Critical points of the area functional of a complex closed curve on the manifold of Kähler metrics

Abstract

We consider a compact complex manifold M M of dimension n n that admits Kähler metrics and we assume that C ↪ M C\hookrightarrow M is a closed complex curve. We denote by K C ( 1 ) \mathcal {KC}(1) the space of classes of Kähler forms [ ω ] ∈ H 1 , 1 ( M , R ) [\omega ]\in H^{1,1}(M,\mathbb {R}) that define Kähler metrics of volume 1 on M M and define A C : K C ( 1 ) → R \mathbf {A}_{C}:\mathcal {KC}(1)\to \mathbb {R} by A C ( [ ω ] ) = ∫ C ω = area of C in the induced metric by ω \mathbf {A}_{C}([\omega ])=\int _{C} \omega =\text {area of }C\text { in the induced metric by }\omega . We show how the Riemann-Hodge bilinear relations imply that any critical point of A C \mathbf {A}_{C} is the strict global minimum and we give conditions under which there is such a critical point [ ω ] [\omega ] : A positive multiple of [ ω ] n − 1 ∈ H 2 n − 2 ( M , R ) [\omega ]^{n-1}\in H^{2n-2}(M,\mathbb {R}) is the Poincaré dual of the homology class of C C . Applying this to the Abel-Jacobi map of a curve into its Jacobian, C ↪ J ( C ) C\hookrightarrow J(C) , we obtain that the Theta metric minimizes the area of C C within all Kähler metrics of volume 1 on J ( C ) J(C) .