Vanishing and injectivity theorems for Hodge modules

Wu, Lei

doi:10.1090/tran/6869

Vanishing and injectivity theorems for Hodge modules
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Trans. Amer. Math. Soc. 369 (2017), 7719-7736 Request permission

Abstract:

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

Y. Brunebarbe, Symmetric differential forms, variations of Hodge structures and fundamental groups of complex varieties, Thèse de doctorat, (2014).
Eduardo Cattani, Aroldo Kaplan, and Wilfried Schmid, Degeneration of Hodge structures, Ann. of Math. (2) 123 (1986), no. 3, 457–535. MR 840721, DOI 10.2307/1971333
Hélène Esnault and Eckart Viehweg, Lectures on vanishing theorems, DMV Seminar, vol. 20, Birkhäuser Verlag, Basel, 1992. MR 1193913, DOI 10.1007/978-3-0348-8600-0
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, DOI 10.4171/PRIMS/125
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, DOI 10.1007/978-0-8176-4523-6
Yujiro Kawamata, Characterization of abelian varieties, Compositio Math. 43 (1981), no. 2, 253–276. MR 622451
János Kollár, Higher direct images of dualizing sheaves. II, Ann. of Math. (2) 124 (1986), no. 1, 171–202. MR 847955, DOI 10.2307/1971390
János Kollár, Shafarevich maps and automorphic forms, M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 1995. MR 1341589, DOI 10.1515/9781400864195
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, DOI 10.1007/978-3-642-18808-4
M. Popa, Kodaira-Saito vanishing and applications, preprint, to appear in L’Enseignement Mathématique.
Morihiko Saito, Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci. 24 (1988), no. 6, 849–995 (1989) (French). MR 1000123, DOI 10.2977/prims/1195173930
Morihiko Saito, Mixed Hodge modules, Publ. Res. Inst. Math. Sci. 26 (1990), no. 2, 221–333. MR 1047415, DOI 10.2977/prims/1195171082
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.
Masa-Hiko Sait\B{o}, Applications of Hodge modules—Kollár conjecture and Kodaira vanishing, Sūrikaisekikenkyūsho K\B{o}kyūroku 803 (1992), 107–124. Algebraic geometry and Hodge theory (Japanese) (Kyoto, 1991). MR 1227176
Christian Schnell, On Saito’s vanishing theorem, Math. Res. Lett. 23 (2016), no. 2, 499–527. MR 3512896, DOI 10.4310/MRL.2016.v23.n2.a10
—, An overview of Morihiko Saito’s theory of mixed Hodge modules, preprint arXiv:1405.3096 (2014).
J. Suh, Vanishing theorems for mixed Hodge modules and applications, preprint, to appear in J. Eur. Math. Soc., 2015.