Mixed Hodge structure

In algebraic geometry, a mixed Hodge structure is an algebraic structure containing information about the cohomology of general algebraic varieties. It is a generalization of a Hodge structure, which is used to study smooth projective varieties.
In mixed Hodge theory, where the decomposition of a cohomology group formula_1 may have subspaces of different weights, i.e. as a direct sum of Hodge structures
where each of the Hodge structures have weight formula_3. One of the early hints that such structures should exist comes from the long exact sequence of a pair of smooth projective varieties formula_4. The cohomology groups formula_5 (for formula_6) should have differing weights coming from both formula_7 and formula_8.
Motivation.
Originally, Hodge structures were introduced as a tool for keeping track of abstract Hodge decompositions on the cohomology groups of smooth projective algebraic varieties. These structures gave geometers new tools for studying algebraic curves, such as the Torelli theorem, Abelian varieties, and the cohomology of smooth projective varieties. One of the chief results for computing Hodge structures is an explicit decomposition of the cohomology groups of smooth hypersurfaces using the relation between the Jacobian ideal and the Hodge decomposition of a smooth projective hypersurface through Griffith's residue theorem. Porting this language to smooth non-projective varieties and singular varieties requires the concept of mixed Hodge structures.
Definition.
A mixed Hodge structure (MHS) is a triple formula_9 such that
where the induced filtration of formula_16 on the graded piecesformula_21are pure Hodge structures of weight formula_22.
Remark on filtrations.
Note that similar to Hodge structures, mixed Hodge structures use a filtration instead of a direct sum decomposition since the cohomology groups with anti-holomorphic terms, formula_23 where formula_24, don't vary holomorphically. But, the filtrations can vary holomorphically, giving a better defined structure.
Morphisms of mixed Hodge structures.
Morphisms of mixed Hodge structures are defined by maps of abelian groupsformula_25such thatformula_26and the induced map of formula_27-vector spaces has the propertyformula_28
Further definitions and properties.
Hodge numbers.
The Hodge numbers of a MHS are defined as the dimensionsformula_29since formula_30 is a weight formula_31 Hodge structure, andformula_32is the formula_33-component of a weight formula_31 Hodge structure.
Homological properties.
There is an Abelian category of mixed Hodge structures which has vanishing formula_35-groups whenever the cohomological degree is greater than formula_36: that is, given mixed hodge structures formula_37 the groupsformula_38for formula_39pg 83.
Mixed Hodge structures on bi-filtered complexes.
Many mixed Hodge structures can be constructed from a bifiltered complex. This includes complements of smooth varieties defined by the complement of a normal crossing variety, and log cohomology. Given a complex of sheaves of abelian groups formula_40 and filtrations formula_41 of the complex, meaningformula_42There is an induced mixed Hodge structure on the hyperhomology groupsformula_43from the bi-filtered complex formula_44. Such a bi-filtered complex is called a mixed Hodge complex
Logarithmic complex.
Given a smooth variety formula_45 where formula_46 is a normal crossing divisor (meaning all intersections of components are complete intersections), there are filtrations on the log cohomology complex formula_47 given byformula_48It turns out these filtrations define a natural mixed Hodge structure on the cohomology group formula_49 from the mixed Hodge complex defined on the logarithmic complex formula_47.
Smooth compactifications.
The above construction of the logarithmic complex extends to every smooth variety; and the mixed Hodge structure is isomorphic under any such compactificaiton. Note a smooth compactification of a smooth variety formula_51 is defined as a smooth variety formula_52 and an embedding formula_53 such that formula_46 is a normal crossing divisor. That is, given compactifications formula_55 with boundary divisors formula_56there is an isomorphism of mixed Hodge structureformula_57
formula_58showing the mixed Hodge structure is invariant under smooth compactification.
Example.
For example, on a genus formula_59 plane curve formula_60 logarithmic cohomology of formula_60 with the normal crossing divisor formula_62 with formula_63 can be easily computed since the terms of the complex formula_64 equal toformula_65are both acyclic. Then, the Hypercohomology is justformula_66the first vector space are just the constant sections, hence the differential is the zero map. The second is the vector space is isomorphic to the vector space spanned byformula_67Then formula_68 has a weight formula_69 mixed Hodge structure and formula_70 has a weight formula_59 mixed Hodge structure.
Examples.
Complement of a smooth projective variety by a closed subvariety.
Given a smooth projective variety formula_72 of dimension formula_73 and a closed subvariety formula_4 there is a long exact sequence in cohomologypg7-8formula_75coming from the distinguished triangleformula_76of constructible sheaves. There is another long exact sequenceformula_77from the distinguished triangleformula_78whenever formula_52 is smooth. Note the homology groups formula_80 are called Borel–Moore homology, which are dual to cohomology for general spaces and the formula_81 means tensoring with the Tate structure formula_82 add weight formula_83 to the weight filtration. The smoothness hypothesis is required because Verdier duality implies formula_84, and formula_85 whenever formula_86 is smooth. Also, the dualizing complex for formula_86 has weight formula_73, hence formula_89. Also, the maps from Borel-Moore homology must be twisted by up to weight formula_81 is order for it to have a map to formula_91. Also, there is the perfect duality paringformula_92giving an isomophism of the two groups.
Algebraic torus.
A one dimensional algebraic torus formula_93 is isomorphic to the variety formula_94, hence its cohomology groups are isomorphic toformula_95The long exact exact sequence then readsformula_96Since formula_97 and formula_98 this gives the exact sequenceformula_99since there is a twisting of weights for well-defined maps of mixed Hodge structures, there is the isomorphism formula_100
Quartic K3 surface minus a genus 3 curve.
Given a quartic K3 surface formula_52, and a genus 3 curve formula_102 defined by the vanishing locus of a generic section of formula_103, hence it is isomorphic to a degree formula_104 plane curve, which has genus 3. Then, the Gysin sequence gives the long exact sequenceformula_105But, it is a result that the maps formula_106 take a Hodge class of type formula_107 to a Hodge class of type formula_108. The Hodge structures for both the K3 surface and the curve are well-known, and can be computed using the Jacobian ideal. In the case of the curve there are two zero mapsformula_109 formula_110hence formula_111 contains the weight one pieces formula_112. Because formula_113 has dimension formula_114, but the Leftschetz class formula_115 is killed off by the mapformula_116sending the formula_117 class in formula_118 to the formula_119 class in formula_120. Then the primitive cohomology group formula_121 is the weight 2 piece of formula_111. Therefore,formula_123The induced filtrations on these graded pieces are the Hodge filtrations coming from each cohomology group.