AI 中文总结
研究李代数胚在满射淹没下拉回上同调相关问题，通过直接微分几何证明，利用叶状上同调消失结果及拼接论证，得出纤维特定条件下系数拉回的同构及单射情况，还证明了全纯李代数胚类似结果。
AI 中文摘要
我们给出了Crainic关于李代数胚在满射淹没下的拉回上同调结果的直接微分几何证明：若纤维连通且度数\(n\)以下的上同调消失，对任意表示系数的拉回在度数\(n\)以下是同构，在度数\(n + 1\)是单射。证明归结为叶状上同调的一个消失结果，并使用了Buchdahl的穷竭与拼接论证。不需要纤维的局部平凡性或有限型假设。我们还证明了全纯李代数胚的类似结果。
英文摘要
We give a direct differential-geometric proof of a result of Crainic on the cohomology of pullbacks of Lie algebroids under surjective submersions: If the fibers are connected and have vanishing cohomology through degree $n$, pullback with coefficients in any representation is an isomorphism through degree $n$ and is injective in degree $n+1$. The proof reduces to a vanishing result for leafwise cohomology and uses an exhaustion-and-patching argument due to Buchdahl. No local triviality or finite-type assumption on the fibers is required. We also prove the analogous result for holomorphic Lie algebroids.
