of the de Rham transform is that it preserves a sum rule. This allows comparison of the Hölder regularity of a given subdivision scheme with that of its de Rham transform. A graphical comparison is made for three different families of subdivision schemes, the last one being the generalized four-point scheme.     

The multiple residue and the weight filtration on the logarithmic de Rham complex

We study the multiple residue of logarithmic differential forms with poles along a reducible divisor and compute the kernel and the image of the multiple residue map. As an application we describe the weight filtration on the logarithmic de Rham complex for divisors whose irreducible components are given locally by a regular sequence of holomorphic functions. In particular, this allows us to compute the mixed Hodge structure on the cohomology of the complement of divisors of certain types without the use of theorems on resolution of singularities and the standard reduction to the case of normal crossings.     

The de Rham complex on unbounded domains with Sobolev space topology

This paper studies the operator dd^∗+d^∗d acting on q-forms on an unbounded domain with smooth boundary, where d is the exterior derivative and d^∗ is the adjoint of d calculated using the Sobolev space topology. The domain of d^∗ is determined and an expression for d^∗ is obtained. The operator dd^∗+d^∗d gives rise to a boundary value problem. Global regularity is obtained using weighted norms and global existence is obtained by using the theory of compact operators.     

Sharp de Rham realization

We introduce the sharp (universal) extension of a 1-motive (with additive factors and torsion) over a field of characteristic zero. We define the sharp de Rham realization T by passing to the Lie-algebra. Over the complex numbers we then show a (sharp de Rham) comparison theorem in the category of formal Hodge structures. For a free 1-motive along with its Cartier dual we get a canonical connection on their sharp extensions yielding a perfect pairing on sharp realizations. Thus we show how to provide one-dimensional sharp de Rham cohomology of algebraic varieties.     

Stochastic Algebraic de Rham Complexes

We define completion of the algebraic de Rham complex associated to the algebras of functionals smooth in the Chen–Souriau sense or in the Nualart–Pardoux sense over the loop space. We show that the stochastic algebraic de Rham cohomology groups are equal to the deterministic cohomology groups of the loop space.     

A polar de Rham theorem

We prove an analogue of the de Rham theorem for polar homology; that the polar homology HP_q(X) of a smooth projective variety X is isomorphic to its H^n,n−q Dolbeault cohomology group. This analogue can be regarded as a geometric complexification where arbitrary (sub)manifolds are replaced by complex (sub)manifolds and de Rham's operator d is replaced by Dolbeault's .     

A Conformal de Rham Complex

We introduce the notion of a conformal de Rham complex of a Riemannian manifold. This is a graded differential Banach algebra and it is invariant under quasiconformal maps, in particular the associated cohomology is a new quasiconformal invariant.     

Dagger Formal Geometry and de Rham Cohomology

Let K be a finite extension of Qp with R its ring of integers and k = Fq its residue field.Let π be a uniformizer of R. At first, let us recall some concepts.