Kähler爆破上的纯量曲率与收缩不等式
Scalar curvature on Kähler blow-ups and systolic inequalities
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中文总结 AI 辅助
本文发展针对特定Kähler流形的加权水平集方法，证明其爆破的Kähler度量序列的纯量曲率逼近原度量，建立正纯量曲率Kähler流形的精确2-收缩估计及偶收缩不等式。
中文摘要 AI 辅助
本文针对几乎没有有理曲线的奇异基Z上具有几乎全纯映射的Kähler流形$(X^n,\omega)$，发展了加权水平集方法。作为关键中间结果，证明X沿余维数$\operatorname{codim}S\ge2$的光滑子流形S的任意爆破$\operatorname{Bl}_SX$，都存在Kähler度量序列，其纯量曲率全局且任意$C^0$逼近$\omega$的纯量曲率。由此建立所有正纯量曲率Kähler流形$(X,\omega)$的精确2-收缩估计，证明$\min_XS(\omega)\cdot\operatorname{sys}_2(\omega)\le2\pi r(r+1)$，其中r为X的有理维数，等号当且仅当万有覆叠经正规化后分解为$(\widetilde X,\widetilde \omega)\cong(\mathbb P^r,\omega_{\mathrm{FS}})\times(Y^{n-r},\omega_{\mathrm{RF}})$，这里$\omega_{\mathrm{FS}}$是富比尼-施图迪度量，$\omega_{\mathrm{RF}}$是里奇平坦度量。还证明了当一般纤维为射影空间时该情形下的精确偶收缩不等式。
英文摘要
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.
