Extremal Kaehler metrics and Ray-Singer analytic torsion
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Let (X,[ω]) be a compact Kaehler manifold with a fixed Kaehler class [ω]. Let K_ωbe the set of all Kaehler metrics on X whose Kaehler class equals [ω]. 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_ω, where v is a fixed nonzero vector of the determinant line λ(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)=[ω]. Then the critical points of Q_L are exactly the metrics g\in K_ωof constant scalar curvature. In particular, if c_1(X)=0 or if c_1(X)<0 and 1/(2π)[ω] = -c_1(X), then K_ωcontains a unique Kaehler-Einstein metric g_{KE} and Q_L attains its absolut maximum at g_{KE}.
33 pages, LaTeX2e, to be published in Geometric Aspects of partial differential equations, Proceedings of the Roskilde Conference, Sept. 1998, AMS series Contemporary Mathematics
33 pages, LaTeX2e, to be published in Geometric Aspects of partial differential equations, Proceedings of the Roskilde Conference, Sept. 1998, AMS series Contemporary Mathematics