Abstract
We construct complex projective schemes with Lyubeznik numbers of their cones depending on the choices of projective embeddings. This answers a question of G. Lyubeznik in the characteristic 0 case. It contrasts with a theorem of W. Zhang in the positive characteristic case where the Frobenius endomorphism is used. Reducibility of schemes is essential in our argument. B. Wang recently constructed examples of irreducible projective schemes (which are not normal) from our examples of reducible ones. So the question is still open in the normal singular case.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beilinson, A., Bernstein, J., Deligne, P.: Faisceaux Pervers, Astérisque, vol. 100. Soc. Math. France, Paris (1982)
Borel, A., et al.: Algebraic \(D\)-Modules, Perspectives in Math, vol. 2. Academic Press, Boston (1987)
Brodman, M.P., Sharp, R.Y.: Local Cohomology: An Algebraic Introduction with Geometric Applications, Cambridge Studies in Adv. Math. vol. 60, Cambridge University Press, Cambridge (1998)
Deligne, P.: Théorie de Hodge II. Publ. Math. IHES 40, 5–57 (1971)
Deligne, P.: Théorie de Hodge III. Publ. Math. IHES 44, 5–77 (1974)
Dimca, A.: Sheaves in Topology, Universitext. Springer, Berlin (2004)
Durfee, A.H., Saito, M.: Mixed Hodge structures on the intersection cohomology of links. Compos. Math. 76, 49–67 (1990)
Eisenbud, D.: The Geometry of Syzygies—A Second Course in Commutative Algebra and Algebraic Geometry. Springer, Berlin (2005)
García López, R.: A motivation for some local cohomologies. arXiv:1904.12216
García López, R., Sabbah, C.: Topological computation of local cohomology multiplicities. Collect. Math. 49, 317–324 (1998)
Goresky, M., MacPherson, R.: Intersection homology II. Inv. Math. 71, 77–129 (1983)
Grothendieck, A.: Eléments de géométrie algébrique II. Publ. Math. IHES 8, pp. 222 (1961)
Grothendieck, A.: On the de Rham cohomology of algebraic varieties. Publ. Math. IHES 29, 95–103 (1966)
Hartshorne, R.: Local Cohomology (A seminar given by A. Grothendieck, Harvard University, Fall, 1961), Lect. Notes in Math. vol. 41, Springer, Berlin (1967)
Hartshorne, R.: Algebraic Geometry. Springer, New York (1977)
Hernández, D.J., Núñez-Betancourt, L., Pérez, F., Witt, E.E.: Lyubeznik numbers and injective dimension in mixed characteristic. Trans. Am. Math. Soc. 371, 7533–7557 (2019)
Huneke, C.L., Sharp, R.Y.: Bass numbers of local cohomology modules. Trans. Am. Math. Soc. 339, 765–779 (1993)
Iyengar, S.B.: et al., Twenty-Four Hours of Local Cohomology, Graduate Studies in Math. vol. 87, American Mathematical Society, Providence (2007)
Kashiwara, M.: The Riemann-Hilbert problem for holonomic systems. Publ. RIMS, Kyoto Univ. 20, 319–365 (1984)
Kashiwara, M., Kawai, T.: On holonomic systems of microdifferential equations III. Systems with regular singularities. Publ. RIMS, Kyoto Univ. 17, 813–979 (1981)
Kashiwara, M., Schapira, P.: Sheaves on Manifolds. Springer, Berlin (1994)
Kochmann, S.: Bordism, Stable Homotopy and Adams Spectral Sequences. American Mathematical Society, Providence (1996)
Lyubeznik, G.: Finiteness properties of local cohomology modules (an application of \(D\)-modules to commutative algebra). Inv. Math. 113, 41–55 (1993)
Lyubeznik, G.: A Partial Survey of Local Cohomology, Local Cohomology and its Applications (Guanajuato, 1999), Lect. Notes in Pure and Appl. Math. vol. 226, pp. 121–154. Dekker, New York (2002)
Mebkhout, Z.: Une autre équivalence de catégories. Compos. Math. 51, 63–88 (1984)
Núñez-Betancourt, L., Witt, E.E.: Lyubeznik numbers in mixed characteristic. Math. Res. Lett. 20, 1125–1143 (2013)
Núñez-Betancourt, L., Witt, E.E., Zhang, W.: A survey on the Lyubeznik numbers. Contemp. Math. 657, 154–181 (2016)
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)
Switala, N.: Lyubeznik numbers for nonsingular projective varieties. Bull. Lond. Math. Soc. 47, 1–6 (2015)
Switzer, R.M.: Algebraic Topology-Homotopy and Homology, Classics in Mathematics. Springer, Berlin (2002)
Verdier, J.-L.: Dualité dans la cohomologie des espaces localement compacts, Séminaire Bourbaki, Vol. 9, Exp. No. 300, pp. 337–349. Soc. Math. France, Paris (1995)
Verdier, J.-L.: Des catégories dérivées des catégories abéliennes. Astérisque 239, pp. 253 (1996)
Wang, B.: Lyubeznik numbers of irreducible projective varieties depend on the embedding. Proc. Am. Math. Soc. 148, 2091–2096 (2020)
Zhang, W.: Lyubeznik numbers of projective schemes. Adv. Math. 228, 575–616 (2011)
Acknowledgements
The first named author was supported by a DFG Emmy-Noether-Fellowship (RE 3567/1-1) and acknowledges partial support by the project SISYPH: ANR-13-IS01-0001-01/02 and DFG Grant HE 2287/4-1 & SE 1114/5-1. He would like to thank Duco van Straten for a stimulating discussion. The second named author is partially supported by Kakenhi 15K04816. The third named author is supported in part by NFS Grant DMS-1401392 and by Simons Foundation Collaboration Grant for Mathematicians #580839. He thanks Nick Switala for stimulating conversations. The authors thank the referees for their useful comments to improve the paper.
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
Reichelt, T., Saito, M. & Walther, U. Dependence of Lyubeznik numbers of cones of projective schemes on projective embeddings. Sel. Math. New Ser. 27, 6 (2021). https://doi.org/10.1007/s00029-020-00612-3
Accepted:
Published:
DOI: https://doi.org/10.1007/s00029-020-00612-3