论文标题
Kähler-Einstein指标和圆形束的阻塞平坦度
Kähler-Einstein metrics and obstruction flatness of circle bundles
论文作者
论文摘要
Obstruction flatness of a strongly pseudoconvex hypersurface $Σ$ in a complex manifold refers to the property that any (local) Kähler-Einstein metric on the pseudoconvex side of $Σ$, complete up to $Σ$, has a potential $-\log u$ such that $u$ is $C^\infty$-smooth up to $Σ$.通常，$ u $只有有限的平滑度高达$σ$。在本文中，我们研究了Hypersurfaces $σ$的阻塞平坦度，因为单位圆圈捆绑$ s（l）$ s（l）负Hermitian Line Bundles $（l，h）$（l，h）$（l，h）$（M，g）。此外，如果所有这些特征值严格少于一个和$（m，g）$的完成，那么我们表明相应的磁盘捆绑包允许完整的Kähler-Einstein公制。最后，当$（m，g）$是kählersurface $（\ dim m = 2 $）时，我们给出了$ s（l）$的阻塞平坦度的必要条件，并具有恒定的标量曲率。
Obstruction flatness of a strongly pseudoconvex hypersurface $Σ$ in a complex manifold refers to the property that any (local) Kähler-Einstein metric on the pseudoconvex side of $Σ$, complete up to $Σ$, has a potential $-\log u$ such that $u$ is $C^\infty$-smooth up to $Σ$. In general, $u$ has only a finite degree of smoothness up to $Σ$. In this paper, we study obstruction flatness of hypersurfaces $Σ$ that arise as unit circle bundles $S(L)$ of negative Hermitian line bundles $(L, h)$ over Kähler manifolds $(M, g).$ We prove that if $(M,g)$ has constant Ricci eigenvalues, then $S(L)$ is obstruction flat. If, in addition, all these eigenvalues are strictly less than one and $(M,g)$ is complete, then we show that the corresponding disk bundle admits a complete Kähler-Einstein metric. Finally, we give a necessary and sufficient condition for obstruction flatness of $S(L)$ when $(M, g)$ is a Kähler surface $(\dim M=2$) with constant scalar curvature.
