The Hodge theory or the set of Hodge is a central message of the Hodge theory. This theory combines the mathematical sub-areas of analysis , differential geometry and algebraic topology . The Hodge decomposition and the Hodge theory are named after the mathematician William Vallance Douglas Hodge , who developed them in the 1930s as an extension of the De Rham cohomology .
Elliptical complex
With smooth to cuts referred to in a vector bundle. Let be an oriented Riemannian manifold and a sequence of vector bundles . An elliptic complex is a sequence of first-order
partial differential operators
Γ
∞
{\ displaystyle \ Gamma ^ {\ infty}}
(
M.
,
G
)
{\ displaystyle (M, g)}
(
E.
i
)
i
{\ displaystyle (E_ {i}) _ {i}}
(
D.
i
)
i
{\ displaystyle (D_ {i}) _ {i}}
0
⟶
Γ
∞
(
E.
0
)
⟶
D.
0
Γ
∞
(
E.
1
)
⟶
D.
1
...
⟶
D.
m
-
1
Γ
∞
(
E.
m
)
⟶
0
,
{\ displaystyle 0 \ longrightarrow \ Gamma ^ {\ infty} (E_ {0}) {\ stackrel {D_ {0}} {\ longrightarrow}} \ Gamma ^ {\ infty} (E_ {1}) {\ stackrel { D_ {1}} {\ longrightarrow}} \ ldots {\ stackrel {D_ {m-1}} {\ longrightarrow}} \ Gamma ^ {\ infty} (E_ {m}) \ longrightarrow 0,}
so the following properties apply.
The result is a coquette complex , that is, it applies to everyone and
(
Γ
∞
(
E.
i
)
,
D.
i
)
{\ displaystyle (\ Gamma ^ {\ infty} (E_ {i}), D_ {i})}
D.
i
∘
D.
i
-
1
=
0
{\ displaystyle D_ {i} \ circ D_ {i-1} = 0}
1
≤
i
≤
m
{\ displaystyle 1 \ leq i \ leq m}
for each is the sequence of the main symbols
(
x
,
ξ
)
∈
T
∗
M.
∖
{
0
}
{\ displaystyle (x, \ xi) \ in T ^ {*} M \ backslash \ {0 \}}
0
⟶
π
(
E.
0
)
⟶
σ
D.
0
π
(
E.
1
)
⟶
σ
D.
1
...
⟶
σ
D.
m
-
1
π
(
E.
m
)
⟶
0
{\ displaystyle 0 \ longrightarrow \ pi (E_ {0}) {\ stackrel {\ sigma _ {D_ {0}}} {\ longrightarrow}} \ pi (E_ {1}) {\ stackrel {\ sigma _ {D_ {1}}} {\ longrightarrow}} \ ldots {\ stackrel {\ sigma _ {D_ {m-1}}} {\ longrightarrow}} \ pi (E_ {m}) \ longrightarrow 0}
exactly. The term bundle projection.
π
:
E.
i
→
M.
{\ displaystyle \ pi \ colon E_ {i} \ to M}
The spaces can be understood, for example, as the spaces of differential forms .
Γ
∞
(
E.
i
)
{\ displaystyle \ Gamma ^ {\ infty} (E_ {i})}
Hodge's theorem
Let now be a compact , oriented Riemannian manifold and the i-th cohomology group of the elliptic complex . Also define a (Laplace) operator
M.
{\ displaystyle M}
H
i
(
E.
.
,
D.
.
)
{\ displaystyle H ^ {i} (E _ {.}, D_ {.})}
(
Γ
∞
(
E.
i
)
,
D.
i
)
{\ displaystyle (\ Gamma ^ {\ infty} (E_ {i}), D_ {i})}
Δ
i
:
Γ
∞
(
E.
i
)
→
Γ
∞
(
E.
i
)
.
{\ displaystyle \ Delta _ {i}: \ Gamma ^ {\ infty} (E_ {i}) \ to \ Gamma ^ {\ infty} (E_ {i}).}
by
Δ
i
=
D.
i
∗
∘
D.
i
+
D.
i
-
1
∘
D.
i
-
1
∗
.
{\ displaystyle \ Delta _ {i} = D_ {i} ^ {*} \ circ D_ {i} + D_ {i-1} \ circ D_ {i-1} ^ {*}.}
This is an elliptic operator . Now applies:
The -th cohomology group is for all isomorphic to the kernel of , that is
i
{\ displaystyle i}
H
i
(
E.
.
,
D.
.
)
{\ displaystyle H ^ {i} (E., D_ {.})}
i
∈
Z
{\ displaystyle i \ in \ mathbb {Z}}
Δ
i
{\ displaystyle \ Delta _ {i}}
∀
i
:
H
i
(
E.
.
,
D.
.
)
≅
ker
(
Δ
i
)
⊂
Γ
∞
(
E.
i
)
.
{\ displaystyle \ forall i: \ H ^ {i} (E., D _ {.}) \ cong \ ker (\ Delta _ {i}) \ subset \ Gamma ^ {\ infty} (E_ {i}). }
The dimension of the -th cohomology group is finite for all
i
{\ displaystyle i}
i
∈
Z
{\ displaystyle i \ in \ mathbb {Z}}
dim
H
i
(
E.
.
,
D.
.
)
<
∞
.
{\ displaystyle \ dim H ^ {i} (E., D _ {.}) <\ infty.}
Γ
∞
(
E.
i
)
=
ker
(
Δ
i
)
⊕
R.
(
D.
i
-
1
)
⊕
R.
(
D.
i
∗
)
.
{\ displaystyle \ Gamma ^ {\ infty} (E_ {i}) = \ ker (\ Delta _ {i}) \ oplus R (D_ {i-1}) \ oplus R (D_ {i} ^ {*} ).}
It denotes the core and the image of an operator.
ker
{\ displaystyle \ ker}
R.
{\ displaystyle R}
Example: De Rham cohomology
The De Rham Complex
0
→
A.
0
(
M.
)
→
d
0
A.
1
(
M.
)
→
d
1
...
→
d
m
-
1
A.
m
(
M.
)
→
0
{\ displaystyle 0 \ to {\ mathcal {A}} ^ {0} (M) {\ xrightarrow {\ mathrm {d_ {0}}}} {\ mathcal {A}} ^ {1} (M) {\ xrightarrow {\ mathrm {d_ {1}}}} \ ldots {\ xrightarrow {\ mathrm {d} _ {m-1}}} {\ mathcal {A}} ^ {m} (M) \ to 0}
is an elliptical complex. The spaces are again the spaces of the differential forms of the i-th degree and is the external derivative . The associated sequence of main symbols is the Koszul complex . The operator is the Hodge-Laplace operator . The core of this operator is called the space of harmonic differential forms , since this is defined analogously to the space of harmonic functions . According to Hodge's theorem, there is now an isomorphism between the i-th De Rham cohomology group and the space of harmonic differential forms of degree .
A.
i
{\ displaystyle {\ mathcal {A}} ^ {i}}
d
i
{\ displaystyle \ mathrm {d} _ {i}}
Δ
=
d
∗
d
+
d
d
∗
{\ displaystyle \ Delta = \ mathrm {d} ^ {*} \ mathrm {d} + \ mathrm {d} \ mathrm {d} ^ {*}}
H
d
R.
i
(
A.
(
M.
)
,
d
)
{\ displaystyle H _ {\ mathrm {dR}} ^ {i} ({\ mathcal {A}} (M), \ mathrm {d})}
ker
(
Δ
i
)
{\ displaystyle \ ker (\ Delta _ {i})}
i
{\ displaystyle i}
Also are
b
i
(
M.
)
: =
dim
(
H
d
R.
i
(
A.
(
M.
)
,
d
)
)
{\ displaystyle b_ {i} (M): = \ dim (H _ {\ mathrm {dR}} ^ {i} ({\ mathcal {A}} (M), \ mathrm {d}))}
well-defined numbers, since the De Rham cohomology groups have finite dimensions. These numbers are called Betti numbers . The Hodge star operator also induces an isomorphism between the spaces and . This is the Poincaré duality and applies to the Betti numbers
⋆
:
A.
i
(
M.
)
→
A.
n
-
i
(
M.
)
{\ displaystyle \ star: {\ mathcal {A}} ^ {i} (M) \ to {\ mathcal {A}} ^ {ni} (M)}
ker
(
Δ
i
)
{\ displaystyle \ ker (\ Delta _ {i})}
ker
(
Δ
n
-
i
)
{\ displaystyle \ ker (\ Delta _ {ni})}
b
i
(
M.
)
=
b
n
-
i
(
M.
)
.
{\ displaystyle b_ {i} (M) = b_ {ni} (M).}
literature
Liviu I. Nicolaescu: Lectures on the geometry of manifolds. 2nd edition. World Scientific, Singapore et al. 2007, ISBN 978-981-270853-3 .
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