A rational mixed Hodge structure is a finite-dimensional rational vector space 
with an increasing weight filtration 
and a decreasing Hodge filtration 
on 
, such that every 
with its induced Hodge filtration is a pure Hodge
structure of weight 
. Pure Hodge structures
are the special case with only one nonzero weight grade. The cohomology
of complex algebraic varieties carries mixed
Hodge structures, including when the varieties are not smooth or not proper. The
character-variety weight filtration occurs in the P=W
conjecture.
Mixed Hodge Structure
See also
Cohomology, Filtration, Hodge Structure, P=W ConjectureExplore with Wolfram|Alpha
References
Deligne, P. "Théorie de Hodge: II." Publ. Math. I.H.E.S. 40, 5-57, 1971. https://doi.org/10.1007/BF02684692.Deligne, P. "Théorie de Hodge: III." Publ. Math. I.H.E.S. 44, 5-77, 1974. https://doi.org/10.1007/BF02685881.Cite this as:
Weisstein, Eric W. "Mixed Hodge Structure." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MixedHodgeStructure.html