arXiv research

The paper studies perturbations of de Rham Hodge operators on manifolds with or without boundaries.

Local index density of perturbed de Rham complex is invariant under certain conditions.

Paper introduces a new multilinear functional for spectral triples and computes its properties.

It is shown that if one uses the notion of infinity nilpotent elements due to Moerdijk and Reyes, instead of the usual definition of nilpotents to define reduced $C^\infty$-schemes, the resulting de Rham spaces are given as quotients by actions of germs of diagonals, instead of the formal neighbourhoods of the diagonal…

The paper analyzes heat trace asymptotics for de Rham and Dolbeault complexes in both real and complex settings.

The main result is a version of Morse inequalities for the minimum and maximum ideal boundary conditions of the de Rham complex on strata of compact Thom-Mather stratifications, endowed with adapted metrics. An adaptation of the analytic method of Witten is used in the proof, as well as certain perturbation of the harm…

Getzler-Jones-Petrack introduced $A_\infty$ structures on the equivariant complex for manifold $M$ with smooth $\mathbb{S}^1$ action, motivated by geometry of loop spaces. Applying Witten's deformation by Morse functions followed by homological perturbation we obtained a new set of $A_\infty$ structures. We extend and …

This work is dedicated to some new exotic homological constructions associated with the different Morse-type inequalities for differential forms and vector fields. It contains also survey of ideas developed by the present author in 1986 for this goal.

The goal of the present paper is to calculate the limit spectrum of the Hodge-de Rham operator under the perturbation of collapsing one part of a manifold obtained by gluing together two manifolds with the same boundary. It appears to take place in the general problem of blowing up conical singularities as introduced i…

We study general conditions under which the computations of the index of a perturbed Dirac operator $D_{s}=D+sZ$ localize to the singular set of the bundle endomorphism $Z$ in the semi-classical limit $s\to \infty$. We show how to use Witten's method to compute the index of $D$ by doing a combinatorial computation inv…

In this paper, we study the perturbative aspects of the half-twisted variant of Witten's topological A-model coupled to a non-dynamical gauge field with Kahler target space X being a G-manifold. Our main objective is to furnish a purely physical interpretation of the equivariant cohomology of the chiral de Rham complex…

Two de Rham complexes in diffeology are compared using a factor map.

Let $M$ be a stratum of a compact stratified space $A$. It is equipped with a general adapted metric $g$, which is slightly more general than the adapted metrics of Nagase and Brasselet-Hector-Saralegi. In particular, $g$ has a general type, which is an extension of the type of an adapted metric. A restriction on this …

Compact moduli space shown for Seiberg-Witten on flat scalar curvature manifold.

Study of de Rham cohomology on non-Hausdorff manifolds.

The paper explores de Rham theory for singular spaces and stacks.

New Lipschitz de Rham theorem for $L_p$-cohomology.

We consider open manifolds which are interiors of a compact manifold with boundary, and Riemannian metrics asymptotic to a conformally cylindrical metric near the boundary. We show that the essential spectrum of the Laplace operator on functions vanishes under the presence of a magnetic field which does not define an i…

The paper proves formulas and theorems for statistical de Rham Hodge operators on manifolds with boundary.

We prove an analogue of the de Rham theorem for the extended L^2-cohomology introduced by M. Farber. This is done by establishing that the de Rham complex over a compact closed manifold with coefficients in a flat Hilbert bundle E of A-modules over a finite von Neumann algebra A is chain-homotopy equivalent (with bound…

Study de Rham homomorphism for Lipschitz cohomologies on metric simplicial complexes.

Study describes global sections of chiral de Rham complex on compact Ricci-flat Kähler manifolds.

Study mapping class group action on de Rham quasimorphisms, finding no fixed points.

De Rham theorem extended to Orlicz cohomology.

Authors prove de Rham cohomology of Poisson and Jacobi manifolds is trivial.

The Burde--de Rham theorem is extended to finitely presented pro-$p$ groups with specific conditions.

This paper extends de Rham theory of smooth manifolds to exploded manifolds. Included are versions of Stokes' theorem, De Rham cohomology, Poincare duality, and integration along the fiber. The resulting cohomology theory is used to define Gromov Witten invariants of exploded manifolds in a separate paper.

Promotes spectral functionals to noncommutative fields and proves a theorem.

The primitive cohomology of Calabi-Yau intersections is described using a twisted de Rham complex.

Paper defines spectral triple and computes functional for nonminimal de Rham-Hodge operator.

Study calculates global sections on complex curves.

Researchers compute de Rham cohomology of geodesic flow foliations on hyperbolic surfaces.

Study Hodge-de Rham numbers for almost complex 4-manifolds, extending properties from complex surfaces.

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 …

A De Rham model for string topology based on the theory of iterated integrals is presented.

We prove a vanishing theorem for the twisted de Rham cohomology of a compact manifold.

Investigates Künneth formula for foliated de Rham cohomology, overcoming non-Hausdorff issues.

We derive a blow-up formula for the de Rham cohomology of a local system of complex vector spaces on a compact complex manifold. As an application, we obtain the blow-up invariance of $E_{1}$-degeneracy of the Hodge-de Rham spectral sequence associated to a local system of complex vector spaces.

In this paper, we study the uniqueness in the de Rham-Wu decomposition for pseudo-Riemannian manifolds.

We extend the study of the de Rham operator with ideal boundary conditions from the case of isolated conic singularities, as analyzed by Cheeger, to the case of arbitrary stratified pseudomanifolds. We introduce a class of ideal boundary operators and the notion of mezzoperversity, which intermediates between the stand…

The chiral de Rham complex of Malikov, Schechtman, and Vaintrob, is a sheaf of differential graded vertex algebras that exists on any smooth manifold $Z$, and contains the ordinary de Rham complex at weight zero. Given a closed 3-form $H$ on $Z$, we construct the twisted chiral de Rham differential $D_H$, which coincid…

We show that the de Rham theorem, interpreted as the isomorphism between distributional de Rham cohomology and simplicial homology in the dual dimension for a simplicial decomposition of a compact oriented manifold, is a straightforward consequence of elementary properties of currents. The explicit construction of this…

Using the language and terminology of relative homological algebra, in particular that of derived functors, we introduce equivariant cohomology over a general Lie-Rinehart algebra and equivariant de Rham cohomology over a locally trivial Lie groupoid in terms of suitably defined monads (also known as triples) and the a…

In this paper, we extend the classical de Rham decomposition theorem to the case of Riemannian manifolds with boundary by using the trick of development of curves.

Study new invariants in complex geometry using Bott-Chern hypercohomology.

In the present paper, we consider the Hodge-de Rham Laplacian that acts on conformal Killing and projective Killing one-forms of a compact Riemannian manifold.

Study shows gap between de Rham and symplectic-Bott-Chern harmonic forms for specific almost-Kähler manifolds.

We exhibit a cocycle in the simplicial de Rham complex which represents the Euler class. As an application, we construct a Lie algebra cocycle on $L\mathfrak{so}(4)$.