Scalar curvature on Kähler blow-ups and systolic inequalities

Abstract

In this paper, we develop the weighted level set method for a Kähler manifold $(X^n,ω)$ admitting an almost holomorphic map to a possibly singular base $Z$, which is not uniruled. As a key intermediate result, we prove that any blowup $\operatorname{Bl}_SX$ of $X$ along smooth submanifolds $S$ of $ \operatorname{codim} S\ge2$ admits a sequence of Kähler metrics with scalar curvature globally and arbitrarily $C^0$-close to the scalar curvature of $ω$. As a consequence, we establish the sharp \(2\)-systole estimate for every positive scalar curvature Kähler manifold $(X,ω)$ and prove $\min_XS(ω) \cdot\operatorname{sys}_2(ω) \le 2πr(r+1)$, where \(r\) is the rational dimension of $X$, with equality if and only if the universal cover splits as $(\widetilde X,\widetilde ω) \cong (\mathbb P^r,ω_{\mathrm{FS}}) \times(Y^{n-r},ω_{\mathrm{RF}})$ up to normalization where $ω_{\mathrm{FS}}$ is the Fubini-Study metric and $ω_{\mathrm{RF}}$ is Ricci-flat. We also show a sharp even-systolic inequality in the same setting when the general fibre is the projective space.