Scalar curvature on K\"ahler blow-ups and systolic inequalities
In this paper, we develop the weighted level set method for a K\"ahler manifold $(X^n,\omega)$ 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 K\"ahler metrics with scalar curvature globally and arbitrarily $C^0$-close to the scalar curvature of $\omega$. As a consequence, we establish the sharp \(2\)-systole estimate for every positive scalar curvature K\"ahler manifold $(X,\omega)$ and prove $\min_XS(\omega) \cdot\operatorname{sys}_2(\omega) \le 2\pi r(r+1)$, where \(r\) is the rational dimension of $X$, with equality if and only if the universal cover splits as $(\widetilde X,\widetilde \omega) \cong (\mathbb P^r,\omega_{\mathrm{FS}}) \times(Y^{n-r},\omega_{\mathrm{RF}})$ up to normalization where $\omega_{\mathrm{FS}}$ is the Fubini-Study metric and $\omega_{\mathrm{RF}}$ is Ricci-flat. We also show a sharp even-systolic inequality in the same setting when the general fibre is the projective space.