Scalar curvature of blow-ups of compact K\"ahler manifolds along complex submanifolds

Abstract

Let $(M,\omega)$ be a compact K\"ahler 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 $\sigma: Bl_{X} M \rightarrow M$ denote the blow-up of $M$ along $X$. We show that $Bl_{X} M$ admits a sequence of K\"ahler metrics $\{\widetilde{\omega}_{i}\}_{i \geq 1}$ whose scalar curvatures $S(\widetilde{\omega}_i)$ converge to $\sigma^{\ast} (S(\omega))$ 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 K\"ahler metrics on blow-ups, together with new analytic tools and several modifications needed in our setting.