Scalar curvature of blow-ups of compact Kähler manifolds along complex submanifolds

Abstract

Let $(M,\omega )$ be a compact Kähler manifold with its scalar curvature $S(\omega )$, Assume that $M$ has complex dimension at least $3$ and contains a complex submanifold $X$ of complex codimension at least $2$. Let $\sigma :B{l}_{X}M\to M$ denote the blow-up of $M$ along $X$. We show that $B{l}_{X}M$ admits a sequence of Kähler metrics $\{{\tilde{\omega }}_i{\}}_{i\ge 1}$ whose scalar curvatures $S({\tilde{\omega }}_i)$ converge to ${\sigma }^{\ast }(S(\omega ))$ in the $C^0(B{l}_{X}M)$ 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.

46 pages