Hodge theory

Hodge theory

Hodge theory

is a branch of

algebraic geometry

and complex manifold theory that deals with the decomposition of the

cohomology groups

of an complex

projective algebraic variety

.

Let

Ω

k

⁢

(

M

)

denote the space of

differential

k

-forms. We know that there is a map, called the

exterior derivative

d

:

Ω

k

→

Ω

k

+

1

and that

d

2

=

0

. This forms a complex and the

cohomology

of this complex,

H

D

⁢

R

*

⁢

(

M

)

is called the

de Rham cohomology

of

M

. It is

isomorphic

to the singular cohomology of

M

.

For a complex manifold, it is more natural to have complex coordinates for our

differential forms

. Writing

z

j

=

x

j

+

i

⁢

y

j

we obtain two 1-forms,

d

⁢

z

j

and

d

⁢

z

j

¯

. These forms serve as a basis for all

k

-forms,

d

⁢

z

i

1

∧

d

⁢

z

i

2

∧

…

∧

d

⁢

z

i

p

∧

d

⁢

z

j

1

¯

∧

…

∧

d

⁢

z

j

q

¯

, where

p

+

q

=

k

. We call forms written like this to be

(

p

,

q

)

-forms. Let

Ω

p

,

q

be the space of

(

p

,

q

)

-forms. Then we have a natural decomposition

Ω

k

⁢

(

M

)

=

⊕

p

+

q

=

k

Ω

p

,

q

⁢

(

M

)

.

The question then becomes , does this decomposition pass to the level of cohomology? The answer in general is no. If it were true in general, it would force all

manifolds

to have even first cohomology, which is obviously not true. But fortunately there are some nice classes of manifolds for which is does hold.

The Laplacian opererator

Δ

=

d

⁢

ϕ

+

ϕ

⁢

d

where

ϕ

is the adjoint operator (with respect to the

Riemannian metric

) to

d

. A form is called harmonic if its Laplacian is 0. The motivation for considering the Laplacian comes from looking for differential forms of

minimal

length. Because a basis on

T

⁢

M

induces a basis on

T

*

⁢

M

then we have an metric on

⋀

k

T

*

⁢

M

Due to this, we can define a norm on

Ω

k

. It turns out that for

∥

ω

∥

to be minimal, then

(

ω

,

d

⁢

γ

)

=

0

for all

γ

∈

Ω

k

-

1

. This is

equivalent

to

(

ϕ

⁢

ω

,

γ

)

=

0

. So a

closed form

with minimal norm is harmonic.

The most fundamental theorem of Hodge theory states that the space of harmonic

k

-forms,

H

Δ

k

⁢

(

M

)

≅

H

D

⁢

R

k

⁢

(

M

)

.

Now for local coordinates, a

1

-form is written as

ω

=

∑

f

i

⁢

d

⁢

z

i

+

∑

g

j

⁢

d

⁢

z

j

¯

. We define two new

operators

∂

and

∂

¯

such that

∂

⁡

ω

=

∑

∂

⁡

f

i

∂

⁡

z

k

⁢

d

⁢

z

k

∧

d

⁢

z

i

+

∑

∂

⁡

g

j

∂

⁡

z

k

⁢

d

⁢

z

k

∧

d

⁢

z

j

¯

∂

¯

⁢

ω

=

∑

∂

⁡

f

i

∂

⁡

z

k

¯

⁢

d

⁢

z

k

¯

∧

d

⁢

z

i

+

∑

∂

⁡

g

j

∂

⁡

z

k

¯

⁢

d

⁢

z

k

∧

d

⁢

z

j

¯

These operators also

satisfy

the

relations

∂

2

=

∂

2

¯

=

0

and

∂

⁡

∂

¯

+

∂

¯

⁢

∂

=

0

. These operators also have

adjoints

and they as well have the relation that

d

=

∂

*

+

∂

¯

*

. Writing these in terms of the laplacian we obtain that

Δ

=

(

∂

⁡

∂

*

+

∂

*

⁡

∂

)

+

(

∂

¯

⁢

∂

¯

*

+

∂

¯

*

⁢

∂

¯

)

+

𝕆

where

𝕆

denotes some cross terms.
For this to be of any interest to us in terms on Hodge theory, these cross terms must vanish. Assuming they do, then if a form is harmonic, then each of the components of the

(

p

,

q

)

-form are harmonic. Therefore, we obtain a decomposition of the space of harmonic forms

H

Δ

k

=

⊕

p

+

q

=

k

H

Δ

p

,

q

.
This gives us a Hodge decomposition on the level of cohomology due to the

isomorphism

between the space of harmonic forms and the de Rham cohomology groups.

This has some immediate and interesting

consequences

. It tells us that any smooth complex

projective variety

has an even first

betti number

. With some work, one could also show that the second cohomology group is non-empty.

We worked under the

assumption

that the cross terms in the above expression vanished. A example of a class of manifolds where this holds true is

Kahler Manifolds

, which are complex manifolds with a Riemannian metric that is

compatible

with the

complex structure

.

This is just a small portion of the large topic of Hodge theory.

Title

Hodge theory

Canonical name

HodgeTheory

Date of creation

2013-03-22 15:21:39

Last modified on

2013-03-22 15:21:39

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

12

Author

bwebste (988)

Entry type

Definition

Classification

msc 14N05

Classification

msc 58A14

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