CHAPTER 2
The variation formula
Our general set-up is this: let M be an n-dimensional complex manifold (com-
pact or not) equipped with a Hermitian metric
ds2
=
n
a,b=1
gabdza dzb
and let ω := i
∑n
a,b=1
gabdza dzb be the associated real (1, 1) form. As in the
introduction, we take n 2. We write gab := (gab)−1 for the elements of the
inverse matrix to (gab) and G := det(gab). Note that ωn = 2nn!Gdx1 · · · dx2n
locally where zk = x2k−1 + ix2k. For a domain W M, we let Lp,q(W ) denote the
(p, q) forms on W with complex-valued, C∞(W )-functions as coefficients. We have
the standard linear operators
:
Lp,q(W
)
Ln−q,n−p(W
),
:
Lp,q(W
)
Lp+1,q(W
),
:
Lp,q(W
)
Lp,q+1(W
),
δ := ∂∗ :
Lp,q(W
)
Lp,q−1(W
),
δ := ∂∗ :
Lp,q(W
)
Lp−1,q(W
),
and d = + ∂. We get the box Laplacian operator
δ∂ + ∂δ :
Lp,q(W
)
Lp,q(W
)
and its conjugate
δ∂ + ∂δ :
Lp,q(W
)
Lp,q(W
).
Adding these, we obtain the Laplacian operator
Δ = δ∂ + ∂δ + δ∂ + ∂δ
which is a real operator; in local coordinates acting on functions this has the form
Δu = −2
n
a,b=1
gba
∂2u
∂zb∂za
+
1
2
n
a,b=1
(
1
G
∂(Ggba)
∂za
∂u
∂zb
+
1
G
∂(Ggab)
∂za
∂u
∂zb
)
=: −2[Pu + Ru]. (2.1)
We call Pu the principal part of Δu. Also, we remark that if
ds2
is K¨ahler,
i.e., if = 0, then Δu = −2Pu = −2
∑n
a,b=1
gba
∂2u
∂zb∂za
. As usual, we set, for any
α Lp,q(W ), α 2
W
=
W
α ∗α 0.
Given a nonnegative C∞ function c = c(z) on M, we call a C∞ function u
on an open set D M c-harmonic on D if Δu + cu = 0 on D. Choosing local
coordinates near a fixed point p0 M and a coordinate neighborhood U of p0 such
5
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