4

1. INTRODUCTION

multiple of the volume form, this cohomology class, and all its powers up to the

top dimension, are non-zero.

Note that it follows from the definitions that every holomorphic submanifold

of a Kahler manifold is again Kahler. Thus, to see that smooth complex projective

varieties are Kahler, it suffices to show that CPn is. We also see that holomor-

phic submanifolds of compact Kahler manifolds carry non-zero homology classes,

because the integral of some power of a; is a non-zero multiple of the volume and

is also a homological intersection number.

One can also describe the Kahler condition completely in terms of submanifold

geometry. In fact, uo has the property that its restriction to any oriented real 2-

dimensional submanifold is less than or equal the volume form, and it is equal

to the volume form precisely on the holomorphic curves. This means that a; is a

calibration of the complex structure J. Harvey-Lawson [67] have proved that the

converse is also true, namely every calibrated complex manifold is Kahler.

We construct

CPn

as (C

n + 1

\{0})/C*. Its complex structure is uniquely deter-

mined by the requirement that the projection C n + 1 \ {0} — CPn be holomorphic.

This projection restricts to the Hopf map g 2 n + 1 — CPn. Requiring the differential

of the Hopf map to be isometric on the orthogonal complement of the fiber gives

CPn a unique Riemannian metric g. Now we define the Kahler form on CPn by

formula (3). One has to check that it is closed. Pulling back via the Hopf map,

we have

TT*LO

= da, where a is a 1-form of constant length whose kernel is the

orthogonal complement of the fiber of the Hopf map.

The metric we have described is usually rescaled so that

/ u = 1 .

JCP1

This means that the cohomology class [LJ] is integral and is a generator of

H2(CPn,

Z).

The rescaled metric is called the Fubini-Study metric.

We have seen that smooth complex projective varieties are Kahler manifolds,

but the converse is not true. One can have Kahler metrics whose Kahler forms have

irrational periods, and so cannot be pull-backs of the Fubini-Study metric. When

dim H2(X, E) 1, this problem cannot usually be solved by rescaling. In fact, the

integrality of the Kahler class characterises projective algebraic manifolds among

all Kahler manifolds:

THEOREM

1.6 (Kodaira Embedding Theorem). Every compact Kdhler mani-

fold whose Kahler form has integral periods can be holomorphically embedded in

some CPn.

Given such an embedding, a theorem of Chow says that X is an algebraic

variety. In complex dimension one,

H2(X,

R) is always 1-dimensional, so that

all curves are projective. In complex dimension two, Kodaira's classification of

compact complex surfaces implies that on every Kahler surface one can deform the

complex structure so that, for the deformed structure, X is projective algebraic. In

higher dimensions this is not known to be true. On the other hand, nobody knows

an example of a Kahler manifold which cannot be deformed to a projective one.

This has led many authors to propose that the following problem should have a

positive answer:

OPEN PROBLEM

1.7. Does every compact Kdhler manifold deform to one on

which the Kahler form has rational periods?