Abstract
Let X be a closed equidimensional local complete intersection subscheme of a smooth projective scheme Y over a field, and let \(X_t\) denote the t-th thickening of X in Y. Fix an ample line bundle \(\mathcal {O}_Y(1)\) on Y. We prove the following asymptotic formulation of the Kodaira vanishing theorem: there exists an integer c, such that for all integers \(t \geqslant 1\), the cohomology group \(H^k(X_t,\mathcal {O}_{X_t}(j))\) vanishes for \(k < \dim X\) and \(j < -ct\). Note that there are no restrictions on the characteristic of the field, or on the singular locus of X. We also construct examples illustrating that a linear bound is indeed the best possible, and that the constant c is unbounded, even in a fixed dimension.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arapura, D., Jaffe, D.B.: On Kodaira vanishing for singular varieties. Proc. Am. Math. Soc. 105, 911–916 (1989)
Bhatt, B., Blickle, M., Lyubeznik, G., Singh, A.K., Zhang, W.: Stabilization of the cohomology of thickenings. Am. J. Math. 141, 531–561 (2019)
Cowsik, R.C., Nori, M.V.: On the fibres of blowing up. J. Indian Math. Soc. 40, 217–222 (1976)
Cutkosky, S.D.: Asymptotic multiplicities of graded families of ideals and linear series. Adv. Math. 264, 55–113 (2014)
Cutkosky, S.D., Hà, H.T., Srinivasan, H., Theodorescu, E.: Asymptotic behavior of the length of local cohomology. Can. J. Math. 57, 1178–1192 (2005)
Dao, H., Montaño, J.: Length of local cohomology of powers of ideals. Trans. Am. Math. Soc. 371, 3483–3503 (2019)
Dao, H., Montaño, J.: On asymptotic vanishing behavior of local cohomology. Math. Z. 295, 73–86 (2020)
Illusie, L.: Complexe Cotangent et Déformations. I. Lecture Notes in Mathematics, vol. 239. Springer, Berlin (1971)
Lauritzen, N., Rao, A.P.: Elementary counterexamples to Kodaira vanishing in prime characteristic. Proc. Indian Acad. Sci. Math. Sci. 107, 21–25 (1997)
Lurie, J.: Spectral algebraic geometrie. https://www.math.ias.edu/~lurie/
Quillen, D.: Homology of Commutative Rings. Mimeographed Notes. MIT Press, New York (1968)
Raynaud, M.: Contre-exemple au “vanishing theorem” en caractéristique \(p>0\). C. P. Ramanujam—A Tribute, Tata Institute of Fundamental Research. Studies in Mathematics, vol. 8, pp. 273–278. Springer, Berlin (1978)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Vasudevan Srinivas.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
B.B. was supported by NSF Grant DMS 1801689, a Packard fellowship, and the Simons Foundation Grant 622511, M.B. by DFG Grant SFB/TRR45, G.L. by NSF Grant DMS 1800355, A.K.S. by NSF Grant DMS 1801285, and W.Z. by NSF Grant DMS 1752081. The authors are grateful to the Institute for Advanced Study for support through the Summer Collaborators Program, to the American Institute of Mathematics for support through the SQuaREs Program, and to the referee for several helpful comments.
Rights and permissions
About this article
Cite this article
Bhatt, B., Blickle, M., Lyubeznik, G. et al. An asymptotic vanishing theorem for the cohomology of thickenings. Math. Ann. 380, 161–173 (2021). https://doi.org/10.1007/s00208-020-02140-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-020-02140-z