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Log $\mathscr{D}$-modules and index theorems

Part of: Singularities (Co)homology theory Cycles and subschemes

Published online by Cambridge University Press:  11 January 2021

Lei Wu  and
Peng Zhou
Lei Wu
Affiliation:
Department of Mathematics, University of Utah, 155 South 1400 E, Salt Lake City, UT84112, USA; E-mail: lwu@math.utah.edu.
Peng Zhou
Affiliation:
Department of Mathematics, University of California, Berkeley, 970 Evans Hall, Berkeley, CA94720-3840, USA; E-mail: pzhou.math@berkeley.edu.

Abstract

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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.

MSC classification

Primary: 14F10: Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
Secondary: 32S40: Monodromy; relations with differential equations and $D$-modules 32S60: Stratifications; constructible sheaves; intersection cohomology 14C17: Intersection theory, characteristic classes, intersection multiplicities 14C40: Riemann-Roch theorems
Type
Algebraic and Complex Geometry
Information
Forum of Mathematics, Sigma , Volume 9 , 2021 , e3
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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