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 may have subspaces of different weights, i.e. as a direct sum of Hodge structures

where each of the Hodge structures have weight . One of the early hints that such structures should exist comes from the long exact sequence associated to a pair of smooth projective varieties . This sequence suggests that the cohomology groups (for ) should have differing weights coming from both and .

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 such that

is a -module of finite type is an increasing -filtration on , is a decreasing -filtration on ,

where the induced filtration of on the graded piecesare pure Hodge structures of weight .

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, where , 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 groupssuch thatand the induced map of -vector spaces has the property

Further definitions and properties

Hodge numbers The Hodge numbers of a MHS are defined as the dimensionssince is a weight Hodge structure, andis the -component of a weight Hodge structure.

Homological properties There is an Abelian category of mixed Hodge structures which has vanishing -groups whenever the cohomological degree is greater than : that is, given mixed hodge structures the groupsfor pg 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. Given a complex of sheaves of abelian groups and filtrations of the complex, meaningThere is an induced mixed Hodge structure on the hyperhomology groupsfrom the bi-filtered complex . Such a bi-filtered complex is called a mixed Hodge complex.

Logarithmic complex Given a smooth variety where is a normal crossing divisor (meaning all intersections of components are complete intersections), there are filtrations on the logarithmic de Rham complex given byIt turns out these filtrations define a natural mixed Hodge structure on the cohomology group from the mixed Hodge complex defined on the logarithmic complex .

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 is defined as a smooth variety and an embedding such that is a normal crossing divisor. That is, given compactifications with boundary divisors there is an isomorphism of mixed Hodge structureshowing the mixed Hodge structure is invariant under smooth compactification.

Example For example, on a genus plane curve logarithmic cohomology of with the normal crossing divisor with can be easily computedNote we are using Bézout's theorem since this can be given as the complement of the intersection with a hyperplane. since the terms of the complex equal toare both acyclic. Then, the Hypercohomology is justthe 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 byThen has a weight mixed Hodge structure and has a weight mixed Hodge structure.

Examples

Complement of a smooth projective curve by two points

We give here the simplest example which isolates the phenomenon at hand. We will explain the MHS on (a twist of) where is the complement of a smooth projective curve by two points.

Let be a smooth proper (hence projective) curve over and let be two points in . We get a so called recollement situation where .

One has an exact triangle of constructible sheaves on

The purity isomorphism (coming from Verdier duality) identifies , whence the relevant terms of the exact triangle upon taking cohomology give us the exact sequence of MHS

Rationalizing, and noting that , we get a sequence of rational MHS

We may now read off the weight filtration on as follows: in filtered degree we have , in filtered degree we have and in filtered degree we have .

It remains to check that the graded pieces have the correct weights. But it is immediate from the exact sequence of rational MHS above that is pure of weight for . Note that if the genus of the curve is greater than , then indeed the graded for is non-zero.

Complement of a smooth projective variety by a closed subvariety Given a smooth projective variety of dimension and a closed subvariety there is a long exact sequence in cohomologypg7-8coming from the distinguished triangleof constructible sheaves. There is another long exact sequencefrom the distinguished trianglewhenever is smooth. Note the homology groups are called Borel–Moore homology, which are dual to cohomology for general spaces and the means tensoring with the Tate structure add weight to the weight filtration. The smoothness hypothesis is required because Verdier duality implies , and whenever is smooth. Also, the dualizing complex for has weight , hence . Also, the maps from Borel-Moore homology must be twisted by up to weight is order for it to have a map to . Also, there is the perfect duality pairinggiving an isomorphism of the two groups.

Algebraic torus A one dimensional algebraic torus is isomorphic to the variety , hence its cohomology groups are isomorphic toThe long exact exact sequence then readsSince and this gives the exact sequencesince there is a twisting of weights for well-defined maps of mixed Hodge structures, there is the isomorphism

Quartic K3 surface minus a genus 3 curve Given a quartic K3 surface , and a genus 3 curve defined by the vanishing locus of a generic section of , hence it is isomorphic to a degree plane curve, which has genus 3. Then, the Gysin sequence gives the long exact sequenceBut, it is a result that the maps take a Hodge class of type to a Hodge class of type . 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 maps hence contains the weight one pieces . Because has dimension , but the Leftschetz class is killed off by the mapsending the class in to the class in . Then the primitive cohomology group is the weight 2 piece of . Therefore,The induced filtrations on these graded pieces are the Hodge filtrations coming from each cohomology group.

Mixed Hodge structure

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