De Rham Complex Research Articles

On a topological counterpart of regularization for holonomic 𝒟-modules

On a complex manifold, the embedding of the category of regular holonomic D-modules into that of holonomic D-modules has a left quasi-inverse functor $\mathcal{M}\mapsto\mathcal{M}_{\mathrm{reg}}$, called regularization. Recall that $\mathcal{M}_{\mathrm{reg}}$ is reconstructed from the de Rham complex of $\mathcal{M}$ by the regular Riemann-Hilbert correspondence. Similarly, on a topological space, the embedding of sheaves into enhanced ind-sheaves has a left quasi-inverse functor, called here sheafification. Regularization and sheafification are intertwined by the irregular Riemann-Hilbert correspondence. Here, we study some of their properties. In particular, we provide a germ formula for the sheafification of enhanced specialization and microlocalization.

A Perturbation of the de Rham Complex

We consider a perturbation of the de Rham complex on a compact manifold with boundary. This perturbation goes beyond the framework of complexes, and so cohomology does not apply to it. On the other hand, its curvature is "small", hence there is a natural way to introduce an Euler characteristic and develop a Lefschetz theory for the perturbation. This work is intended as an attempt to develop a cohomology theory for arbitrary sequences of linear mappings

An algorithm for a lifted Massey triple product of a smooth projective plane curve

We provide an explicit algorithm to compute a lifted Massey triple product relative to a defining system for a smooth projective plane curve [Formula: see text] defined by a homogeneous polynomial [Formula: see text] over a field. The main idea is to use the description (due to Carlson and Griffiths) of the cup product for [Formula: see text] in terms of the multiplications inside the Jacobian ring of [Formula: see text] and the Cech–deRham complex of [Formula: see text]. Our algorithm gives a criterion whether a lifted Massey triple product vanishes or not in [Formula: see text] under a particular nontrivial defining system of the Massey triple product and thus can be viewed as a generalization of the vanishing criterion of the cup product in [Formula: see text] of Carlson and Griffiths. Based on our algorithm, we provide explicit numerical examples by running the computer program.

EVOLUTIONARY DE RHAM-HODGE METHOD.

The de Rham-Hodge theory is a landmark of the 20th Century's mathematics and has had a great impact on mathematics, physics, computer science, and engineering. This work introduces an evolutionary de Rham-Hodge method to provide a unified paradigm for the multiscale geometric and topological analysis of evolving manifolds constructed from a filtration, which induces a family of evolutionary de Rham complexes. While the present method can be easily applied to close manifolds, the emphasis is given to more challenging compact manifolds with 2-manifold boundaries, which require appropriate analysis and treatment of boundary conditions on differential forms to maintain proper topological properties. Three sets of unique evolutionary Hodge Laplacians are proposed to generate three sets of topology-preserving singular spectra, for which the multiplicities of zero eigenvalues correspond to exactly the persistent Betti numbers of dimensions 0, 1 and 2. Additionally, three sets of non-zero eigenvalues further reveal both topological persistence and geometric progression during the manifold evolution. Extensive numerical experiments are carried out via the discrete exterior calculus to demonstrate the potential of the proposed paradigm for data representation and shape analysis of both point cloud data and density maps. To demonstrate the utility of the proposed method, the application is considered to the protein B-factor predictions of a few challenging cases for which existing biophysical models break down.

A Poisson transform adapted to the Rumin complex

Let G be a semisimple Lie group with finite center, [Formula: see text] a maximal compact subgroup, and [Formula: see text] a parabolic subgroup. Following ideas of P. Y. Gaillard, one may use G-invariant differential forms on [Formula: see text] to construct G-equivariant Poisson transforms mapping differential forms on G/P to differential forms on G/K. Such invariant forms can be constructed using finite-dimensional representation theory. In this general setting, we first prove that the transforms that always produce harmonic forms are exactly those that descend from the de Rham complex on G/P to the associated Bernstein–Gelfand–Gelfand (or BGG) complex in a well defined sense. The main part of this paper is devoted to an explicit construction of such transforms with additional favorable properties in the case that [Formula: see text]. Thus, G/P is [Formula: see text] with its natural CR structure and the relevant BGG complex is the Rumin complex, while G/K is complex hyperbolic space of complex dimension [Formula: see text]. The construction is carried out both for complex and for real differential forms and the compatibility of the transforms with the natural operators that are available on their sources and targets are analyzed in detail.

Simplicial cochain algebras for diffeological spaces

The original de Rham cohomology due to Souriau and the singular cohomology in diffeology are not isomorphic to each other in general. This manuscript introduces a singular de Rham complex endowed with an integration map into the singular cochain complex which gives the de Rham theorem for every diffeological space. It is also proved that a morphism called the factor map from the original de Rham complex to the new one is a quasi-isomorphism for a manifold and, more general, a space with singularities. Moreover, Chen’s iterated integrals are considered in a diffeological framework. As a consequence, we deduce that the bar complex of the original de Rham complex of a simply-connected diffeological space is quasi-isomorphic to the singular de Rham complex of the diffeological free loop space provided the factor map for the underlying diffeological space is a quasi-isomorphism. The process for proving the assertion yields the Leray–Serre spectral sequence and the Eilenberg–Moore spectral sequence in diffeology.

Hodge theorem for the logarithmic de Rham complex via derived intersections

In a beautiful paper, Deligne and Illusie (Invent Math 89(2):247–270, 1987) proved the degeneration of the Hodge-to-de Rham spectral sequence using positive characteristic methods. Kato (in: Igusa (ed) ALG analysis, geographic and numbers theory, Johns Hopkins University Press, Baltimore, 1989) generalized this result to logarithmic schemes. In this paper, we use the theory of twisted derived intersections developed in Arinkin et al. (Algebraic Geom 4:394–423, 2017) and the author of this paper to give a new, geometric interpretation of the Hodge theorem for the logarithmic de Rham complex.

Gauge Modules for the Lie Algebras of Vector Fields on Affine Varieties

For a smooth irreducible affine algebraic variety we study a class of gauge modules admitting compatible actions of both the algebra \(\mathcal {A}\) of functions and the Lie algebra \(\mathcal {V}\) of vector fields on the variety. We prove that a gauge module corresponding to a simple \(\mathfrak {gl}_{N}\)-module is irreducible as a module over the Lie algebra of vector fields unless it appears in the de Rham complex.

The nonconforming virtual element method for the Darcy–Stokes problem

We present the nonconforming virtual element method for the Darcy–Stokes problem. By using a gradient projection operator and a special polynomial subspace orthogonal to the gradient of some polynomial space, we construct a H1-nonconforming but H(div)-conforming virtual element that allows us to compute the L2-projection. The optimal convergence is proved under the assumption of sufficient regularity, and the uniform convergence is also obtained for the lowest-order case. Besides, we establish a discrete exact sequence of de Rham complex related to the nonconforming virtual element. Finally, we carry out some numerical tests to make a comparison of the convergence between different nonconforming virtual elements and confirm the optimal and uniform convergence of the nonconforming virtual element.

Application of a minimal compatible element to incompressible and nearly incompressible continuum mechanics

In this note we will explore some applications of the recently constructed piecewise affine, H1-conforming element that fits in a discrete de Rham complex (Christiansen and Hu, 2018). In particular we show how the element leads to locking free methods for incompressible elasticity and viscosity robust methods for the Brinkman model.

Differential calculus over double Lie algebroids

M. Van den Bergh [20] defined the notion of a double Lie algebroid and showed that a double quasi-Poisson algebra gives rise to a double Lie algebroid.We give new examples of double Lie algebroids and develop a differential calculus in that context. We recover the non commutative Karoubi–de Rham complex [7, 9] and the double Poisson–Lichnerowicz cohomology [16] as particular cases of our construction.

Hypercohomologies of truncated twisted holomorphic de Rham complexes

We investigate the hypercohomologies of truncated twisted holomorphic de Rham complexes on (not necessarily compact) complex manifolds. In particular, we generalize Leray-Hirsch, K\"{u}nneth and Poincar\'{e}-Serre duality theorems on them. At last, a blowup formula is given, which affirmatively answers a question posed by Chen, Y. and Yang, S. in \cite{CY}.

Poincaré–Friedrichs type constants for operators involving grad, curl, and div: Theory and numerical experiments

We give some theoretical as well as computational results on Laplace and Maxwell constants, i.e., on the smallest constants cn>0 arising in estimates of the form |u|L2(Ω)≤c0|gradu|L2(Ω),|E|L2(Ω)≤c1|curlE|L2(Ω),|H|L2(Ω)≤c2|divH|L2(Ω).Besides the classical de Rham complex we investigate the complex of elasticity and the complex related to the biharmonic equation and general relativity as well using the general functional analytical concept of Hilbert complexes. We consider mixed boundary conditions and bounded Lipschitz domains of arbitrary topology. Our numerical aspects are presented by examples for the de Rham complex in 2D and 3D which not only confirm our theoretical findings but also indicate some interesting conjectures.

On $E_1$-degeneration for the special fiber of a semistable family

We study the $E_1$-degeneration of the logarithmic Hodge to de Rham spectral sequence of the special fiber of a semistable family over a discrete valuation ring. On the one hand, we prove that the $E_1$-degeneration property is invariant under admissible blow-ups. Assuming functorial resolution of singularities over $\mathbb{Z}$, this implies that the $E_1$-degeneration property depends only on the generic fiber. On the other hand, we show by explicit examples that the decomposability of the logarithmic de Rham complex is not invariant under admissible blow-ups, which answer negatively an open problem of L. Illusie (Problem 7.14 \cite{Illusie2002}). We also give an algebraic proof of an $E_1$-degeneration result in characteristic zero due to Steenbrink and Kawamata-Namikawa.

Simple Curl-Curl-Conforming Finite Elements in Two Dimensions

We construct smooth finite element de Rham complexes in two space dimensions. This leads to three families of curl-curl-conforming finite elements, two of which contain two existing families. The simplest triangular and rectangular finite elements have only six and eight degrees of freedom, respectively. Numerical experiments for each family demonstrate the convergence and efficiency of the elements for solving the quad-curl problem.

Deformations of overconvergent isocrystals on the projective line

Let k be a perfect field of positive characteristic and Z an effective Cartier divisor in the projective line over k with complement U. In this note, we establish some results about the formal deformation theory of overconvergent isocrystals on U with fixed “local monodromy” along Z. En route, we show that a Hochschild cochain complex governs deformations of a module over an arbitrary associative algebra. We also relate this Hochschild cochain complex to a de Rham complex in order to understand the deformation theory of a differential module over a differential ring.

Superstring field theory, superforms and supergeometry

Inspired by superstring field theory, we study differential, integral, and inverse forms and their mutual relations on a supermanifold from a sheaf-theoretical point of view. In particular, the formal distributional properties of integral forms are recovered in this scenario in a geometrical way. Further, we show how inverse forms “extend” the ordinary de Rham complex on a supermanifold, thus providing a mathematical foundation of the Large Hilbert Space used in superstrings. Last, we briefly discuss how the Hodge diamond of a supermanifold looks like, and we explicitly compute it for super Riemann surfaces.

Multipatch approximation of the de Rham sequence and its traces in isogeometric analysis

We define a conforming B-spline discretisation of the de Rham complex on multipatch geometries. We introduce and analyse the properties of interpolation operators onto these spaces which commute w.r.t. the surface differential operators. Using these results as a basis, we derive new convergence results of optimal order w.r.t. the respective energy spaces and provide approximation properties of the spline discretisations of trace spaces for application in the theory of isogeometric boundary element methods. Our analysis allows for a straight forward generalisation to finite element methods.

Quantum de Rham complex on 𝔸h1|2

Introducing [Formula: see text]- and [Formula: see text]-deformations of [Formula: see text]-graded ([Formula: see text])- and ([Formula: see text])-spaces, denoted by [Formula: see text] and [Formula: see text], a two-parameter differential calculus, the quantum de Rham complex, on [Formula: see text] are explicitly constructed. It is shown that in contrast to the standard [Formula: see text]-deformation of [Formula: see text], the above complex is unique for [Formula: see text]. The quantum de Rham complex of [Formula: see text] is discussed via a contraction using standard one.

Twisted de Rham Complex on Line and Singular Vectors in sl<sub>2</sub> Verma Modules

We consider two complexes. The first complex is the twisted de Rham complex of scalar meromorphic differential forms on projective line, holomorphic on the complement to a finite set of points. The second complex is the chain complex of the Lie algebra of $\mathfrak{sl}_2$-valued algebraic functions on the same complement, with coefficients in a tensor product of contragradient Verma modules over the affine Lie algebra $\hat{{\mathfrak{sl}_2}}$. In [Schechtman V., Varchenko A., Mosc. Math. J. 17 (2017), 787-802] a construction of a monomorphism of the first complex to the second was suggested and it was indicated that under this monomorphism the existence of singular vectors in the Verma modules (the Malikov-Feigin-Fuchs singular vectors) is reflected in the relations between the cohomology classes of the de Rham complex. In this paper we prove these results.