In mathematics, a Hodge structure, named after W. V. D. Hodge, is an algebraic structure at the level of linear algebra, similar to the one that Hodge theory gives to the cohomology groups of a smooth and compact Kähler manifold. Hodge structures have been generalized for all complex varieties (even if they are singular and non-complete) in the form of mixed Hodge structures, defined by Pierre Deligne (1970). A variation of Hodge structure is a family of Hodge structures parameterized by a manifold, first studied by Phillip Griffiths (1968). All these concepts were further generalized to mixed Hodge modules over complex varieties by Morihiko Saito (1989).
A pure Hodge structure of integer weight n consists of an abelian group
H\Z
H
Hp,q
p+q=n
Hp,q
Hq,p
H:=H\Z ⊗ \Z\Complex=oplus\nolimitsp+q=nHp,q,
\overline{Hp,q
An equivalent definition is obtained by replacing the direct sum decomposition of
H
H
FpH(p\in\Z),
\forallp,q : p+q=n+1, FpH\cap\overline{FqH}=0 and FpH ⊕ \overline{FqH}=H.
The relation between these two descriptions is given as follows:
Hp,q=FpH\cap\overline{FqH},
FpH=oplus\nolimitsi\geqHi,n-i.
For example, if
X
H\Z=Hn(X,\Z)
n
H=Hn(X,\Complex)
n
H
n
Hn
FpH
For applications in algebraic geometry, namely, classification of complex projective varieties by their periods, the set of all Hodge structures of weight
n
H\Z
(H\Z,Hp,q)
Q
H\Z
H
\begin{align} Q(\varphi,\psi)&=(-1)nQ(\psi,\varphi)&&for\varphi\inHp,q,\psi\inHp',q';\\ Q(\varphi,\psi)&=0&&for\varphi\inHp,q,\psi\inHp',q',p\neq';\\ ip-qQ\left(\varphi,\bar{\varphi}\right)&>0&&for\varphi\inHp,q, \varphi\ne0. \end{align}
In terms of the Hodge filtration, these conditions imply that
\begin{align} Q\left(Fp,Fn-p+1\right)&=0,\\ Q\left(C\varphi,\bar{\varphi}\right)&>0&&for\varphi\ne0, \end{align}
where
C
H
C=ip-q
Hp,q
Yet another definition of a Hodge structure is based on the equivalence between the
\Z
\Complex*
H
Hp,q
z\in\Complex*
zp{\bar{z}}q.
In the theory of motives, it becomes important to allow more general coefficients for the cohomology. The definition of a Hodge structure is modified by fixing a Noetherian subring A of the field
\R
A ⊗ \Z\R
\Z
See main article: Mixed Hodge structure.
It was noticed by Jean-Pierre Serre in the 1960s based on the Weil conjectures that even singular (possibly reducible) and non-complete algebraic varieties should admit 'virtual Betti numbers'. More precisely, one should be able to assign to any algebraic variety X a polynomial PX(t), called its virtual Poincaré polynomial, with the properties
The existence of such polynomials would follow from the existence of an analogue of Hodge structure in the cohomologies of a general (singular and non-complete) algebraic variety. The novel feature is that the nth cohomology of a general variety looks as if it contained pieces of different weights. This led Alexander Grothendieck to his conjectural theory of motives and motivated a search for an extension of Hodge theory, which culminated in the work of Pierre Deligne. He introduced the notion of a mixed Hodge structure, developed techniques for working with them, gave their construction (based on Heisuke Hironaka's resolution of singularities) and related them to the weights on l-adic cohomology, proving the last part of the Weil conjectures.
To motivate the definition, consider the case of a reducible complex algebraic curve X consisting of two nonsingular components,
X1
X2
Q1
Q2
P1,...,Pn
\alphai
Pi
\betaj
Xk\subsetX
k=1,2
\alpha1,...,\alphan
\gamma
Q1
Q2
X1
X2
H1(X)
0\subsetW0\subsetW1\subset
1(X), | |
W | |
2=H |
whose successive quotients Wn/Wn−1 originate from the cohomology of smooth complete varieties, hence admit (pure) Hodge structures, albeit of different weights. Further examples can be found in "A Naive Guide to Mixed Hodge Theory".[3]
A mixed Hodge structure on an abelian group
H\Z
H\Z
H\Q=H\Z ⊗ \Z\Q
H\Q
W | |
\operatorname{gr} | |
n |
H=Wn ⊗ \Complex/Wn-1 ⊗ \Complex
is defined by
Fp
W | |
\operatorname{gr} | |
n |
H=\left(Fp\capWn ⊗ \Complex+Wn-1 ⊗ \Complex\right)/Wn-1 ⊗ \Complex.
One can define a notion of a morphism of mixed Hodge structures, which has to be compatible with the filtrations F and W and prove the following:
Theorem. Mixed Hodge structures form an abelian category. The kernels and cokernels in this category coincide with the usual kernels and cokernels in the category of vector spaces, with the induced filtrations.
The total cohomology of a compact Kähler manifold has a mixed Hodge structure, where the nth space of the weight filtration Wn is the direct sum of the cohomology groups (with rational coefficients) of degree less than or equal to n. Therefore, one can think of classical Hodge theory in the compact, complex case as providing a double grading on the complex cohomology group, which defines an increasing filtration Fp and a decreasing filtration Wn that are compatible in certain way. In general, the total cohomology space still has these two filtrations, but they no longer come from a direct sum decomposition. In relation with the third definition of the pure Hodge structure, one can say that a mixed Hodge structure cannot be described using the action of the group
\Complex*.
Moreover, the category of (mixed) Hodge structures admits a good notion of tensor product, corresponding to the product of varieties, as well as related concepts of inner Hom and dual object, making it into a Tannakian category. By Tannaka–Krein philosophy, this category is equivalent to the category of finite-dimensional representations of a certain group, which Deligne, Milne and et el. has explicitly described, see [4] and . The description of this group was recast in more geometrical terms by . The corresponding (much more involved) analysis for rational pure polarizable Hodge structures was done by .
Deligne has proved that the nth cohomology group of an arbitrary algebraic variety has a canonical mixed Hodge structure. This structure is functorial, and compatible with the products of varieties (Künneth isomorphism) and the product in cohomology. For a complete nonsingular variety X this structure is pure of weight n, and the Hodge filtration can be defined through the hypercohomology of the truncated de Rham complex.
The proof roughly consists of two parts, taking care of noncompactness and singularities. Both parts use the resolution of singularities (due to Hironaka) in an essential way. In the singular case, varieties are replaced by simplicial schemes, leading to more complicated homological algebra, and a technical notion of a Hodge structure on complexes (as opposed to cohomology) is used.
Using the theory of motives, it is possible to refine the weight filtration on the cohomology with rational coefficients to one with integral coefficients.[5]
\Z(1)
\Z
2\pii\Z
\Complex
\Z(1) ⊗ \Complex=H-1,-1.
\Z(n);
X
X\subsetPn+1
d
f\in\Complex[x0,\ldots,xn+1]
X
X
g=
4 | |
x | |
0 |
+ … +
4 | |
x | |
3 |
d=4
n=2
R(g)4
H1,1(X)
[L]
Hk,k(X)
1
d
xd+yd+zd
g
The machinery based on the notions of Hodge structure and mixed Hodge structure forms a part of still largely conjectural theory of motives envisaged by Alexander Grothendieck. Arithmetic information for nonsingular algebraic variety X, encoded by eigenvalue of Frobenius elements acting on its l-adic cohomology, has something in common with the Hodge structure arising from X considered as a complex algebraic variety. Sergei Gelfand and Yuri Manin remarked around 1988 in their Methods of homological algebra, that unlike Galois symmetries acting on other cohomology groups, the origin of "Hodge symmetries" is very mysterious, although formally, they are expressed through the action of the fairly uncomplicated group
RC/R{C}*
A variation of Hodge structure is a family of Hodge structuresparameterized by a complex manifold X. More precisely a variation of Hodge structure of weight n on a complex manifold X consists of a locally constant sheaf S of finitely generated abelian groups on X, together with a decreasing Hodge filtration F on S ⊗ OX, subject to the following two conditions:
Fn
Fn-1 ⊗
1 | |
\Omega | |
X. |
Here the natural (flat) connection on S ⊗ OX induced by the flat connection on S and the flat connection d on OX, and OX is the sheaf of holomorphic functions on X, and
1 | |
\Omega | |
X |
A variation of mixed Hodge structure can be defined in a similar way, by adding a grading or filtration W to S. Typical examples can be found from algebraic morphisms
f:\Complexn\to\Complex
\begin{cases} f:\Complex2\to\Complex\\ f(x,y)=y6-x6 \end{cases}
Xt=f-1(\{t\})=\left\{(x,y)\in\Complex2:y6-x6=t\right\}
t ≠ 0
t=0.
\R
i | |
f | |
* |
\left(
\underline{\Q} | |
\Complex2 |
\right)
See main article: Mixed Hodge module. Hodge modules are a generalization of variation of Hodge structures on a complex manifold. They can be thought of informally as something like sheaves of Hodge structures on a manifold; the precise definition is rather technical and complicated. There are generalizations to mixed Hodge modules, and to manifolds with singularities.
For each smooth complex variety, there is an abelian category of mixed Hodge modules associated with it. These behave formally like the categories of sheaves over the manifolds; for example, morphisms f between manifolds induce functors f∗, f*, f!, f! between (derived categories of) mixed Hodge modules similar to the ones for sheaves.
p+q | |
E | |
1=H |
W | |
(\operatorname{gr} | |
nH) ⇒ |
Hp+q,
using notations in
. The important fact is that this is degenerate at the term E1, which means the Hodge–de Rham spectral sequence, and then the Hodge decomposition, depends only on the complex structure not Kähler metric on M.
\Complex
\R;
\R,
A ⊗ \Complex.
S(\R)
\Complex*