Recent work of Saper, Zucker,
and others indicates that Kähler metrics with appropriate growth rates on the
nonsingular set of a compact Kähler variety are useful in describing the geometry of
such a variety. It has been conjectured that for every complex algebraic variety X
there exists a Kähler metric on the nonsingular set of X whose L2-cohomology is
isomorphic to the intersection cohomology of X. Saper proved this conjecture for
varieties with isolated singularities, using a complete Kähler metric. Similar results
have been obtained by others using incomplete metrics. We give natural and
explicit constructions of three types of Kähler metrics on the nonsingular set
X − Xsing of a subvariety X of a compact Kähler manifold. No restrictions on
the type of singularities of X are assumed. Similar constructions can be
done for nonembedded compact Kähler varieties. The first metric is Hodge
if X is algebraic but is not complete on X − Xsing if X is singular. The
completion of X − Xsing under this metric is a desingularization of X. The second
metric is complete and generalizes Saper’s metric for varieties with isolated
singularities. Moreover each incomplete metric of the first type is naturally
associated with a complete metric of the second type. The third metric is a sum
of the first two and has Poincaré-type growth near the singular locus of
X.
