Algebraic 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
![{\displaystyle H^{k}(X)=\bigoplus _{i}(H_{i},F_{i}^{\bullet })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/667dc2b7d871207df2999ed2386fc9a9e9fe650c)
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[1] (MHS) is a triple
such that
is a
-module of finite type
is an increasing
-filtration on
, ![{\displaystyle \cdots \subset W_{0}\subset W_{1}\subset W_{2}\subset \cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bcea87f3d10438f24ab4212cde21b476b9153379)
is a decreasing
-filtration on
, ![{\displaystyle H_{\mathbb {C} }=F^{0}\supset F^{1}\supset F^{2}\supset \cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/846ca7d75315eb432cdcdcfebebfe47ef06e53f9)
where the induced filtration of
on the graded pieces
![{\displaystyle {\text{Gr}}^{W_{\bullet }}H_{\mathbb {Q} }={\frac {W_{k}H_{\mathbb {Q} }}{W_{k-1}H_{\mathbb {Q} }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a99e0adf99b12ba373ca8dc7ad3e924fb5c485ab)
are pure Hodge structures of weight
.
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 groups
![{\displaystyle f:(H_{\mathbb {Z} },W_{\bullet },F^{\bullet })\to (H_{\mathbb {Z} }',W_{\bullet }',F'^{\bullet })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1dda94f70180102ef07bcc82ab0a3d8adf28627)
such that
![{\displaystyle f(W_{l})\subset W'_{l}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/450098ddf8a98064979499f55185a7208ce5f57a)
and the induced map of
-vector spaces has the property
![{\displaystyle f_{\mathbb {C} }(F^{p})\subset F'^{p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/130b5631b5d3356fdd7a52b1cb877f3dca68f904)
Further definitions and properties
Hodge numbers
The Hodge numbers of a MHS are defined as the dimensions
![{\displaystyle h^{p,q}(H_{\mathbb {Z} })=\dim _{\mathbb {C} }{\text{Gr}}_{F^{\bullet }}^{p}{\text{Gr}}_{p+q}^{W_{\bullet }}H_{\mathbb {C} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f5f202109b00c37a5c71358d11d3ad1bca959ae)
since
is a weight
Hodge structure, and
![{\displaystyle {\text{Gr}}_{p}^{F^{\bullet }}={\frac {F^{p}}{F^{p+1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27aec1135ae06b27c5ecf4950a312563443032ba)
is the
-component of a weight
Hodge structure.
Homological properties
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
![{\displaystyle \operatorname {Ext} _{MHS}^{p}((H_{\mathbb {Z} },W_{\bullet },F^{\bullet }),(H_{\mathbb {Z} }',W_{\bullet }',F'^{\bullet }))=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5400d46dcf169a37c745b0c5c68a605f2e208b7)
for
[2]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
[1] of the complex, meaning
![{\displaystyle {\begin{aligned}d(W_{i}A^{\bullet })&\subset W_{i}A^{\bullet }\\d(F^{i}A^{\bullet })&\subset F^{i}A^{\bullet }\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76d2981dd2f37bec74de35e5706f431625648def)
There is an induced mixed Hodge structure on the hyperhomology groups
![{\displaystyle (\mathbb {H} ^{k}(X,A^{\bullet }),W_{\bullet },F^{\bullet })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be4f383b09bdc3e293cdc580a305b8fe7ec8fb3a)
from the bi-filtered complex
. Such a bi-filtered complex is called a mixed Hodge complex[1]: 23
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 by
![{\displaystyle {\begin{aligned}W_{m}\Omega _{X}^{i}(\log D)&={\begin{cases}\Omega _{X}^{i}(\log D)&{\text{ if }}i\leq m\\\Omega _{X}^{i-m}\wedge \Omega _{X}^{m}(\log D)&{\text{ if }}0\leq m\leq i\\0&{\text{ if }}m<0\end{cases}}\\[6pt]F^{p}\Omega _{X}^{i}(\log D)&={\begin{cases}\Omega _{X}^{i}(\log D)&{\text{ if }}p\leq i\\0&{\text{ otherwise}}\end{cases}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29c06302ad297443b9f5441450b7cf5dffaf4d6e)
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
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
![{\displaystyle (\mathbb {H} ^{k}(X,\Omega _{X}^{\bullet }(\log D)),W_{\bullet },F^{\bullet })\cong (\mathbb {H} ^{k}(X',\Omega _{X'}^{\bullet }(\log D')),W_{\bullet },F^{\bullet })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/272afdaa9d23d3f9726e4faad7e96d0774b995a6)
showing the mixed Hodge structure is invariant under smooth compactification.[2]
Example
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
![{\displaystyle {\mathcal {O}}_{C}\xrightarrow {d} \Omega _{C}^{1}(\log D)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38678f19619e3ff1f6f27fdd09c2d5a4856fdedd)
are both acyclic. Then, the Hypercohomology is just
![{\displaystyle \Gamma ({\mathcal {O}}_{\mathbb {P} ^{1}})\xrightarrow {d} \Gamma (\Omega _{\mathbb {P} ^{1}}(\log D))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfbc0694b25b1fa798d485fdd3f5c4c78f7df505)
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
![{\displaystyle \mathbb {C} \cdot {\frac {dx}{x-p_{1}}}\oplus \cdots \oplus \mathbb {C} {\frac {dx}{x-p_{k-1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40ec1c9ddcbc53123af86bb8b853e0e352caeb24)
Then
has a weight
mixed Hodge structure and
has a weight
mixed Hodge structure.
Examples
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 cohomology[4]pg7-8
![{\displaystyle \cdots \to H_{c}^{m}(U;\mathbb {Z} )\to H^{m}(X;\mathbb {Z} )\to H^{m}(Y;\mathbb {Z} )\to H_{c}^{m+1}(U;\mathbb {Z} )\to \cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/10c0fe3a1daa6e0d040d035a3b3321cff5504370)
coming from the distinguished triangle
![{\displaystyle \mathbf {R} j_{!}\mathbb {Z} _{U}\to \mathbb {Z} _{X}\to i_{*}\mathbb {Z} _{Y}\xrightarrow {[+1]} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5fe1fe5b354eec9dda1d9021f0abf2a806378c3)
of constructible sheaves. There is another long exact sequence
![{\displaystyle \cdots \to H_{2n-m}^{BM}(Y;\mathbb {Z} )(-n)\to H^{m}(X;\mathbb {Z} )\to H^{m}(U;\mathbb {Z} )\to H_{2n-m-1}^{BM}(Y;\mathbb {Z} )(-n)\to \cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/32cc0d8d658e391f2402ceb02a7fe9ba7a91866d)
from the distinguished triangle
![{\displaystyle i_{*}i^{!}\mathbb {Z} _{X}\to \mathbb {Z} _{X}\to \mathbf {R} j_{*}\mathbb {Z} _{U}\xrightarrow {[+1]} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bbda8e74aa93d4c01b80258deb3cb8974f5ed46)
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
![{\displaystyle H_{2n-m}^{BM}(Y)\times H^{m}(Y)\to \mathbb {Z} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3117aba592c012bb0438fa128a6c923cea1a202d)
giving an isomorphism of the two groups.
Algebraic torus
A one dimensional algebraic torus
is isomorphic to the variety
, hence its cohomology groups are isomorphic to
![{\displaystyle {\begin{aligned}H^{0}(\mathbb {T} )\oplus H^{1}(\mathbb {T} )&\cong \mathbb {Z} \oplus \mathbb {Z} \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c573affe868312a49288baed6a7e8e5d8fd6cc4b)
The long exact exact sequence then reads
![{\displaystyle {\begin{matrix}&H_{2}^{BM}(Y)(-1)\to H^{0}(\mathbb {P} ^{1})\to H^{0}(\mathbb {G} _{m})\to {\text{ }}\\&H_{1}^{BM}(Y)(-1)\to H^{1}(\mathbb {P} ^{1})\to H^{1}(\mathbb {G} _{m})\to {\text{ }}\\&H_{0}^{BM}(Y)(-1)\to H^{2}(\mathbb {P} ^{1})\to H^{2}(\mathbb {G} _{m})\to 0\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ac1f3d095e5c9998c79ef3677cfb88dcc1979a2)
Since
and
this gives the exact sequence
![{\displaystyle 0\to H^{1}(\mathbb {G} _{m})\to H_{0}^{BM}(Y)(-1)\to H^{2}(\mathbb {P} ^{1})\to 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/244dd8a8dec0ee6edf875e8d7841880ab8ecbcaa)
since there is a twisting of weights for well-defined maps of mixed Hodge structures, there is the isomorphism
![{\displaystyle H^{1}(\mathbb {G} _{m})\cong \mathbb {Z} (-1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1bb43d11383522caca67c89649503cca49e012cd)
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 sequence
![{\displaystyle \to H^{k-2}(C)\xrightarrow {\gamma _{k}} H^{k}(X)\xrightarrow {i^{*}} H^{k}(U)\xrightarrow {R} H^{k-1}(C)\to }](https://wikimedia.org/api/rest_v1/media/math/render/svg/122a2ba8ab08905bb514307a35b95ee418e2dcd6)
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
![{\displaystyle \gamma _{3}:H^{0,1}(C)\to H^{1,2}(X)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91bf90fd574adf46cdb2f85e51ae23145561e135)
hence
contains the weight one pieces
. Because
has dimension
, but the Leftschetz class
is killed off by the map
![{\displaystyle \gamma _{2}:H^{0}(C)\to H^{2}(X)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ae880718b64752a1c6449563baeb819851fa492)
sending the
class in
to the
class in
. Then the primitive cohomology group
is the weight 2 piece of
. Therefore,
![{\displaystyle {\begin{aligned}{\text{Gr}}_{2}^{W_{\bullet }}H^{2}(U)&=H_{\text{prim}}^{2}(X)\\{\text{Gr}}_{1}^{W_{\bullet }}H^{2}(U)&=H^{1}(C)\\{\text{Gr}}_{k}^{W_{\bullet }}H^{2}(U)&=0&k\neq 1,2\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72729daf16c051190ba6aca8c18862458d84b69e)
The induced filtrations on these graded pieces are the Hodge filtrations coming from each cohomology group.
See also
References
- ^ 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.
- ^ a b c Peters, C. (Chris) (2008). Mixed hodge structures. Steenbrink, J. H. M. Berlin: Springer. ISBN 978-3-540-77017-6. OCLC 233973725.
- ^ Note we are using Bézout's theorem since this can be given as the complement of the intersection with a hyperplane.
- ^ Corti, Alessandro. "Introduction to mixed Hodge theory: a lecture to the LSGNT" (PDF). Archived (PDF) from the original on 2020-08-12.
- ^ Griffiths; Schmid (1975). Recent developments in Hodge theory: a discussion of techniques and results. Oxford University Press. pp. 31–127.
- 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.
Examples
- A Naive Guide to Mixed Hodge Theory
- Introduction to Limit Mixed Hodge Structures
- Deligne’s Mixed Hodge Structure for Projective Varieties with only Normal Crossing Singularities
In Mirror Symmetry
- Local B-Model and Mixed Hodge Structure