Hostname: page-component-f554764f5-wjqwx Total loading time: 0 Render date: 2025-04-10T12:06:42.192Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected].
Peng Zhou
Affiliation:
Department of Mathematics, University of California, Berkeley, 970 Evans Hall, Berkeley, CA94720-3840, USA; E-mail: [email protected].

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

Baum, Paul, Fulton, William, and MacPherson, Robert, Riemann-Roch for singular varieties, Publications Mathématiques de l'Institut des Hautes Études Scientifiques 45 (1975), no. 1, 101–145.CrossRefGoogle Scholar
Beilinson, Alexander, Bernstein, Joseph and Deligne, Pierre, Faisceaux pervers, Astérisque 100 (1982), 3–171.Google Scholar
Beilinson, Alexander and Gaitsgory, Dennis, A corollary of the b-function lemma, Selecta Mathematica 18 (2012), no. 2, 319–327.CrossRefGoogle Scholar
Bjork, Jan-Erik, Analytic $D$-modules and applications, Mathematics and Its Applications, vol. 247, Springer Science & Business Media, Dordrecht, 2013. MR 1232191Google Scholar
Budur, Nero, Bernstein-sato ideals and local systems, Annales de l'Institut Fourier 65 (2015), no. 2, 549–603 (en).CrossRefGoogle Scholar
Budur, Nero, Veer, Robin, Wu, Lei, and Zhou, Peng, Zero loci of Bernstein-Sato ideals, arXiv preprint arXiv:1907.04010 (2019).Google Scholar
Deligne, Pierre, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, Vol. 163, Springer-Verlag, Berlin-New York, 2006. MR 0417174Google Scholar
Dubson, Alberto S., Classes caractéristiques des variétés singulières, C. R. Acad. Sci. Paris Sér. A-B 287 (1978), no. 4, A237–A240. MR 0499290Google Scholar
Franecki, J. and Kapranov, M., The Gauss map and a noncompact Riemann-Roch formula for constructible sheaves on semiabelian varieties, Duke Math. J. 104 (2000), no. 1, 171–180. MR 1769729Google Scholar
Fulton, William, Intersection Theory, second ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas]. 3rd Series. vol. 2, Springer Science & Business Media, Berlin, 2013. MR 1644323Google Scholar
Ginsburg, Victor, Characteristic varieties and vanishing cycles, Invent. Math. 84 (1986), no. 2, 327–402. MR 833194CrossRefGoogle Scholar
Ginsburg, Victor, Admissible modules on a symmetric space, Astérisque (1989), no. 173 -174, 9–10, 199–255, Orbites unipotentes et représentations, III. MR 1021512Google Scholar
Hotta, Ryoshi, Takeuchi, Kiyoshi, and Tanisaki, Toshiyuki, $D$-modules, perverse sheaves, and representation theory, Progress in Mathematics, vol. 236, Springer Science & Business Media, 2007, Translated from the 1995 Japanese edition by Takeuchi. MR 2357361Google Scholar
Kashiwara, Masaki, Index theorem for constructible sheaves, Systèmes différentiels et singularités, Astérisque, no. 130, Société mathématique de France, 1985, pp. 193–209 (en). MR 804053Google Scholar
Kashiwara, Masaki, $B$-functions and holonomic systems. Rationality of roots of $B$-functions, Invent. Math. 38 (1976/77), no. 1, 33–53. MR 0430304CrossRefGoogle Scholar
Koppensteiner, Clemens and Talpo, Mattia, Holonomic and perverse logarithmic d-modules, Advances in Mathematics 346 (2019), 510–545.CrossRefGoogle Scholar
Laumon, G., Sur la catégorie dérivée des $D$-modules filtrés, Algebraic geometry (Tokyo/Kyoto, 1982), Lecture Notes in Math., vol. 1016, Springer, Berlin, 1983, pp. 151–237. MR 726427Google Scholar
Maisonobe, Philippe, Filtration relative, l'Idéal de Bernstein et ses pentes, arXiv preprint arXiv:1610.03354 (2016).Google Scholar
Malgrange, B., Polynômes de Bernstein-Sato et cohomologie évanescente, Analysis and topology on singular spaces, II, III (Luminy, 1981), Astérisque, vol. 101, Soc. Math. France, Paris, 1983, pp. 243–267. MR 737934Google Scholar
Nadler, D. and Zaslow, E., Constructible sheaves and the Fukaya category, J. Amer. Math. Soc. 22 (2009), 233–286.CrossRefGoogle Scholar
Sabbah, C., Proximité évanescente. I. La structure polaire d'un $D$-module, Compositio Math. 62 (1987), no. 3, 283–328. MR 901394Google Scholar
Sabbah, C., Proximité évanescente. II. Équations fonctionnelles pour plusieurs fonctions analytiques, Compositio Math. 64 (1987), no. 2, 213–241. MR 916482Google Scholar