向量丛的弱正性与弱平坦性
Weak positivity and weak flatness of vector bundles
- Institute of Mathematics, University of Warsaw（华沙大学数学研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究诺特环上拟射影概形向量丛的弱正性，证明其在多种运算下保持，引入弱平坦丛并建立相关定理，最终用于刻画阿贝尔簇的商。
AI中文摘要：
我们研究诺特环上拟射影概形上向量丛的Viehweg弱正性，包括混合特征情形。我们证明弱正性在张量积、对称幂、分配幂和外幂下保持。我们引入弱平坦丛，推广了数值平坦丛，并利用它们为具有小平紧化（小射影紧化）的正规簇构造S-基本群概形。我们将弱平坦性与强数值平坦性进行比较，并建立了Demailly-Peternell-Schneider定理的类似物。对于具有小平紧化的光滑复簇，我们证明弱平坦丛范畴中的半单对象恰为酉平坦丛。我们还证明，配备可积代数连接且具有酉平坦商的不变滤过的向量丛是弱平坦的。作为应用，我们根据某些标准向量丛的弱正性来刻画阿贝尔簇的商。
英文摘要：
We study Viehweg's weak positivity for vector bundles on quasi-projective schemes over noetherian rings, including mixed characteristic. We prove that weak positivity is preserved under tensor products, symmetric powers, divided powers, and exterior powers. We introduce weakly flat bundles, generalizing numerically flat bundles, and we use them to construct an S-fundamental group scheme for normal varieties admitting a small projective compactification. We compare weak flatness with strong numerical flatness and establish analogues of the Demailly-Peternell-Schneider theorem. For smooth complex varieties admitting a small compactification, we prove that the semisimple objects in the category of weakly flat bundles are precisely the unitary flat bundles. We also show that a vector bundle equipped with an integrable algebraic connection and an invariant filtration with unitary flat quotients is weakly flat. As applications, we characterize quotients of abelian varieties in terms of weak positivity of some standard vector bundles.
