Hodge structure

In mathematics , a Hodge structure is an algebraic structure that generalizes the Hodge decomposition of the cohomology of compact Kähler manifolds. Hodge structures have a variety of uses in complex and algebraic geometry.

Definitions

A Hodge decomposition of a real vector space is a decomposition ${\displaystyle V}$

${\displaystyle V\otimes _{\mathbb {R} }\mathbb {C} =\bigoplus _{p,q\in \mathbb {Z} ^{2}}V^{p,q}}$

with for everyone . ${\displaystyle {\overline {V^{q,p}}}=V^{p,q}}$ ${\displaystyle p,q}$

A Hodge structure is a real vector space together with a Hodge decomposition.

A pure Hodge structure by weight is a Hodge structure with ${\displaystyle n}$

${\displaystyle V\otimes _{\mathbb {R} }\mathbb {C} =\bigoplus _{p+q=n}V^{p,q}.}$

In general, one has a weight breakdown for a Hodge structure

${\displaystyle V=\bigoplus _{n}V_{n}}$

With ${\displaystyle V_{n}=\bigoplus _{p+q=n}V^{p,q}.}$

A whole Hodge structure (or rational Hodge structure ) is a finitely generated free module ${\displaystyle \mathbb {Z} }$ (or a finitely generated vector space) with a Hodge decomposition of (or ), so that the weight decomposition is defined by. ${\displaystyle \mathbb {Q} }$ ${\displaystyle A}$ ${\displaystyle V=A\otimes _{\mathbb {Z} }\mathbb {R} }$ ${\displaystyle V=A\otimes _{\mathbb {Q} }\mathbb {R} }$ ${\displaystyle \mathbb {Q} }$

Examples

Hodge-Tate structures

Z (n)

${\displaystyle \mathbb {Z} (1)}$is the whole Hodge structure with module ${\displaystyle \mathbb {Z} }$

${\displaystyle 2\pi i\mathbb {Z} \subset \mathbb {C} }$

and . It is the only 1-dimensional Hodge structure weighing -2. ${\displaystyle \mathbb {Z} (1)\otimes _{\mathbb {Z} }\mathbb {C} =H^{-1,-1}}$

With becomes the -fold tensor product ${\displaystyle \mathbb {Z} (n)}$ ${\displaystyle n}$

${\displaystyle \mathbb {Z} (n):=\mathbb {Z} (1)\otimes \ldots \otimes \mathbb {Z} (1)}$

designated.

Q (n)

${\displaystyle \mathbb {Q} (1)}$is the rational Hodge structure with -Vector space ${\displaystyle \mathbb {Q} }$

${\displaystyle 2\pi i\mathbb {Q} \subset \mathbb {C} }$

and . is -fold tensor product . ${\displaystyle \mathbb {Q} (1)\otimes _{\mathbb {Q} }\mathbb {C} =H^{-1,-1}}$ ${\displaystyle \mathbb {Q} (n)}$ ${\displaystyle n}$ ${\displaystyle \mathbb {Q} (n):=\mathbb {Q} (1)\otimes \ldots \otimes \mathbb {Q} (1)}$

R (n)

${\displaystyle \mathbb {R} (1)}$is the Hodge structure with - vector space ${\displaystyle \mathbb {R} }$

${\displaystyle i\mathbb {R} \subset \mathbb {C} }$

and . is -fold tensor product . ${\displaystyle \mathbb {R} (1)\otimes _{\mathbb {R} }\mathbb {C} =H^{-1,-1}}$ ${\displaystyle \mathbb {R} (n)}$ ${\displaystyle n}$ ${\displaystyle \mathbb {R} (n):=\mathbb {R} (1)\otimes \ldots \otimes \mathbb {R} (1)}$

Hodge decomposition theorem for Kähler manifolds

The cohomology of a compact Kähler manifold has a Hodge structure: according to Hodge's theorem one can identify the -th cohomology with the space of harmonic differential forms and it holds ${\displaystyle M}$ ${\displaystyle n}$ ${\displaystyle H^{n}(M;\mathbb {R} )}$ ${\displaystyle {\mathcal {H}}^{n}(M)}$

${\displaystyle {\mathcal {H}}^{n}(M)\otimes _{\mathbb {R} }\mathbb {C} =\bigoplus _{p+q=n}{\mathcal {H}}^{p,q}(M)}$

where denotes the harmonic (p, q) forms . It applies . ${\displaystyle {\mathcal {H}}^{p,q}(M)}$ ${\displaystyle {\overline {{\mathcal {H}}^{p,q}(M)}}={\mathcal {H}}^{q,p}(M)}$

Hodge filtration

To a pure Hodge structure by weight one describes the filtration ${\displaystyle n}$

${\displaystyle \ldots \supset F^{p}\supset F^{p+1}\supset \ldots }$

With

${\displaystyle F^{p}=\bigoplus _{r\geq p}V^{r,s}\subset V\otimes _{\mathbb {R} }\mathbb {C} }$

as an associated Hodge filtration.

The Hodge filtration determines the Hodge decomposition

${\displaystyle V^{p,q}=F^{p}\cap {\overline {F^{q}}}.}$

The existence of a pure Hodge decomposition of weight is therefore equivalent to the existence of a filtration of with for sufficiently large and ${\displaystyle n}$ ${\displaystyle (F^{p})_{p\in \mathbb {Z} }}$ ${\displaystyle V\otimes _{\mathbb {R} }\mathbb {C} }$ ${\displaystyle F^{p}=0}$ ${\displaystyle p}$

${\displaystyle F^{p}\oplus {\overline {F^{q+1}}}=V\otimes _{\mathbb {R} }\mathbb {C} }$

for everyone with . ${\displaystyle p,q}$ ${\displaystyle p+q=n}$

literature

• Wells RO: Differential Analysis on Complex Manifolds (3rd ed.), Springer 2008, ISBN 978-0-387-73891-8