CHAPTER 1

Introduction

1. Kahler geometry

1.1. Historical introduction. In 1932, Erich Kahler submitted a remarkable

paper, which appeared the following year [78]. The German title translates as: "On

a remarkable Hermitian metric". In this paper, Kahler sets out to derive invariants

of a Hermitian metric

n

9=22 9ikdxidxk

i,k=l

using the then novel calculus of differential forms. The first thing he does is to

write down the fundamental 2-form

n

u = i ^2 ^ikdxi A dxk

i,k=l

of the metric which we now call the Kahler form. Next, he writes down the exterior

derivative dw—this is the first invariant. He says that the case when

(1) duj = 0

"presents itself as a remarkable exception", and immediately starts to write down

some consequences, the first of which is that one can express the metric through a

potential u, now called the Kahler potential, in the following manner:

n

rP-

9= ^2 a ct-

dxi^k

•

*-* oxidxk

i,k=l

Kahler then points out that the (local) existence of the potential is equivalent to the

closedness of a;. At various points he remarks that these notions have an intrinsic

meaning1.

What Kahler did not say, although he could have, and we suspect he was aware

of it, is that du = 0 is also equivalent to the existence of holomorphic coordinates

in which the metric has the form

n

(2) g= 5 ^ ( % + [2])dxidSife ,

i,k=l

meaning that up to and including terms of first order, the metric looks like the fiat

metric on C

n

. This immediately implies the following:

SCHOLIUM 1.1. An identity involving only the metric and its first derivatives

holds on any Kahler manifold if and only if it holds in flat C

n

.

1See

[20] for more on the history, and for a discussion of the importance of Kahler's paper

for differential geometry.

1

http://dx.doi.org/10.1090/surv/044/01