Abstract
One of the most mysterious aspects of Saito’s theory of Hodge modules are the Hodge and weight filtrations that accompany the pushforward of a Hodge module under an open embedding. In this paper we consider the open embedding in a product of complementary Grassmannians given by pairs of transverse subspaces. The push-forward of the structure sheaf under this open embedding is an important Hodge module from the viewpoint of geometric representation theory and homological knot invariants. We compute the associated graded of this push-forward with respect to the induced Hodge filtration as well as the resulting weight filtration. The main tool is a categorical \({\mathfrak {sl}}_2\) action on the category of \({\mathcal {D}}_h\)-modules on Grassmannians. Along the way we also clarify the interaction of kernels for \({\mathcal {D}}_h\)-modules with the associated graded functor. Both of these results may be of independent interest.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beilinson, A., Lusztig, G., MacPherson, R.: A geometric setting for the quantum deformation of \(GL_n\). Duke Math. J. 61(2), 655–677 (1990)
Bezrukavnikov, R., Losev, I.: Etingof conjecture for quantized quiver varieties. arXiv:1309.1716
Bjork, J.: Analytic \({\cal{D}}\)-Modules and Applications, Mathematics and its Applications, vol. 247. Kluwer Academic Publishers, Berlin (1993)
Brundan, J.: On the definition of Kac-Moody 2-category. Math. Ann. 364(1–2), 353–372 (2016). arXiv:1501.00350
Cautis, S.: Equivalences and stratified flops. Compositio Math. 148(1), 185–209 (2012). arXiv:0909.0817
Cautis, S.: Rigidity in higher representation theory. arXiv:1409.0827
Cautis, S., Kamnitzer, J.: Knot homology via derived categories of coherent sheaves I, \({sl}_{2}\) case. Duke Math. J. 142(3), 511–588 (2008)
Cautis, S., Kamnitzer, J.: Knot homology via derived categories of coherent sheaves II. Invent. Math. 174(1), 165–232 (2008). arXiv:0710.3216
Cautis, S., Kamnitzer, J.: Braiding via geometric categorical Lie algebra actions. Compositio Math. 148(2), 464–506 (2012). arXiv:1001.0619
Cautis, S., Kamnitzer, J.: Knot homology via derived categories of coherent sheaves IV, coloured links. Quantum Topol. 8(2), 381–411 (2017). arXiv:1410.7156
Cautis, S., Kamnitzer, J., Licata, A.: Derived equivalences for cotangent bundles of Grassmannians via categorical \({\mathfrak{sl}}_{2}\) actions. J. Reine Angew. Math. 675, 53–99 (2013). arXiv:0902.1797
Cautis, S., Kamnitzer, J., Licata, A.: Coherent sheaves on quiver varieties and categorification. Math. Ann. 357(3), 805–854 (2013). arXiv:1104.0352
Cautis, S., Lauda, A.: Implicit structure in 2-representations of quantum groups. Selecta Math. 21(1), 201–244 (2015). arXiv:1111.1431
Chuang, J., Rouquier, R.: Derived equivalences for symmetric groups and \({sl}_{2}\)-categorification. Ann. Math. 167, 245–298 (2008)
Conrad, B.: Grothendieck Duality and Base Change. Lecture Notes in Mathematics, vol. 1750. Springer, New York (2000)
Hartshorne, R.: Residues and Duality. Lecture Notes in Mathematics, vol. 20. Springer, New York (1966)
Hotta, R., Takeuchi, K., Tanisaki, T.: \(D\)-Modules, Perverse Sheaves, and Representation Theory, Progress in Mathematics, vol. 236. Birkhauser, Boston (2008)
Lauda, A.: A categorification of quantum \({sl}_{2}\). Adv. Math. 225(6), 3327–3424 (2010). arXiv:0803.3652v2
Laumon, G.: Sur la catégorie dérivée des \({\cal{D}}\)-Modules filtrés. Lecture Notes in Mathematics, vol. 1016. Springer, Berlin (1983)
Orlov, D.: Equivalences of derived categories and K3 surfaces. J. Math. Sci. (New York) 84(5), 1361–1381 (1997)
Peters, C., Steenbrink, J.: Mixed Hodge Structures, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 52. Springer, Berlin (2008)
Saito, M.: Modules de Hodge polarisables. Publ. RIMS. Kyoto Univ. 24, 849–995 (1988)
Saito, M.: Mixed Hodge modules. Publ. RIMS. Kyoto Univ. 26, 221–333 (1990)
Saito, M.: Induced \(D\)-modules and differential complexes. Bulletin de la S.M.F. tome 117(3), 361–387 (1989)
Schnell, C.: An overview of Morihiko Saito’s theory of mixed Hodge modules. arXiv:1405.3096
The Stacks Project Authors, The Stacks Project. https://stacks.math.columbia.edu (2018)
Webster, B.: A categorical action on quantized quiver varieties. arXiv:1208.5957
Webster, B., Williamson, G.: A geometric construction of colored HOMFLYPT homology. arXiv:0905.0486
Zelevinsky, A.: Small resolutions of singularities of Schubert varieties. Funct. Anal. Appl. 17(2), 142–144 (1983)
Zheng, H.: A geometric categorification of tensor products of \(U_q({sl}_{2})\)-modules. arXiv:0705.2630
Zheng, H.: Categorification of integrable representations of quantum groups. Acta Math. Sin. 30(6), 899–932 (2014). arXiv:0803.3668
Acknowledgements
We would like to thank Sergey Arkhipov, Roman Bezrukavnikov, Thomas Bitoun, Alexander Braverman, Ivan Losev, David Nadler, Raphael Rouquier, Christian Schnell, and Ben Webster for helpful discussions. We also thank the anonymous referee for helpful suggestions. S.C. was supported by NSERC. J.K. was supported by a Sloan Fellowship and by NSERC.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Cautis, S., Dodd, C. & Kamnitzer, J. Associated graded of Hodge modules and categorical \({\mathfrak {sl}}_2\) actions. Sel. Math. New Ser. 27, 22 (2021). https://doi.org/10.1007/s00029-021-00639-0
Accepted:
Published:
DOI: https://doi.org/10.1007/s00029-021-00639-0