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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.
We show that a closed almost Kahler 4-manifold of globally constant holomorphic sectional curvature $kgeq 0$ with respect to the canonical Hermitian connection is automatically Kahler. The same result holds for $k<0$ if we require in addition that th
Two results regarding Kahler supermanifolds with potential $K=A+Cthetabartheta$ are shown. First, if the supermanifold is Kahler-Einstein, then its base (the supermanifold of one lower fermionic dimension and with Kahler potential $A$) has constant s
We classify quadruples $(M,g,m,tau)$ in which $(M,g)$ is a compact Kahler manifold of complex dimension $m>2$ with a nonconstant function $tau$ on $M$ such that the conformally related metric $g/tau^2$, defined wherever $tau e 0$, is Einstein. It tur
We derive some elliptic differential inequalities from the Weitzenbock formulas for the traceless Ricci tensor of a Kahler manifold with constant scalar curvature and the Bochner tensor of a Kahler-Einstein manifold respectively. Using elliptic estim
We prove a uniform diameter bound for long time solutions of the normalized Kahler-Ricci flow on an $n$-dimensional projective manifold $X$ with semi-ample canonical bundle under the assumption that the Ricci curvature is uniformly bounded for all ti