论文标题
Kahler-Einstein Currents III的Riemannian几何形状：负标量曲率的Kahler-Einstein歧管的紧凑性
Riemannian geometry of Kahler-Einstein currents III: compactness of Kahler-Einstein manifolds of negative scalar curvature
论文作者
论文摘要
令$ \ mathcal {k}（n，v）$为$ n $二维紧凑型kahler-ineinstein歧管$（x，g）$满足$ ric（g）= - g $，$ v $ bumbore toble bumbor。我们证明，将任何顺序$ \ {（x_j，g_j）\} _ {j = 1}^\ infty $ in $ \ mathcal {k}（k}（n，n，v）$收敛，在尖锐的gromov-hausdorff拓扑结合中，没有完整的kahler-kahler-space inseinseinseinseinseinsinmetric space，融合的封闭式单数不超过$ 2N-4 $的封闭式单数，而有限的度量空间对$ n $ n $维的半维度半log典型模型具有双态形态，其非数量端子位点的复杂尺寸的非日志端子位点不超过$ n-1 $。我们还表明，Weil-Petersson度量标准量唯一地扩展到具有局部电势的Kahler电流，该电位在规范上极化的歧管的模量空间的KSBA紧凑型上。特别是，相对于Weil-Petersson度量，粗糙的KSBA模量空间具有有限的体积。我们的结果是对固定属的紧凑型riemann表面上的双曲线指标的高度泛化结果大于一个。
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.
