Let $mathcal{K}(n, V)$ be the set of $n$-dimensional compact Kahler-Einstein manifolds $(X, g)$ satisfying $Ric(g)= - g$ with volume bounded above by $V$. We prove that after passing to a subsequence, any sequence ${ (X_j, g_j)}_{j=1}^infty$ in $mathcal{K}(n, V)$ converges, in the pointed Gromov-Hausdorff topology, to a finite union of complete Kahler-Einstein metric spaces without loss of volume. The convergence is smooth off a closed singular set of Hausdorff dimension no greater than $2n-4$, and the limiting metric space is biholomorphic to an $n$-dimensional semi-log canonical model with its non log terminal locus of complex dimension no greater than $n-1$ removed. We also show that the Weil-Petersson metric extends uniquely to a Kahler current with bounded local potentials on the KSBA compactification of the moduli space of canonically polarized manifolds. In particular, the coarse KSBA moduli space has finite volume with respect to the Weil-Petersson metric. Our results are a high dimensional generalization of the well known compactness results for hyperbolic metrics on compact Riemann surfaces of fixed genus greater than one.
تحميل البحث