224

T . Ohsawa

for ail feDomdnDomd*nC'^-''{r)nCo{Vy satisfying (/,м)ко = 0 for any иеЖ,"'^^ Here ~^ denotes the completion with respect to || II«, and we are not distinguishing the minimal and the maximal closed extensions of d since they coinside by the completeness of the metric ds^. Similarly there must exist a constant C5 such that

\\ç : 'fLuCsi\\dfL + \\d''f\\e ) foraüfeDomrfnDomrf*nC"-4nnCo(FlPsatisfyingaM)Ko=OforanyMe^-4 Therefore, for every /бС"(Г)пСо(П which satisfies (/,w)xo = 0 for all иеЖ", one can find g^eDomd* and h^eDomd with respect to ds^, such that

\\cp ; 'g , \UuCjfKj .

and

\\ç ; 'K\\sèCs\\fKoL^

Now suppose that dimJf"ydimJf{^ = k, Then, by Aronszajn's unique tion theorem there must exist/S ...,/^+^6jr" such that/i,, ...Jko' are ly independent. Let Lcz^" be the linear subspace spanned by {f\ ,..J^^^}. Then we can choose a sequence {/^; m=l, 2, ...jcL such that ||/^ll = l and 1тКо^Щт for all m. Thus, using what we have mentioned above, we can find g^eDomd* and h^eDomd satisfying

lmKo = d*gm + dh^

\\ ( Pl / lgm\\nmuC4 . \\lmKo\\l / m Wl/lthmWi/muCsWlmKoWllm-

Letting m-^00, we choose a weakly convergent subsequence of /^, g^ and h^ and finally obtain /eL,g6DomCax and heDomd^^^ satisfying /Ko = Caxg + d^^^h, Wcpy^gWuC^ and ||<pf ^/i|| ^Cg. By Proposition 1 we have gGDom<i„ and heDomd^,^. Therefore \\1ко\\^ = {1ко^ П ={d^ing + d^,^ 0 = 0. On the other hand /Ф0 since dimL< 00 and ||/^|1 = L But it is a contradiction to Aronszajn's theorem or that f^^,. • •, /ко" ^ were linearly independent. Q.E.D.

References

1 . Cheeger, J.: On the Hodge theory of Riemannian pseudomanifolds. Proc. Symp. Pure Math. ' 36, 91-146 (1980)

2 . Cheeger, J., Goreski, M., MacPherson, R.: L^-cohomology and intersection homology for singular varieties. Ann. Math. Stud. 102, Seminar on Differential Geometry, 303-340 (1982)

3 . Hsiang, W.C, Pati, V.: L^-cohomology of normal algebraic surfaces I. Invent. Math. 81, 395-412 (1985)

4Nagase , M.: Remarks on the L^-cohomology of singular algebraic surfaces. J. Math. Soc. Japan 41,97-116 (1989)

5 . Ohsawa, T.: Hodge spectral sequence on compact Kahler spaces. Publ. RIMS, 23, 613-615 (1987) Suppl. : Publ. RIMS, 26 (to appear)

6 . Saper, L.: L^-cohomology and intersection homology of certain algebraic varieties with lated singularities. Invent. Math. 82, 207-255 (1985). L^-cohomology of Kahler varieties with isolated singularities. Preprint

Note ddded in proof

The proof of Theorem 7 is incomplete. Corrected argument is given in [5], Supplement. The author is grateful to colleagues including G. Marinescu and M. Stern who have pointed out this gap.