Cheeger-Goreski-MacPherson's conjecture for the varieties with isolated singularities

Abstract

In [2] Cheeger-Goreski-MacPherson asked the following question. Let VclPN(IE) be a projective variety and let V' be the set of regular points of V. Is the L 2 de Rham cohomology group of V' with respect to the Fubini-Study metric canonically isomorphic to the intersection cohomology group of V? In virtue of the works of Cheeger [1], Hsian-Pati [3] and Nagase [4], we know that the conjecture is true if dim V < 2. Our proof will be based on the previous work [5] which complements a result of L. Saper asserting that there exists a complete Kfihler metric ds 2 on V' such that the L 2 de Rham cohomology group of V' with respect to ds 2 is canonically isomorphic to the intersection cohomology group of V. To relate Saper's L 2 cohomology group to the original one, the two metrics ds 2 and say ds 2 shall be connected by a path ds 2 + e d s 2, eel0, 1]. Then we shall see that a sharpened version of H6rmanderDonnelly-Fefferman's L 2 estimate is true for this family (cf. Prop. 10), which enables us to compare the space of harmonic forms with respect to ds2v and ds 2. In short, it will be shown that the ordinary L 2 cohomology is canonically isomorphic to certain L 2 cohomology which is already known to be isomorphic to the intersection cohomology. The manuscript of the present work was written during the author's stay at Bergische Universitfit Gesamthochschule Wuppertal on the occasion of Aktivitfit in Komplex Analysis in MPI at Bonn. He would like to express deep gratitude to the hospitality of these institutions.