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
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