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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

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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

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A mixed Hodge structure[1] (MHS) is a triple   such that

  1.   is a  -module of finite type
  2.   is an increasing  -filtration on  ,  
  3.   is a decreasing  -filtration on  ,  

where the induced filtration of   on the graded pieces

 

are pure Hodge structures of weight  .

Remark on filtrations

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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

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Morphisms of mixed Hodge structures are defined by maps of abelian groups

 

such that

 

and the induced map of  -vector spaces has the property

 

Further definitions and properties

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Hodge numbers

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The Hodge numbers of a MHS are defined as the dimensions

 

since   is a weight   Hodge structure, and

 

is the  -component of a weight   Hodge structure.

Homological properties

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There is an Abelian category[2] of mixed Hodge structures which has vanishing  -groups whenever the cohomological degree is greater than  : that is, given mixed hodge structures   the groups

 

for  [2]pg 83.

Mixed Hodge structures on bi-filtered complexes

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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  [1] of the complex, meaning

 

There is an induced mixed Hodge structure on the hyperhomology groups

 

from the bi-filtered complex  . Such a bi-filtered complex is called a mixed Hodge complex[1]: 23 

Logarithmic complex

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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 by

 

It 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

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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 structure

 

showing the mixed Hodge structure is invariant under smooth compactification.[2]

Example

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For example, on a genus   plane curve   logarithmic cohomology of   with the normal crossing divisor   with   can be easily computed[3] since the terms of the complex   equal to

 

are both acyclic. Then, the Hypercohomology is just

 

the 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 by

 

Then   has a weight   mixed Hodge structure and   has a weight   mixed Hodge structure.

Examples

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Complement of a smooth projective variety by a closed subvariety

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Given a smooth projective variety   of dimension   and a closed subvariety   there is a long exact sequence in cohomology[4]pg7-8

 

coming from the distinguished triangle

 

of constructible sheaves. There is another long exact sequence

 

from the distinguished triangle

 

whenever   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 pairing

 

giving an isomorphism of the two groups.

Algebraic torus

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A one dimensional algebraic torus   is isomorphic to the variety  , hence its cohomology groups are isomorphic to

 

The long exact exact sequence then reads

 

Since   and   this gives the exact sequence

 

since 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

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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 sequence

 

But, it is a result that the maps   take a Hodge class of type   to a Hodge class of type  .[5] 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 map

 

sending 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.

See also

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References

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  1. ^ a b c Filippini, Sara Angela; Ruddat, Helge; Thompson, Alan (2015). "An Introduction to Hodge Structures". Calabi-Yau Varieties: Arithmetic, Geometry and Physics. Fields Institute Monographs. Vol. 34. pp. 83–130. arXiv:1412.8499. doi:10.1007/978-1-4939-2830-9_4. ISBN 978-1-4939-2829-3. S2CID 119696589.
  2. ^ a b c Peters, C. (Chris) (2008). Mixed hodge structures. Steenbrink, J. H. M. Berlin: Springer. ISBN 978-3-540-77017-6. OCLC 233973725.
  3. ^ Note we are using Bézout's theorem since this can be given as the complement of the intersection with a hyperplane.
  4. ^ Corti, Alessandro. "Introduction to mixed Hodge theory: a lecture to the LSGNT" (PDF). Archived (PDF) from the original on 2020-08-12.
  5. ^ Griffiths; Schmid (1975). Recent developments in Hodge theory: a discussion of techniques and results. Oxford University Press. pp. 31–127.

Examples

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In Mirror Symmetry

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