Extremal Kaehler metrics and Ray-Singer analytic torsion

Abstract

Let (X,[\omega]) be a compact Kaehler manifold with a fixed Kaehler class [\omega]. Let K_\omega be the set of all Kaehler metrics on X whose Kaehler class equals [\omega]. In this paper we investigate the critical points of the functional Q(g)= |v|_g T_0(X,g)^{1/2} for g \in K_\omega, where v is a fixed nonzero vector of the determinant line \lambda(X) associated to H^*(X) and T_0(X,g) is the Ray-Singer analytic torsion. For a polarized algebraic manifold (X,L) we consider a twisted version Q_L(g) of this functional and assume that c_1(L)=[\omega]. Then the critical points of Q_L are exactly the metrics g\in K_\omega of constant scalar curvature. In particular, if c_1(X)=0 or if c_1(X)<0 and 1/(2\pi)[\omega] = -c_1(X), then K_\omega contains a unique Kaehler-Einstein metric g_{KE} and Q_L attains its absolut maximum at g_{KE}.