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Hodge Theory (MN-49)$
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Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, and Lê Dung Tráng

Print publication date: 2014

Print ISBN-13: 9780691161341

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691161341.001.0001

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Mixed Hodge Structures

Mixed Hodge Structures

Chapter:
(p.123) Chapter Three Mixed Hodge Structures
Source:
Hodge Theory (MN-49)
Author(s):

Fouad El Zein

Lˆe D˜ung Tr ´ang

, Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, Lê Dũng Tráng
Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691161341.003.0003

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.

Keywords:   mixed Hodge structures, spectral sequences, cohomology, Lefschetz decomposition, linear algebra, algebraic homology, Kähler manifolds, mixed Hodge complex

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