In the previous part of this diptych, we defined the notion of an admissible simplicial connection, as well as explaining how H.I. Green constructed a resolution of coherent analytic sheaves by locally free sheaves on the Čech nerve. This paper seeks to apply these abstract formalisms, by showing that Green’s barycentric simplicial connection is indeed admissible, and that this condition is exactly what we need in order to be able to apply Chern–Weil theory and construct characteristic classes. We show that, in the case of (global) vector bundles, the simplicial construction agrees with what one might construct manually: the explicit Čech representatives of the exponential Atiyah classes of a vector bundle agree. Finally, we summarise how all the preceding theory fits together to allow us to define Chern classes of coherent analytic sheaves, as well as showing uniqueness in the compact case.

在这幅双联画的前一部分中，我们定义了可接受的单纯连接的概念，并解释了 HI Green 如何通过 Çech 神经上的局部自由层构建相干分析层的分辨率。本文试图通过展示格林的重心理论来应用这些抽象形式主义单纯联系确实是可以接受的，并且这个条件正是我们能够应用 Chern-Weil 理论和构造特征类所需要的。我们表明，在（全局）向量丛的情况下，单纯构造与手动构造的构造一致：向量丛的指数 Atiyah 类的显式 Čech 代表一致。最后，我们总结了所有前面的理论如何结合在一起，使我们能够定义相干解析层的 Chern 类，并在紧的情况下显示唯一性。