沿复子流形爆破的紧凯勒流形的纯量曲率
Scalar curvature of blow-ups of compact Kähler manifolds along complex submanifolds
AI总结：
针对复维数至少3且含复余维数至少2子流形的紧凯勒流形，证明其沿该子流形爆破所得流形上存在凯勒度量列，其纯量曲率按$C^0$范数收敛到原度量拉回的纯量曲率，推广了点爆破的对应结果。
AI中文摘要：
设$(M,\omega)$为具有纯量曲率$S(\omega)$的紧凯勒流形，假设$M$的复维数至少为3，且包含复余维数至少为2的复子流形$X$。令$\sigma: Bl_{X} M \rightarrow M$表示$M$沿$X$的爆破映射。我们证明，$Bl_{X} M$存在一列凯勒度量$\{\widetilde{\omega}_{i}\}_{i \geq 1}$，其纯量曲率$S(\widetilde{\omega}_i)$在$Bl_{X} M$上的$C^0$范数下收敛到$\sigma^{*} (S(\omega))$。本研究受Brown近期结果的启发，该结果建立了在点处爆破的对应结论；证明基于构造爆破上极端凯勒度量的胶合方法，结合了新的分析工具及针对本研究情形所需的若干修改。
英文摘要：
Let $(M,ω)$ be a compact Kähler manifold with its scalar curvature $S(ω)$, Assume that $M$ has complex dimension at least $3$ and contains a complex submanifold $X$ of complex codimension at least $2$. Let $σ: Bl_{X} M \rightarrow M$ denote the blow-up of $M$ along $X$. We show that $Bl_{X} M$ admits a sequence of Kähler metrics $\{\widetildeω_{i}\}_{i \geq 1}$ whose scalar curvatures $S(\widetildeω_i)$ converge to $σ^{\ast} (S(ω))$ in the $C^0(Bl_{X} M)$ norm. Our work is motivated by a recent result of Brown, who established the corresponding result for blow-ups at a point. The proof is based on the gluing method for constructing extremal Kähler metrics on blow-ups, together with new analytic tools and several modifications needed in our setting.
