### The moduli space of extremal compact Kähler manifolds and generalized Weil-Petersson metrics

Akira Fujiki, Georg Schumacher
1990 Publications of the Research Institute for Mathematical Sciences
where j is the natural inclusion. Proposition 1.1. Let S Teg be the set of smooth points of S. Let (p e F(S, ^S(R)) be a C°° function whose restriction to S reg is pluriharmonic. Then (p is pluriharmonic on the whole S. Proof. We extend the argument in the proof of [Fu2; Lemma 6]. Since the problem is local, we fix a point o e S and prove that we can ^n^ a holomorphic function / on S with Im / = <p, determined up to additive real constants on each connected component of S. By adjusting these
more » ... stants we may assume that / descends to a (continuous) meromorphic function f on S whose Imaginary part coincides with <p. Since cp is of class C°°? by a theorem of Spallek [Sp; Satz 4.2] / Is holomorphic, and hence cp is pluriharmonic, on the whole S. It remains to prove the surjectivity of a. First note that from the compactness of E the surjectivity of a follows at once. Hence, fe, and therefore fcjS = yb also, is the zero map. On the other hand, If we replace S by a suitable neighborhood of o, we may assume that y is isomorphic. Thus b is the zero map and a is surjective. Corollary 1.2. 1) Two locally dd-exact real C°° (1, l)-forms which coincide on S reg coincides on the whole S. 2) The map d<3:^s-»^J !l descends to dd: ^s-» ^l with respect to e: ^s -» @ s . Proof. 1) The problem Is local. Let \l/ 1 and \j/ 2 be R-valued C°° functions on S with dd\// l = dd\l/ 2 on S Teg . Then by applying Proposition 0.1 to ^ := i// l -\l/ 2 we get that the equality is even true on the whole S. 2) Since the support of the kernel of £ Is contained in S -S Teg , dd vanishes identically on Ker e by the above proposition. We denote by