Esnault, Hélène; Sabbah, Claude

Good Lattices of Algebraic Connections (with an Appendix by Claude Sabbah)

Doc. Math. 24, 271-301 (2019)
DOI: 10.25537/dm.2019v24.271-301
Communicated by Takeshi Saito

Summary

We construct a logarithmic model of connections on smooth quasi-projective \(n\)-dimensional geometrically irreducible varieties defined over an algebraically closed field of characteristic \(0\). It consists of a good compactification of the variety together with \((n+1)\) lattices on it which are stabilized by log differential operators, and compute algebraically de Rham cohomology. The construction is derived from the existence of good Deligne-Malgrange lattices, a theorem of Kedlaya and Mochizuki which consists first in eliminating the turning points. Moreover, we show that a logarithmic model obtained in this way, called a good model, yields a formula predicted by Michael Groechenig, computing the class of the characteristic variety of the underlying \(\mathcal{D}\)-module in the \(K\)-theory group of the variety.

Mathematics Subject Classification

14F40, 14F10, 32C38

Keywords/Phrases

flat connection, good lattice, good model, de Rham cohomology, characteristic variety

References

  • 1. Beilinson A., Bloch, S., Esnault, H.: $\epsilon$-factors for Gauß-Manin determinants, Moscow Mathematical Journal, 2 no. 3 (2002), 1-56. zbl 1061.14010; MR1988970.
  • 2. Beilinson, A., Bloch, S., Deligne, P., Esnault, H.: Periods for irregular connections on curves, preprint 2003, 59 pages, unpublished, http://www.mi.fu-berlin.de/users/esnault/preprints/helene/69-preprint-per051206.pdf.
  • 3. Bloch, S., Esnault, H.: Local Fourier transforms and rigidity for D-modules, Asian Journal of Mathematics, special issue dedicated to Armand Borel, 8 no. 4 (2004), 587-606. DOI 10.4310/AJM.2004.v8.n4.a16; zbl 1082.14506; MR2127940; arxiv math/0312343.
  • 4. Deligne, P.: Équations différentielles à points singuliers réguliers, Lecture Notes in Math. 163, Springer Verlag, 1970. DOI 10.1007/BFb0061194; zbl 0244.14004; MR0417174.
  • 5. Laumon, G.: Sur la catégorie dérivée des $\mathcal{D}$-modules filtrés, in Algebraic geometry (Tokyo/Kyoto, 1982), Lect. Notes in Math. 1016, Springer-Verlag, 1983, 151-237. zbl 0551.14006; MR0726427.
  • 6. Kashiwara, M.: $D$-modules and microlocal calculus, Translations of Mathematical Monographs 217, American Mathematical Society, Providence, R.I., 2003. zbl 1017.32012; MR1943036.
  • 7. Kawamata, Y.: Characterization of abelian varieties, Compos. Math. 43 (1981), 253-276. zbl 0471.14022; MR0622451.
  • 8. Kedlaya, K.: Good formal structures for flat meromorphic connections, II: excellent schemes, J. Amer. Math. Soc. 24 (2011), 183-229. DOI 10.1090/S0894-0347-2010-00681-9; zbl 1282.14037; MR2726603 arxiv 1001.0544.
  • 9. Malgrange, B.: Connexions méromorphes, II: le réseau canonique, Invent. Math. 124 (1996), 367-387. DOI 10.1007/s002220050057; zbl 0849.32003; MR1369422.
  • 10. Matsuki, K.: Introduction to the Mori program, Universitext, Springer Verlag, 2002. zbl 0988.14007; MR1875410.
  • 11. Mochizuki, T.: On Deligne-Malgrange lattices, resolution of turning points and harmonic bundles, Annales de l'Institut Fourier 59 no. 7 (2009), 2819-2837. DOI 10.5802/aif.2509; zbl 1202.32008; MR2649340.
  • 12. Mochizuki, T.: Wild harmonic bundles and wild pure twistor D-modules, Astérisque 340, Société Mathématique de France, Paris 2011. zbl 1245.32001; MR2919903; arxiv 0803.1344.
  • 13. Mochizuki, T.: Mixed twistor D-Modules, Lect. Notes in Math. 2125, Springer, Heidelberg, New York, 2015. DOI 10.1007/978-3-319-10088-3; zbl 1356.32002; MR3381953; arxiv 1104.3366.
  • 14. Sabbah, C.: \\'Equations différentielles à points singuliers irréguliers et phénomènes de Stokes en dimension $2$, Astérisque 263, Société Mathématique de France, Paris, 2000. MR1741802.
  • 15. Sabbah, C., Schnell, C.: The MHM project, http://www.math.polytechnique.fr/~sabbah/MHMProject/mhm.html.
  • 16. Wei, C.: Logarithmic comparison with smooth boundary divisor in mixed Hodge modules, eprint, 2017. arxiv 1710.07407.

Affiliation

Esnault, Hélène
Freie Universität Berlin, Mathematik, Arnimallee 3, 14195 Berlin, Germany
Sabbah, Claude
CMLS, École Polytechnique, CNRS, Université Paris-Saclay, F-91128 Palaiseau cedex, France

Downloads