Abstract
Abstract 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.
Highlights
Our first theorem is a comparison between a -module M and its lattice M in terms of de Rham (DR) complexes (see (2.1) for its definition)
In the case that M =, the result (1.3) has an application in [BVWZ19] to prove a conjecture of Budur [Bud15] about zero loci of Bernstein-Sato ideals, which generalizes a classical theorem of Malgrange and Kashiwara relating the -function of a multivariate polynomial with the monodromy eigenvalues on the Milnor fibers cohomology
Using -functions for lattices, we show that DR, (M( )) stabilizes for large
Summary
Let ( , ) be a smooth log pair; that is, is a smooth variety over C, and is a reduced normal crossing divisor. Denote the open embedding by : = \ ↩→. The sheaf of log differential operators , is the subsheaf of consisting of differential operators that preserve the defining ideal of the divisor. Log -modules are (left or right) modules over ,. We mainly focus on studying log -modules associated with -modules, called lattices (Definition 2.4), and we use them to study perverse sheaves on
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