AI Chat Paper
Note: Please note that the following content is generated by AMiner AI. SciOpen does not take any responsibility related to this content.
{{lang === 'zh_CN' ? '文章概述' : 'Summary'}}
{{lang === 'en_US' ? '中' : 'Eng'}}
Chat more with AI
Powered by AMiner. Clicking this button will take you away from SciOpen.
Home Peking Mathematical Journal Article
Article Link
Cite
EndNote(RIS) BibTeX
Collect
Show Outline
Outline
Show full outline
Hide outline
Outline
Show full outline
Hide outline
Original Article

De Rham Comparison and Poincaré Duality for Rigid Varieties

University of Minnesota, 127 Vincent Hall, 206 Church Street SE, Minneapolis, MN 55455, USA
Beijing International Center for Mathematical Research, Peking University, 5 Yi He Yuan Road, Beijing 100871, China
California Institute of Technology, 1200 East California Boulevard, Pasadena, CA 91125, USA
Show Author Information

Abstract

Over any smooth algebraic variety over a p-adic local field k, we construct the de Rham comparison isomorphisms for the étale cohomology with partial compact support of de Rham Zp-local systems, and show that they are compatible with Poincaré duality and with the canonical morphisms among such cohomology. We deduce these results from their analogues for rigid analytic varieties that are Zariski open in some proper smooth rigid analytic varieties over k. In particular, we prove finiteness of étale cohomology with partial compact support of any Zp-local systems, and establish the Poincaré duality for such cohomology after inverting p.

Keywords

De Rham comparisonétale cohomologyRigid analytic varietyPoincaré duality

References

1
André, Y., Baldassarri, F.: De Rham Cohomology of Differential Modules on Algebraic Varieties. Progress in Mathematics, Vol. 189. Birkhäuser, Basel (2001)
Crossref
2
André, Y., Baldassarri, F., Cailotto, M.: De Rham Cohomology of Differential Modules on Algebraic Varieties, 2nd Edition. Progress in Mathematics, Vol. 189. Birkhäuser, Basel (2020)
Crossref
3
Artin, M., Grothendieck, A., Verdier, J.-L.: (eds.), Théorie des topos et cohomologie étale des schémas (SGA 4). Lecture Notes in Mathematics, Vols. 269, 270, 305. Springer-Verlag, Berlin-New York, 1972–1973
Crossref
4
Beilinson, A., Bernstein, J., Deligne, P., Gabber, O.: Faisceaux pervers, 2nd Edition. Astérisque, Vol. 100. Société Mathématique de France, Paris (2018)
5
Bernstein, I.N., Gelfand, I.M., Gelfand, S.I.: Differential operators on the base affine space and a study of g-modules. In: Lie Groups and Their Representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971), Halsted, New York, 21–64 (1975)
6

Beyer, P.: On Serre-duality for coherent sheaves on rigid-analytic spaces. Manuscr. Math. 93(2), 219–245 (1997)
Crossref Google Scholar
7

Bierstone, E., Milman, P.D.: Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant. Invent. Math. 128(2), 207–302 (1997)
Crossref Google Scholar
8

Borel, A.: Stable real cohomology of arithmetic groups. Ann. Sci. Ecole Norm. Sup. (4) 7, 235–272 (1974)
Crossref Google Scholar
9
Borel, A.: Stable real cohomology of arithmetic groups, Ⅱ, Manifolds and Lie groups. In: Manifolds and Lie Groups (Notre Dame, Ind., 1980), Progress in Mathematics, Vol. 14, Birkhäuser, Boston, 21–55 (1981)
Crossref
10
Borel, A., Wallach, N.: Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, 2nd Edition. Mathematical Surveys and Monographs, Vol. 67. American Mathematical Society, Providence (2000)
Crossref
11
Brodmann, M.P., Sharp, R.Y.: Local Cohomology: An Algebraic Introduction with Geometric Applications. Cambridge Studies in Advanced Mathematics, Vol. 60. Cambridge University Press, Cambridge (1998)
Crossref
12

Colmez, P., Dospinescu, G., Hauseux, J., Nizioł, W.: p-adic étale cohomology of period domains. Math. Ann. 381, 105–180 (2021)
Crossref Google Scholar
13

Conrad, B.: Irreducible components of rigid spaces. Ann. Inst. Fourier (Grenoble) 49(2), 473–541 (1999)
Crossref Google Scholar
14

Conrad, B.: Relative ampleness in rigid geometry. Ann. Inst. Fourier (Grenoble) 56(4), 1049–1126 (2006)
Crossref Google Scholar
15
Deligne, P.: Équations différentielles à points singuliers réguliers. Lecture Notes in Mathematics, Vol. 163. Springer-Verlag, Berlin (1970)
Crossref
16

Deligne, P., Illusie, L.: Relèvements modulo p2 et décomposition du complexe de de Rham. Invent. Math. 89(2), 247–270 (1987)
Crossref Google Scholar
17
Diao, H.S., Lan, K.-W., Liu, R.C., Zhu, X.W.: Logarithmic adic spaces: some foundational results, arXiv: 1912.09836v2 (2022)
Crossref
18
Diao, H.S., Lan, K.-W., Liu, R.C., Zhu, X.W.: Logarithmic Riemann–Hilbert correspondences for rigid varieties. arXiv: 1803.05786v4 (to appear in J. Am. Math. Soc.)
19
Esnault, H., Viehweg, E.: Lectures on Vanishing Theorems. DMV Seminar, Vol. 20. Birkhäuser Verlag, Basel (1992)
Crossref
20
Faltings, G.: On the cohomology of locally symmetric hermitian spaces. In: Séminaire d’Algèbre Paul Dubreil et Marie–Paule Malliavin, 35ème année (Paris, 1982), Lecture Notes in Mathematics, Vol. 1029, Springer-Verlag, Berlin, 55–98 (1983)
Crossref
21
Faltings, G.: Crystalline cohomology and p-adic Galois-representations. In: Algebraic Analysis, Geometry, and Number Theory (Baltimore, MD, 1988), Johns Hopkins Univ. Press, Baltimore, 25–80 (1989)
22
Faltings, G.: Almost étale extensions. In: Cohomologies p-adiques et applications arithmétiques (Ⅱ), Astérisque, No. 279, Société Mathématique de France, Paris, 185–270 (2002)
23
Faltings, G., Chai, C.-L.: Degeneration of Abelian Varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Vol. 22. Springer-Verlag, Berlin (1990)
Crossref
24

Harris, M.: Arithmetic vector bundles and automorphic forms on Shimura varieties. I. Invent. Math. 82(1), 151–189 (1985)
Crossref Google Scholar
25

Harris, M.: Functorial properties of toroidal compactifications of locally symmetric varieties. Proc. Lond. Math. Soc. (3) 59(1), 1–22 (1989)
Crossref Google Scholar
26
Harris, M.: Automorphic forms and the cohomology of vector bundles on Shimura varieties. In: Automorphic Forms, Shimura Varieties, and L-Functions. Vol. Ⅱ (Ann Arbor, MI, 1988), Perspectives in Mathematics, Vol. 11, Academic Press Inc., Boston, 41–91 (1990)
27
Harris, M., Lan, K.-W., Taylor, R., Thorne, J.: On the rigid cohomology of certain Shimura varieties. Res. Math. Sci. 3, Art. No. 37, 308 pp. (2016)
Crossref
28

Harris, M., Taylor, R.: The Geometry and Cohomology of Some Simple Shimura Varieties. Annals of Mathematics Studies, Vol. 151. Princeton University Press, Princeton (2001)
Crossref
29
Harris, M., Zucker, S.: Boundary Cohomology of Shimura Varieties Ⅲ. Coherent Cohomology on Higher-Rank Boundary Strata and Applications to Hodge Theory. Mémoires de la Société Mathématique de France (Nouvelle Série), Vol. 85. Société Mathématique de France, Paris (2001)
Crossref
30
Hartshorne, R.: Residues and Duality. Lecture Notes in Mathematics, Vol. 20. Springer-Verlag, Berlin (1966)
Crossref
31
Hartshorne, R.: Local Cohomology. Lecture Notes in Mathematics, Vol. 41. Springer-Verlag, Berlin (1967)
Crossref
32

Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero: Ⅰ. Ann. Math. (2) 79(1), 109–203 (1964)
Crossref Google Scholar
33

Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero: Ⅱ. Ann. Math. (2) 79(2), 205–326 (1964)
Crossref Google Scholar
34
Huber, R.: Étale Cohomology of Rigid Analytic Varieties and Adic Spaces. Aspects of Mathematics, Vol. E30. Friedr. Vieweg & Sohn, Braunschweig (1996)
Crossref
35

Huber, R.: A comparison theorem for ℓ-adic cohomology. Compos. Math. 112(2), 217–235 (1998)
Crossref Google Scholar
36

Jannsen, U.: Continuous étale cohomology. Math. Ann. 280(2), 207–245 (1988)
Crossref Google Scholar
37
Kashiwara, M., Schapira, P.: Sheaves on Manifolds. Grundlehren der Mathematischen Wissenschaften, Vol. 292. Springer-Verlag, Berlin (1990)
Crossref
38

Köpf, U.: Über eigentliche Familien algebraischer Varietäten über affinoiden Räumen. Schr. Math. Inst. Univ. Münster (2) 7, ⅳ+72 pp. (1974)
Google Scholar
39

Lan, K.-W.: Comparison between analytic and algebraic constructions of toroidal compactifications of PEL-type Shimura varieties. J. Reine Angew. Math. 664, 163–228 (2012)
Crossref Google Scholar
40

Lan, K.-W.: Toroidal compactifications of PEL-type Kuga families. Algebra Number Theory 6(5), 885–966 (2012)
Crossref Google Scholar
41
Lan, K.-W.: Vanishing theorems for coherent automorphic cohomology. Res. Math. Sci. 3, Art. No. 39, 43 pp. (2016)
Crossref
42

Lan, K.-W., Polo, P.: Dual BGG complexes for automorphic bundles. Math. Res. Lett. 25(1), 85–141 (2018)
Crossref Google Scholar
43
Lan, K.-W., Stroh, B.: Compactifications of subschemes of integral models of Shimura varieties. Forum Math. Sigma 6, e18, 105 pp. (2018)
Crossref
44

Lan, K.-W., Stroh, B.: Nearby cycles of automorphic étale sheaves. Compos. Math. 154(1), 80–119 (2018)
Crossref Google Scholar
45

Li, J.-S., Schwermer, J.: On the Eisenstein cohomology of arithmetic groups. Duke Math. J. 123(1), 141–169 (2004)
Crossref Google Scholar
46

Liu, R.C., Zhu, X.W.: Rigidity and a Riemann-Hilbert correspondence for p-adic local systems. Invent. Math. 207(1), 291–343 (2017)
Crossref Google Scholar
47

Looijenga, E.: L2-cohomology of locally symmetric varieties. Compos. Math. 67(1), 3–20 (1988)
Google Scholar
48
Looijenga, E., Rapoport, M.: Weights in the local cohomology of a Baily–Borel compactification. In: Complex Geometry and Lie Theory (Sundance, UT, 1989), Proceedings of Symposia in Pure Mathematics, Vol. 53, Springer-Verlag, Berlin, 223–260 (1991)
Crossref
49
Milne, J.S.: Canonical models of (mixed) Shimura varieties and automorphic vector bundles. In: Automorphic Forms, Shimura Varieties, and L-Functions. Vol. Ⅰ (Ann Arbor, MI, 1988), Perspectives in Mathematics, Vol. 10, Academic Press, Boston, 283–414 (1990)
50

Mumford, D.: Hirzebruch’s proportionality theorem in the non-compact case. Invent. Math. 42, 239–272 (1977)
Crossref Google Scholar
51

Nagata, M.: Imbedding of an abstract variety in a complete variety. J. Math. Kyoto Univ. 2, 1–10 (1962)
Crossref Google Scholar
52
Pink, R.: Arithmetic Compactification of Mixed Shimura Varieties. Ph.D. Thesis, Rheinischen Friedrich-Wilhelms-Universität, Bonn (1989)
53

Saito, M.: Modules de Hodge polarisables. Publ. Res. Inst. Math. Sci. 24(6), 849–995 (1988)
Crossref Google Scholar
54

Saito, M.: Mixed Hodge modules. Publ. Res. Inst. Math. Sci. 26(2), 221–333 (1990)
Crossref Google Scholar
55

Saper, L., Stern, M.: L2-cohomology of arithmetic varieties. Ann. Math. (2) 132(1), 1–69 (1990)
Crossref Google Scholar
56

Scholze, P.: Perfectoid spaces. Publ. Math. Inst. Hautes Étud. Sci. 116, 245–313 (2012)
Crossref Google Scholar
57
Scholze, P.: p-adic Hodge theory for rigid-analytic varieties. Forum Math. Pi 1, e1, 77 pp. (2013)
Crossref
58
Scholze, P., Weinstein, J.: Berkeley Lectures on p-adic Geometry, Annals of Mathematics Studies, Vol. 207. Princeton University Press, Princeton (2020)
Crossref
59

Schwermer, J.: Eisenstein series and cohomology of arithmetic groups: the generic case. Invent. Math. 116, 481–511 (1994)
Crossref Google Scholar
60

Serre, J.-P.: Geométrie algébrique et géométrie analytique, Ann. Inst. Fourier (Grenoble) 6, 1–42 (1955/1956)
Crossref Google Scholar
61
Su, J.: Coherent Cohomology of Shimura Varieties and Automorphic Forms. Ph.D. Thesis, Princeton University, Princeton (2019)
62

van der Put, M.: Serre duality for rigid analytic spaces. Indag. Math. (N.S) 3(2), 219–235 (1992)
Crossref Google Scholar
63

Zucker, S.: L2 cohomology of warped products and arithmetic groups. Invent. Math. 70(2), 169–218 (1982)
Crossref Google Scholar
Peking Mathematical Journal
Volume 6 Issue 1,
March 2023
Pages 143-216
DOI: 10.1007/s42543-020-00031-5
Cite this article:
Lan, KW., Liu, R. & Zhu, X. De Rham Comparison and Poincaré Duality for Rigid Varieties. Peking Math J 6, 143-216 (2023). https://doi.org/10.1007/s42543-020-00031-5

178

Views

2

Crossref

Altmetrics
Received: 09 February 2020
Revised: 22 August 2020
Accepted: 12 October 2020
Published: 15 December 2022
© Peking University 2022
Return