We study log $\mathscr {D}$ -modules on smooth log pairs and construct a comparison theorem of log de Rham complexes. The proof uses Sabbah’s generalized b-functions. As applications, we deduce a log index theorem and a Riemann-Roch type formula for perverse sheaves on smooth quasi-projective varieties. The log index theorem naturally generalizes the Dubson-Kashiwara index theorem on smooth projective varieties.

中文翻译：

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对数模块和索引定理

我们学习日志 $\mathscr {D}$ - 平滑对数对上的模块，并构建 log de Rham 复合物的比较定理。证明使用 Sabbah 的广义b-职能。作为应用，我们推导出了光滑准射影变体上反滑轮的对数指数定理和黎曼-罗赫型公式。对数索引定理自然地将 Dubson-Kashiwara 索引定理推广到光滑射影簇上。