Page 204 - Curvature and Homology

186 V. COMPLEX MANIFOLDS

2. Consider C,, with the metric

The fundamental 2-form in this case is given by

Clearly, this form is closed, and so the metric defines a Kaehler structure
on C,.
3. Let r be a discrete subgroup of maximal rank of the additive group
of C,, and denote by T, the quotient space Ci/l'; I' is actually the discrete
additive group (over R) generated by 2n independent vectors. It is
clear that l' is a properly discontinuous group without fixed points.
As a topological space, C,/r is homeomorphic with the product of a
torus of dimension 2n and a vector space over R. However, CJI' is
compact since r has rank 2n, and so it is isomorphic as a topological
group with the torus. Since the complex structure on C, is invariant
under l' (cf. 5 5.8) one is able to define a complex structure (and one
only) on the quotient space T,. With this complex structure the
manifold T, = C,/r is called a complex multi-torus.
Let n denote the natural projection of C,, onto T,. Then, n is a
holomorphic map. The metric of Cn defined in example 2 is invariant by
the translations of r. We are therefore able to define a metric on T, in
such a way that n is locally an isometry. Since the property of a
complex manifold which ensures that it be Kaehlerian is a local
property, T, is a Kaehler manifold.
We describe the homology properties of the multi-torus T,: The
projection 7 induces a canonical isomorphism n* of the space of
differential forms on T, onto the space of differential forms on C,
invariant by the translations of I'. Since the isomorphism n* commutes
with the operators d and 8, n* defines an isomorphism of the space
AZ(T,,) of the harmonic forms on T, onto AiC(Cn)-the vector sub-
space of A*c(C,) generated by (dzA) and their exterior products. For,
the elements of A p(C,) are-harmonic and invariant by the translations
of l'. Conversely, every form a on C, may be expressed as

where the coefficients are complex-valued functions. If a is the image
by n* of a harmonic form on T, it is harmonic and invariant by r, that