Twisted De Rham Cohomology Research Articles

Exponentially twisted de Rham cohomology and rigid cohomology

A comparison theorem between exponentially twisted de Rham cohomology and rigid cohomology with coefficients in a Dwork crystal is proved.

On the blow-up formula of twisted de Rham cohomology

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 with a local system of complex vector spaces.

Topology of Pollicott-Ruelle resonant states

We prove that the twisted De Rham cohomology of a flat vector bundleover some smooth manifold is isomorphic to the cohomology of invariant Pollicott–Ruelleresonant states associated with Anosov and Morse–Smale flows. As a consequence, weobtain generalized Morse inequalities for such flows. In the case of Morse–Smale flows,we relate the resonances lying on the imaginary axis with the twisted Fuller measuresused by Fried in his work on Reidemeister torsion. In particular, when V is a nonsingularMorse-Smale flow, we show that the Reidemeister torsion can be recovered from the onlyknowledge of dynamical resonances on the imaginary axis by expressing the torsion as azeta regularized infinite product of these resonances.

Matrix factorizations and higher residue pairings

The periodic cyclic homology of any proper dg category comes equipped with a canonical pairing. We show that in the case of the dg category of matrix factorizations of an isolated singularity the canonical pairing can be identified with Kyoji Saito's higher residue pairing on the twisted de Rham cohomology of the singularity.

Double filtration of twisted logarithmic complex and Gauss–Manin connection

The twisted de Rham complex associated with hypergeometric integral of a power product of polynomials is quasi-isomorphic to the corresponding logarithmic complex. We show in this article that the latter has a double filtration with respect to degrees of polynomials and exterior algebras. By a combinatorial method we prove the quasi-isomorphism between the twisted de Rham cohomology and a specially filtered subcomplex in case of polynomials of the same degree. This fact gives a more detailed structure of a basis for the twisted de Rham cohomology.

On a spectral sequence for twisted cohomologies

Let (Ω*(M), d) be the de Rham cochain complex for a smooth compact closed manifolds M of dimension n. For an odd-degree closed form H, there is a twisted de Rham cochain complex (Ω*(M), d + H ∧) and its associated twisted de Rham cohomology H*(M,H). The authors show that there exists a spectral sequence {E , d r } derived from the filtration $F_p (\Omega ^ * (M)) = \mathop \oplus \limits_{i \geqslant p} \Omega ^i (M)$ of Ω*M, which converges to the twisted de Rham cohomology H*(M,H). It is also shown that the differentials in the spectral sequence can be given in terms of cup products and specific elements of Massey products as well, which generalizes a result of Atiyah and Segal. Some results about the indeterminacy of differentials are also given in this paper.

Non-commutative Hodge structures: Towards matching categorical and geometric examples

The subject of the present work is the de Rham part of non-commutative Hodge structures on the periodic cyclic homology of differential graded algebras and categories. We discuss explicit formulas for the corresponding connection on the periodic cyclic homology viewed as a bundle over the punctured formal disk. Our main result says that for the category of matrix factorizations of a polynomial the formulas reproduce, up to a certain shift, a well-known connection on the associated twisted de Rham cohomology which plays a central role in the geometric approach to the Hodge theory of isolated singularities.

Irregular Hodge filtration on twisted de Rham cohomology

We give a definition and study the basic properties of the irregular Hodge filtration on the exponentially twisted de Rham cohomology of a smooth quasi-projective complex variety.

A vanishing theorem in twisted de Rham cohomology

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

Twisted Poincaré lemma and twisted Čech–de Rham isomorphism in case dimension =1

For a compact Riemann surface, (n+1)-tuple x:=(x0,…,xn) of points on it, and a holomorphic vector bundle with an integrable connection on the open Riemann surface Xx deprived of (n+1) points x0,…,xn, let L be the local system of horizontal sections of the connection. In this article, we give a suitable covering of Xx to calculate the Čech cohomology and describe the isomorphism between the cohomology and the twisted de Rham cohomology, which is the cohomology of the complex with the differentials given by the connection. This isomorphism is given by the integrations over Aomoto’s regularized paths, the so-called Euler type integrals. For the family {Xx}x parametrized by x, we give a variant of the isomorphism.

Twisted de Rham cohomology, homological definition of the integral and “Physics over a ring”

We use the notion of the twisted de Rham cohomology to give meaning to an integral of the form ∫ g ( x ) e f ( x ) d x over an arbitrary ring. We discuss also a definition of a family of integrals and some properties of the above homological definition of the integral. We show how to use the twisted de Rham cohomology to define the Frobenius map on the p-adic cohomology. Finally, we consider two-dimensional topological quantum field theories with general coefficients.

Entire cyclic homology of stable continuous trace algebras

A central result in this paper is the computation of the entire cyclic homology of canonical smooth subalgebras of stable continuous trace C*-algebras having smooth manifolds M as their spectrum. More precisely, the entire cyclic homology is shown to be canonically isomorphic to the continuous periodic cyclic homology for these algebras. By an earlier result of the authors, one concludes that the entire cyclic homology of the algebra is canonically isomorphic to the twisted de Rham cohomology of M.

The twisted de Rham cohomology for basic constructionsof hyperplane arrangements and its applications

In this paper we study the twisted de Rham cohomology for an arrangement of hyperplanes. We prove formulas for basic constructions of arrangements: the twisted Kiinneth formula for products or decompositions, the long exact sequence for triple constructions, and the relation for coning and deconing constructions. Moreover, we obtain the vanishing, dimension, and basis of the twisted de Rham cohomology for an arrangement in general position and a generic arrangement with arbitrary weight.

On a generalised Connes–Hochschild–Kostant–Rosenberg theorem

The central result of this paper is an explicit computation of the Hochschild and cyclic homologies of a natural smooth subalgebra of stable continuous trace algebras having smooth manifolds X as their spectrum. More precisely, the Hochschild homology is identified with the space of differential forms on X, and the periodic cyclic homology with the twisted de Rham cohomology of X, thereby generalising some fundamental results of Connes and Hochschild–Kostant–Rosenberg. The Connes–Chern character is also identified here with the twisted Chern character.

The serre spectral sequence of a noncommutative fibration for de Rham cohomology

For differential calculi on noncommutative algebras, we construct a twisted de Rham cohomology using flat connections on modules. This has propertie s similar, in some respects, to sheaf cohomology on topological spaces. We also discuss generalised mapping properties of these theories, and relations of these properties to corings. Using t his, we give conditions for the Serre spectral sequence to hold for a noncommutative fibration. Th is might be better read as giving the definition of a fibration in noncommutative differential geo metry. We also study the multiplicative structure of such spectral sequences. Finally we show that some noncommutative homogeneous spaces satisfy the conditions to be such a fibration, and in th e process clarify the differential structure on these homogeneous spaces. We also give two explicit examples of differential fibrations: these are built on the quantum Hopf fibration with two differe nt differential structures.

Cohomological Intersection Numbers for the Generalized Airy Functions at Veronese Points

We establish the exterior power structure for the polynomial twisted de Rham cohomology group associated with the generalized Airy functions at a point of extended Veronese variety. Using this structure, we obtain a natural basis of the twisted de Rham cohomology group from that of one dimensional case, which is considered as an analogue of the flat basis of the Jacobi ring of A-type simple singularity. We compute explicitly the cohomological intersection numbers for these basis and extend the result of Iwasaki and Matsumoto.

Generalized Airy Functions and the Cohomological Intersection Numbers

We compute explicitly the cohomological intersection numbers for the basis and extend the result of Iwasaki and Matsumoto. To this end, we establish the exterior power structure for the polynomial twisted de Rham cohomology group associated with the generalized Airy functions at a point of extended Veronese variety. Using this structure, we introduce a natural basis of the twisted de Rham cohomology group coming from that of the one-dimensional case, which is considered as an analogue of a flat basis of the Jacobi ring of A-type simple singularity.

Isolated Singularity, Witten's Laplacian, and Duality for Twisted de Rham Cohomology

In his famous paper (Witten E. Super-symmetry and Morse Theory. J Diff Geom 1982; 17:661–692), Witten used a twisted Laplacian, twisted by a Morse function, to develop his Morse theory as a super-symmetric quantum mechanics. Afterwards, in their studies on puits multiples en mecanique semi-classique, Helffer and Sjöstrand (Helffer B, Sjöstrand J. Puits Multiples en Mecanique Semi-classique IV, Etude du Complexe de Witten. Comm. Partial Differential Equations 1985; 10:245–340) made a detailed investigation of Witten's complexes. In this paper, we use a twisted Laplacian, twisted by a versal deformation of an isolated singularity (invariant under a finite unitary reflection group), to construct a duality between a pair of polynomial twisted de Rham cohomology groups associated with the isolated singularity. The construction is based on a twisted version of Hodge-Kodaira decomposition derived through a pseudo-differential calculus of Witten's twisted Laplacian. Discussion is also made on the real structure and super-symmetry, compatible with the duality constructed, of the twisted de Rham cohomology.

Intersection matrix of a generalized Airy function in terms of skew-Schur polynomials

A duality is introduced between a pair of polynomial twisted de Rham cohomology groups associated with a generalized Airy function in several variables. Natural bases of the twisted de Rham groups are constructed in terms of Schur polynomials. Then the intersection matrix relative to these bases is calculated explicitly in terms of skew-Schur polynomials.

99/02299 The development of video-based sensors for off-and onstream analysis: Cruz, E. B. and Adel, G. T. Proc. Annu. ISA Anal. Div. Symp., 1998, 31, 259–268

The periodic cyclic homology of any proper dg category comes equipped with a canonical pairing. We show that in the case of the dg category of matrix factorizations of an isolated singularity the canonical pairing can be identified with Kyoji Saito's higher residue pairing on the twisted de Rham cohomology of the singularity.