Abstract:We will show that the singular cohomology groups of a smooth quasi-projective complex variety relative to a normal crossing divisor can be described in terms of $\delta$-admissible chains. Roughly speaking, a $\delta$-admissible chain is a simplicial semi-algebraic chain meeting the “faces” properly. As an application, we show that the Abel-Jacobi map for higher Chow cycles can be described via $\delta$-admissible chains. As an example, we will describe the Hodge realization of the polylog cycles constructed by Bloch-Kriz in terms of the Abel-Jacobi map.

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中文翻译：

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射影变体的半代数链和更高周循环的 Abel-Jacobi 映射

摘要：我们将证明，相对于正态交叉因数的光滑准射影复变体的奇异上同调群可以用 $\delta$ 容许链来描述。粗略地说，$\delta$-admissible chain 是一个简单的半代数链，正确地满足“faces”。作为一个应用，我们展示了更高周循环的 Abel-Jacobi 映射可以通过 $\delta$-admissible 链来描述。作为一个例子，我们将根据 Abel-Jacobi 映射描述由 Bloch-Kriz 构建的 polylog 循环的 Hodge 实现。

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