Extremal Kaehler metrics and Ray-Singer analytic torsion

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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 gin K_omega of constant scalar curvature. In particular, if c_1(X)=0 or if c_1(X)<0 and 1/(2pi)[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}.

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