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

Abstract:

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