Mixed Hodge Structures

2017 ◽  
Author(s):  
Fouad El Zein ◽  
Lˆe D˜ung Tr ´ang
Keyword(s):  
Spectral Sequence ◽  
Linear Algebra ◽  
Algebraic Variety ◽  
Spectral Sequences ◽  
Hodge Structures ◽  
De Rham Complex ◽  
Rham Complex ◽  
De Rham ◽  
Definition Of ◽  
Mixed Hodge Structures

This chapter discusses mixed Hodge structures (MHS). It first defines the abstract category of Hodge structures and introduces spectral sequences. The decomposition on the cohomology of Kähler manifolds is used to prove the degeneration at rank 1 of the spectral sequence defined by the filtration F on the de Rham complex in the projective nonsingular case. The chapter then introduces an abstract definition of MHS as an object of interest in linear algebra. It then attempts to develop algebraic homology techniques on filtered complexes up to filtered quasi-isomorphisms of complexes. Finally, this chapter provides the construction of the MHS on any algebraic variety.

scholarly journals Hodge theorem for the logarithmic de Rham complex via derived intersections

2020 ◽  
Vol 7 (3) ◽  
Author(s):  
Márton Hablicsek
Keyword(s):  
Spectral Sequence ◽  
Positive Characteristic ◽  
Geometric Interpretation ◽  
De Rham Complex ◽  
Rham Complex ◽  
Johns Hopkins University ◽  
De Rham ◽  
Characteristic Methods ◽  
Invent Math

Abstract 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.

scholarly journals THE VAN EST SPECTRAL SEQUENCE FOR HOPF ALGEBRAS

2004 ◽  
Vol 01 (01n02) ◽  
pp. 33-48 ◽  
Author(s):  
E. J. BEGGS ◽  
TOMASZ BRZEZIŃSKI
Keyword(s):  
Hopf Algebra ◽  
Spectral Sequence ◽  
Lie Algebra ◽  
Hopf Algebras ◽  
Cohomology Group ◽  
Spectral Sequences ◽  
De Rham Cohomology ◽  
Lie Algebra Cohomology ◽  
De Rham ◽  
Definition Of

Various aspects of the de Rham cohomology of Hopf algebras are discussed. In particular, it is shown that the de Rham cohomology of an algebra with the differentiable coaction of a cosemisimple Hopf algebra with trivial 0-th cohomology group, reduces to the de Rham cohomology of (co)invariant forms. Spectral sequences are discussed and the van Est spectral sequence for Hopf algebras is introduced. A definition of Hopf–Lie algebra cohomology is also given.

scholarly journals Counterexamples to Hochschild-Kostant-Rosenberg in characteristic p

2021 ◽  
Vol 9 ◽  
Author(s):  
Benjamin Antieau ◽  
Bhargav Bhatt ◽  
Akhil Mathew
Keyword(s):  
Spectral Sequence ◽  
Spectral Sequences ◽  
Characteristic P ◽  
De Rham

Abstract We give counterexamples to the degeneration of the Hochschild-Kostant-Rosenberg spectral sequence in characteristic p, both in the untwisted and twisted settings. We also prove that the de Rham-HP and crystalline-TP spectral sequences need not degenerate.

scholarly journals Integral calculus on quantum exterior algebras

2014 ◽  
Vol 11 (04) ◽  
pp. 1450026 ◽  
Author(s):  
Serkan Karaçuha ◽  
Christian Lomp
Keyword(s):  
Differential Calculus ◽  
Polynomial Algebra ◽  
Exterior Algebra ◽  
Integral Calculus ◽  
De Rham Complex ◽  
Integral Forms ◽  
Rham Complex ◽  
De Rham ◽  
Quantum Polynomial

Hom-connections and associated integral forms have been introduced and studied by Brzeziński as an adjoint version of the usual notion of a connection in non-commutative geometry. Given a flat hom-connection on a differential calculus (Ω, d) over an algebra A yields the integral complex which for various algebras has been shown to be isomorphic to the non-commutative de Rham complex (in the sense of Brzeziński et al. [Non-commutative integral forms and twisted multi-derivations, J. Noncommut. Geom.4 (2010) 281–312]). In this paper we shed further light on the question when the integral and the de Rham complex are isomorphic for an algebra A with a flat Hom-connection. We specialize our study to the case where an n-dimensional differential calculus can be constructed on a quantum exterior algebra over an A-bimodule. Criteria are given for free bimodules with diagonal or upper-triangular bimodule structure. Our results are illustrated for a differential calculus on a multivariate quantum polynomial algebra and for a differential calculus on Manin's quantum n-space.

scholarly journals Chiral de Rham Complex over Locally Complete Intersections

2015 ◽  
Vol 15 (2) ◽  
pp. 353-372
Author(s):  
Fyodor Malikov ◽  
Vadim Schechtman
Keyword(s):  
Complete Intersections ◽  
De Rham Complex ◽  
Rham Complex ◽  
De Rham

scholarly journals Duality symmetry of the p-form effective action and supertrace of the twisted de Rham complex

2003 ◽  
Vol 648 (3) ◽  
pp. 542-556 ◽  
Author(s):  
P. Gilkey ◽  
K. Kirsten ◽  
D. Vassilevich ◽  
A. Zelnikov
Keyword(s):  
Effective Action ◽  
De Rham Complex ◽  
Duality Symmetry ◽  
Rham Complex ◽  
De Rham
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