57 pages; v2: decay theory in Section 3.4 now covers the I_b and I_b^* cases as well, various minor corrections; v3: corrected remarks on compactification of ALH spaces (Definition 1.9, Problem 6.1), rewrote Section 2.2, included references to recent work of Biquard and Minerbe (Remark 1.6(iv)), fixed a gap in Section 5.3 (see the new Step 3 there), additional minor corrections
Let $X$ denote the complex projective plane, blown up at the nine base points of a pencil of cubics, and let $D$ be any fiber of the resulting elliptic fibration on $X$. Using ansatz metrics inspired by work of Gross-Wilson and a PDE method due to Tian-Yau, we prove that $X \setminus D$ admits complete Ricci-flat Kähler metrics in most de Rham cohomology classes. If $D$ is smooth, the metrics converge to split flat cylinders $\R^+ \times S^1 \times D$ at an exponential rate. In this case, we also obtain a partial uniqueness result and a local description of the Einstein moduli space, which contains cylindrical metrics whose cross-section does not split off a circle. If $D$ is singular but of finite monodromy, they converge at least quadratically to flat $T^2$-submersions over flat 2-dimensional cones which need not be quotients of $\R^2$. If $D$ is singular of infinite monodromy, their volume growth rates are 4/3 and 2 for the Kodaira types ${\rm I}_b$ and ${{\rm I}_b}^*$, their injectivity radii decay like $r^{-1/3}$ and $(\log r)^{-1/2}$, and their curvature tensors decay like $r^{-2}$ and $r^{-2}(\log r)^{-1}$. In particular, the ${\rm I}_b$ examples show that the curvature estimate from Cheeger-Tian \cite{ct-einstein} cannot be improved in general.
Holomorphic Cartan geometries and Calabi--Yau manifolds
证明了具有全纯Cartan几何的Calabi-Yau射影流形是阿贝尔簇. (Nous démontrons que les seules variétés projectives de Calabi--Yau qui possèdent des géométrie holomorphes de Cartan sont les variétés abéliennes.）
Continuity of Extremal Transitions and Flops for Calabi-Yau Manifolds
本文研究了Calabi-Yau流形上Ricci平坦Kähler度量在代数几何运算下的行为：极值变换或翻转。我们证明了Candelas和de la Ossa猜想的一个版本：与极值变换和翻转相关的Ricci平坦Calabi-Yau流形可以通过由连续的Ricci平坦Calabi-Yau流形族和Gromov-Hausdorff拓扑中的紧度量空间组成的路径连接。在证明主要结果的一个重要步骤中，建立了Calabi-Yau流形上Ricci平坦Kähler度量沿光滑的收敛性，这是一个独立的结果.
