2013/12/04 by Mark Andrea de Cataldo, de Cataldo, Mark Andrea
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #math.AG
paper · pdf · doi:10.48550/arxiv.1312.1169
in J. Singul. 7 (2013), 134-156
arxiv created 2013/12/04 · openalex publication_date 2013/12/04 · arxiv updated 2013/12/05 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28
According to the decomposition and relative hard Lefschetz theorems, given a projective map of complex quasi projective algebraic varieties and a relatively ample line bundle, the rational intersection cohomology groups of the domain of the map split into various direct summands. While the summands are canonical, the splitting is certainly not, as the choice of the line bundle yields at least three different splittings by means of three mechanisms in a triangulated category introduced by Deligne. It is known that these three choices yield splittings of mixed Hodge structures. In this paper, we use the relative hard Lefschetz theorem and elementary linear algebra to construct five distinct splittings, two of which seem to be new, and to prove that they are splittings of mixed Hodge structures.