Scalar curvature on Kähler blow-ups and systolic inequalities

In this paper, we develop the weighted level set method for a Kähler 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 ${\mathrm{Bl}}_{S}X$ of $X$ along smooth submanifolds $S$ of $\mathrm{codim}S\ge 2$ admits a sequence of Kähler 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 Kähler manifold $(X,\omega )$ and prove ${\min }_{X}S(\omega )\cdot {\mathrm{sys}}_2(\omega )\le 2\pi r(r+1)$, where $r$ is the rational dimension of $X$, with equality if and only if the universal cover splits as $(\tilde{X},\tilde{\omega })\cong ({\mathbb{P}}^r,{\omega }_{\mathrm{F}\mathrm{S}})\times (Y^{n-r},{\omega }_{\mathrm{R}\mathrm{F}})$ up to normalization where ${\omega }_{\mathrm{F}\mathrm{S}}$ is the Fubini-Study metric and ${\omega }_{\mathrm{R}\mathrm{F}}$ is Ricci-flat. We also show a sharp even-systolic inequality in the same setting when the general fibre is the projective space.