COMPLEX GEOMETRY Course notes

5.5 Index Theorem (Heat Equation approach)
Let ∆ p : A p (X) −→ A p (X), λ ∈ R ≥0 . Let E p λ denote the λ-eigenspace for ∆p (finite dimensonal). The
square root √ ∆ = δ is called the Dirac operator.
Lemma 5.5.1. The sequence
is exact for λ > 0.
0 −→ E 0 λ
d
−→ Eλ 1 d
−→ · · · −→ Eλ n −→ 0
Proof: ω ∈ E p λ =⇒ ∆p+1 dω = d∆ p ω = λdω =⇒ dω ∈ E p+1
λ
.
Now ω ∈ E p λ and dω = 0 =⇒ λω = ∆p ω = d ∗ d + dd ∗ ω =⇒ ω = d ( 1
λ d∗ ω ) . Then ∆d ∗ ω = d ∗ ∆ω = λd ∗ ω.
Corollary 5.5.1. ∑ p (−1)p dim(E p λ ) = 0.
Corollary 5.5.2. Let {λ p i } be the spectrum of ∆p , with terms repeated n times if multi = n. Then
where ∑ ′
i
is over i where λ(p) i = 0.
∑
(−1) p tre −t∆p = ∑
p
p
(−1) p e −tλ(p) i
= ∑ p
(−1) p ′∑
i
e −λ(p) i (t)
Note that ∑ ′
i e−λ(p) i
(t) = dim(Ker(∆ p )). Hence
X (X) = ∑ p
= ∑ p
(−1) p dim(Ker(∆ p )) = ∑ (−1) p tre −t∆(p) , where e −t∆(p) = T t ,
p
(−1) ∑ ∫
p e (p) (x, x, t)dVol(x).
i X
Proposition 5.5.1. e(x, x, t) ∼ (4πt) −n/2 ∑ ∞
k=0 u k(x, t)t k , where u k (x, t) is explicitly given in terms of
components of curvatures.
Hence, as t −→ 0, we have
(
X ∼ 1 n/2 ∑
∞ ∫
4πt
k=0
X
)
∞∑
(−1) p tru p k (x, x)dVol(x) t k .
p=0
71