Hodge structure
algebraic structure
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).
Hodge structures
Definition of Hodge structures
A pure Hodge structure of integer weight n consists of an abelian group
H
Z
{\displaystyle H_{\mathbb {Z} }}
and a decomposition of its complexification
H
{\displaystyle H}
into a direct sum of complex subspaces
H
p
,
q
{\displaystyle H^{p,q}}
, where
p
+
q
=
n
{\displaystyle p+q=n}
, with the property that the complex conjugate of
H
p
,
q
{\displaystyle H^{p,q}}
is
H
q
,
p
{\displaystyle H^{q,p}}
:
H
:=
H
Z
⊗
Z
C
=
⨁
p
+
q
=
n
H
p
,
q
,
{\displaystyle H:=H_{\mathbb {Z} }\otimes _{\mathbb {Z} }\mathbb {C} =\bigoplus \nolimits _{p+q=n}H^{p,q},}
H
p
,
q
¯
=
H
q
,
p
.
{\displaystyle {\overline {H^{p,q}}}=H^{q,p}.}
An equivalent definition is obtained by replacing the direct sum decomposition of
H
{\displaystyle H}
by the Hodge filtration, a finite decreasing filtration of
H
{\displaystyle H}
by complex subspaces
F
p
H
(
p
∈
Z
)
,
{\displaystyle F^{p}H(p\in \mathbb {Z} ),}
subject to the condition
∀
p
,
q
:
p
+
q
=
n
+
1
,
F
p
H
∩
F
q
H
¯
=
0
and
F
p
H
⊕
F
q
H
¯
=
H
.
{\displaystyle \forall p,q\ :\ p+q=n+1,\qquad F^{p}H\cap {\overline {F^{q}H}}=0\quad {\text{and}}\quad F^{p}H\oplus {\overline {F^{q}H}}=H.}
The relation between these two descriptions is given as follows:
H
p
,
q
=
F
p
H
∩
F
q
H
¯
,
{\displaystyle H^{p,q}=F^{p}H\cap {\overline {F^{q}H}},}
F
p
H
=
⨁
i
≥
p
H
i
,
n
−
i
.
{\displaystyle F^{p}H=\bigoplus \nolimits _{i\geq p}H^{i,n-i}.}
For example, if
X
{\displaystyle X}
is a compact Kähler manifold,
H
Z
=
H
n
(
X
,
Z
)
{\displaystyle H_{\mathbb {Z} }=H^{n}(X,\mathbb {Z} )}
is the
n
{\displaystyle n}
-th cohomology group of X with integer coefficients, then
H
=
H
n
(
X
,
C
)
{\displaystyle H=H^{n}(X,\mathbb {C} )}
is its
n
{\displaystyle n}
-th cohomology group with complex coefficients and Hodge theory provides the decomposition of
H
{\displaystyle H}
into a direct sum as above, so that these data define a pure Hodge structure of weight
n
{\displaystyle n}
. On the other hand, the Hodge–de Rham spectral sequence supplies
H
n
{\displaystyle H^{n}}
with the decreasing filtration by
F
p
H
{\displaystyle F^{p}H}
as in the second definition.
For applications in algebraic geometry, namely, classification of complex projective varieties by their periods, the set of all Hodge structures of weight
n
{\displaystyle n}
on
H
Z
{\displaystyle H_{\mathbb {Z} }}
is too big.
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