Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Vanishing and injectivity theorems for Hodge modules

Author: Lei Wu
Journal: Trans. Amer. Math. Soc. 369 (2017), 7719-7736
MSC (2010): Primary 14D07; Secondary 14F10, 14F17
Published electronically: April 7, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information


We prove a surjectivity theorem for the Deligne canonical extension of a polarizable variation of Hodge structure with quasi-unipotent monodromy at infinity along the lines of Esnault-Viehweg. We deduce from it several injectivity theorems and vanishing theorems for pure Hodge modules. We also give an inductive proof of Kawamata-Viehweg vanishing for the lowest graded piece of the Hodge filtration of a pure Hodge module using mixed Hodge modules of nearby cycles.

References [Enhancements On Off] (What's this?)

  • [1] Y. Brunebarbe, Symmetric differential forms, variations of Hodge structures and fundamental groups of complex varieties, Thèse de doctorat, (2014).
  • [2] Eduardo Cattani, Aroldo Kaplan, and Wilfried Schmid, Degeneration of Hodge structures, Ann. of Math. (2) 123 (1986), no. 3, 457-535. MR 840721,
  • [3] Hélène Esnault and Eckart Viehweg, Lectures on vanishing theorems, DMV Seminar, vol. 20, Birkhäuser Verlag, Basel, 1992. MR 1193913
  • [4] Osamu Fujino, Taro Fujisawa, and Morihiko Saito, Some remarks on the semipositivity theorems, Publ. Res. Inst. Math. Sci. 50 (2014), no. 1, 85-112. MR 3167580,
  • [5] Ryoshi Hotta, Kiyoshi Takeuchi, and Toshiyuki Tanisaki, $ D$-modules, perverse sheaves, and representation theory, Progress in Mathematics, vol. 236, Birkhäuser Boston, Inc., Boston, MA, 2008. Translated from the 1995 Japanese edition by Takeuchi. MR 2357361
  • [6] Yujiro Kawamata, Characterization of abelian varieties, Compositio Math. 43 (1981), no. 2, 253-276. MR 622451
  • [7] János Kollár, Higher direct images of dualizing sheaves. II, Ann. of Math. (2) 124 (1986), no. 1, 171-202. MR 847955,
  • [8] János Kollár, Shafarevich maps and automorphic forms, M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 1995. MR 1341589
  • [9] Robert Lazarsfeld, Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. MR 2095471
  • [10] M. Popa, Kodaira-Saito vanishing and applications, preprint, to appear in L'Enseignement Mathématique.
  • [11] Morihiko Saito, Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci. 24 (1988), no. 6, 849-995 (1989) (French). MR 1000123,
  • [12] Morihiko Saito, Mixed Hodge modules, Publ. Res. Inst. Math. Sci. 26 (1990), no. 2, 221-333. MR 1047415,
  • [13] Morihiko Saito, On Kollár's Conjecture, Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989), Proc. Sympos. Pure Math. 52 (1991), 509-517.
  • [14] Masa-Hiko Saitō, Applications of Hodge modules--Kollár conjecture and Kodaira vanishing, Sūrikaisekikenkyūsho Kōkyūroku 803 (1992), 107-124. Algebraic geometry and Hodge theory (Japanese) (Kyoto, 1991). MR 1227176
  • [15] Christian Schnell, On Saito's vanishing theorem, Math. Res. Lett. 23 (2016), no. 2, 499-527. MR 3512896,
  • [16] -, An overview of Morihiko Saito's theory of mixed Hodge modules, preprint arXiv:1405.3096 (2014).
  • [17] J. Suh, Vanishing theorems for mixed Hodge modules and applications, preprint, to appear in J. Eur. Math. Soc., 2015.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 14D07, 14F10, 14F17

Retrieve articles in all journals with MSC (2010): 14D07, 14F10, 14F17

Additional Information

Lei Wu
Affiliation: Deparment of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, Illinois 60208

Received by editor(s): October 26, 2015
Received by editor(s) in revised form: November 10, 2015
Published electronically: April 7, 2017
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society