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 dω = 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

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