AI 中文总结
研究o-极小几何中复曲线相关问题，通过分析可定义全纯函数边界行为，证明非紧曲线的上同调集中在穿孔处，计算仿射直线上结构层的上同调，刻画可定义黎曼曲面的可定义紧化。
AI 中文摘要
最近在将o-极小性与复解析几何联系起来方面取得了相当大的进展。然而，即使对于曲线，关于凝聚上同调或向量丛的分类几乎一无所知。在实解析结构$\mathbb{R}_{\mathrm{an}}$及类似结构中，我们表明非紧曲线的上同调完全集中在穿孔处。作为应用，我们计算了仿射直线上结构层的上同调，并描述了与丢番图逼近的联系。最后，我们使用类似技术刻画了哪些可定义的黎曼曲面具有可定义的紧化。证明基于对可定义全纯函数边界行为的仔细分析。
英文摘要
There has recently been considerable progress relating o-minimality to complex analytic geometry. Yet almost nothing is known about coherent cohomology or the classification of vector bundles, even for curves. In $\mathbb{R}_{\mathrm{an}}$ and similar structures, we show that cohomology of noncompact curves is concentrated entirely at punctures. As an application, we compute the cohomology of the structure sheaf on the affine line and describe a connection to Diophantine approximation. Finally, we use similar techniques to characterize which definable Riemann surfaces have definable compactifications. The proofs are based on a careful analysis of boundary behavior for definable holomorphic functions.
CommentsUpdated to reflect that question 3.17 was answered; other minor changes
