Non-Abelian Lefschetz hyperplane theorems
Author:
Daniel Litt
Journal:
J. Algebraic Geom. 27 (2018), 593-646
DOI:
https://doi.org/10.1090/jag/704
Published electronically:
May 17, 2018
MathSciNet review:
3846549
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Abstract |
References |
Additional Information
Abstract:
Let $X$ be a smooth projective variety over the complex numbers, and let $D\subset X$ be an ample divisor. For which spaces $Y$ is the restriction map \begin{equation*}r: \mathrm {Hom}(X, Y)\to \mathrm {Hom}(D, Y) \end{equation*} an isomorphism?
Using positive characteristic methods, we give a fairly exhaustive answer to this question. An example application of our techniques is: if $\dim (X)\geq 3$, $Y$ is smooth, $\Omega ^1_Y$ is nef, and $\dim (Y)< \dim (D),$ the restriction map $r$ is an isomorphism. Taking $Y$ to be the classifying space of a finite group $BG$, the moduli space of pointed curves $\mathscr {M}_{g,n}$, the moduli space of principally polarized Abelian varieties $\mathscr {A}_g$, certain period domains, and various other moduli spaces, one obtains many new and classical Lefschetz hyperplane theorems.
References
- Donu Arapura, Frobenius amplitude and strong vanishing theorems for vector bundles, Duke Math. J. 121 (2004), no. 2, 231–267. With an appendix by Dennis S. Keeler. MR 2034642, DOI https://doi.org/10.1215/S0012-7094-04-12122-0
- Donu Arapura, Partial regularity and amplitude, Amer. J. Math. 128 (2006), no. 4, 1025–1056. MR 2251592
- Donu Arapura, Frobenius amplitude, ultraproducts, and vanishing on singular spaces, Illinois J. Math. 55 (2011), no. 4, 1367–1384 (2013). MR 3082873
- W. L. Baily Jr. and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2) 84 (1966), 442–528. MR 216035, DOI https://doi.org/10.2307/1970457
- Mauro C. Beltrametti and Paltin Ionescu, A view on extending morphisms from ample divisors, Interactions of classical and numerical algebraic geometry, Contemp. Math., vol. 496, Amer. Math. Soc., Providence, RI, 2009, pp. 71–110. MR 2555950, DOI https://doi.org/10.1090/conm/496/09719
- Mauro C. Beltrametti and Andrew J. Sommese, The adjunction theory of complex projective varieties, De Gruyter Expositions in Mathematics, vol. 16, Walter de Gruyter & Co., Berlin, 1995. MR 1318687
- Bhargav Bhatt and Aise Johan de Jong, Lefschetz for local Picard groups, Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), no. 4, 833–849 (English, with English and French summaries). MR 3250065, DOI https://doi.org/10.24033/asens.2228
- Richard E. Borcherds, Ludmil Katzarkov, Tony Pantev, and N. I. Shepherd-Barron, Families of $K3$ surfaces, J. Algebraic Geom. 7 (1998), no. 1, 183–193. MR 1620702
- Damian Brotbek, Projective varieties with ample cotangent bundle, Université Rennes 1, 2001.
- Roger W. Carter and George Lusztig, On the modular representations of the general linear and symmetric groups, Math. Z. 136 (1974), 193–242. MR 354887, DOI https://doi.org/10.1007/BF01214125
- François Charles, Birational boundedness for holomorphic symplectic varieties, Zarhin’s trick for $K3$ surfaces, and the Tate conjecture, Ann. of Math. (2) 184 (2016), no. 2, 487–526. MR 3548531, DOI https://doi.org/10.4007/annals.2016.184.2.4
- Brian Conrad, Deligne’s notes on Nagata compactifications, J. Ramanujan Math. Soc. 22 (2007), no. 3, 205–257. MR 2356346
- Olivier Debarre, Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, 2001. MR 1841091
- Olivier Debarre, Varieties with ample cotangent bundle, Compos. Math. 141 (2005), no. 6, 1445–1459. MR 2188444, DOI https://doi.org/10.1112/S0010437X05001399
- Pierre Deligne and Luc Illusie, Relèvements modulo $p^2$ et décomposition du complexe de de Rham, Invent. Math. 89 (1987), no. 2, 247–270 (French). MR 894379, DOI https://doi.org/10.1007/BF01389078
- A. Grothendieck, Éléments de géométrie algébrique. I. Le langage des schémas, Inst. Hautes Études Sci. Publ. Math. 4 (1960), 228 (French). MR 217083
- N. Fakhruddin, Restriction of sections of abelian schemes, arXiv e-prints (2003-10), available at math/0310405.
- Phillip A. Griffiths, Periods of integrals on algebraic manifolds. III. Some global differential-geometric properties of the period mapping, Inst. Hautes Études Sci. Publ. Math. 38 (1970), 125–180. MR 282990
- Alexander Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux $(SGA$ $2)$, North-Holland Publishing Co., Amsterdam; Masson & Cie, Éditeur, Paris, 1968 (French). Augmenté d’un exposé par Michèle Raynaud; Séminaire de Géométrie Algébrique du Bois-Marie, 1962; Advanced Studies in Pure Mathematics, Vol. 2. MR 0476737
- Andreas Höring, Manifolds with nef cotangent bundle, Asian J. Math. 17 (2013), no. 3, 561–568. MR 3119799, DOI https://doi.org/10.4310/AJM.2013.v17.n3.a7
- Daniel Huybrechts, Moduli spaces of hyperkähler manifolds and mirror symmetry, Intersection theory and moduli, ICTP Lect. Notes, XIX, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004, pp. 185–247. MR 2172498
- Kelly Jabbusch, Positivity of cotangent bundles, Michigan Math. J. 58 (2009), no. 3, 723–744. MR 2595561, DOI https://doi.org/10.1307/mmj/1260475697
- Seán Keel, Basepoint freeness for nef and big line bundles in positive characteristic, Ann. of Math. (2) 149 (1999), no. 1, 253–286. MR 1680559, DOI https://doi.org/10.2307/121025
- Seán Keel and Shigefumi Mori, Quotients by groupoids, Ann. of Math. (2) 145 (1997), no. 1, 193–213. MR 1432041, DOI https://doi.org/10.2307/2951828
- Dennis S. Keeler, Ample filters and Frobenius amplitude, J. Algebra 323 (2010), no. 10, 3039–3053. MR 2609191, DOI https://doi.org/10.1016/j.jalgebra.2010.02.028
- János Kollár, Projectivity of complete moduli, J. Differential Geom. 32 (1990), no. 1, 235–268. MR 1064874
- János Kollár, Rational curves on algebraic varieties, 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. 32, Springer-Verlag, Berlin, 1996. MR 1440180
- Sándor J. Kovács, Families over a base with a birationally nef tangent bundle, Math. Ann. 308 (1997), no. 2, 347–359. MR 1464907, DOI https://doi.org/10.1007/s002080050079
- Henrik Kratz, Compact complex manifolds with numerically effective cotangent bundles, Doc. Math. 2 (1997), 183–193. MR 1464070
- Robert Lazarsfeld, Positivity in algebraic geometry. II, 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. 49, Springer-Verlag, Berlin, 2004. Positivity for vector bundles, and multiplier ideals. MR 2095472
- Daniel Litt, Non-abelian Lefschetz hyperplane theorems II,\nopunct. in preparation.
- Martin Möller, Eckart Viehweg, and Kang Zuo, Special families of curves, of abelian varieties, and of certain minimal manifolds over curves, Global aspects of complex geometry, Springer, Berlin, 2006, pp. 417–450. MR 2264111, DOI https://doi.org/10.1007/3-540-35480-8_11
- Laurent Moret-Bailly, Familles de courbes et de variétés abeliennes sur $\mathbb {P}^1.$ I. Descente des polarisations (1981) (French).
- Martin C. Olsson, On proper coverings of Artin stacks, Adv. Math. 198 (2005), no. 1, 93–106. MR 2183251, DOI https://doi.org/10.1016/j.aim.2004.08.017
- Michèle Raynaud, Théoremes de Lefschetz en cohomologie des faisceaux cohérents et en cohomologie étale. Application au groupe fondamental, Ann. Sci. École Norm. Sup. (4) 7 (1974), 29–52 (French). MR 379503
- Alessandro Silva, Relative vanishing theorems. I. Applications to ample divisors, Comment. Math. Helv. 52 (1977), no. 4, 483–489. MR 460733, DOI https://doi.org/10.1007/BF02567380
- Carlos T. Simpson, Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. Amer. Math. Soc. 1 (1988), no. 4, 867–918. MR 944577, DOI https://doi.org/10.1090/S0894-0347-1988-0944577-9
- Carlos T. Simpson, Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 5–95. MR 1179076
- Andrew John Sommese, On manifolds that cannot be ample divisors, Math. Ann. 221 (1976), no. 1, 55–72. MR 404703, DOI https://doi.org/10.1007/BF01434964
- Michael J. Spurr, Nef cotangent bundles of branched coverings, Proc. Amer. Math. Soc. 118 (1993), no. 1, 57–66. MR 1124150, DOI https://doi.org/10.1090/S0002-9939-1993-1124150-6
- The Stacks Project Authors, Stacks Project.
- Andrey Todorov, Local and global theory of the moduli of polarized Calabi-Yau manifolds, Proceedings of the International Conference on Algebraic Geometry and Singularities (Spanish) (Sevilla, 2001), 2003, pp. 687–730. MR 2023203, DOI https://doi.org/10.4171/RMI/365
- Misha Verbitsky, Mapping class group and a global Torelli theorem for hyperkähler manifolds, Duke Math. J. 162 (2013), no. 15, 2929–2986. Appendix A by Eyal Markman. MR 3161308, DOI https://doi.org/10.1215/00127094-2382680
- Eckart Viehweg, Quasi-projective moduli for polarized manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 30, Springer-Verlag, Berlin, 1995. MR 1368632
- Eckart Viehweg and Kang Zuo, On the isotriviality of families of projective manifolds over curves, J. Algebraic Geom. 10 (2001), no. 4, 781–799. MR 1838979
- Eckart Viehweg and Kang Zuo, On the Brody hyperbolicity of moduli spaces for canonically polarized manifolds, Duke Math. J. 118 (2003), no. 1, 103–150. MR 1978884, DOI https://doi.org/10.1215/S0012-7094-03-11815-3
References
- Donu Arapura, Frobenius amplitude and strong vanishing theorems for vector bundles, with an appendix by Dennis S. Keeler, Duke Math. J. 121 (2004), no. 2, 231–267. MR 2034642 (2005d:14025)
- Donu Arapura, Partial regularity and amplitude, Amer. J. Math. 128 (2006), no. 4, 1025–1056. MR 2251592 (2007h:14053)
- Donu Arapura, Frobenius amplitude, ultraproducts, and vanishing on singular spaces, Illinois J. Math. 55 (2011), no. 4, 1367–1384 (2013). MR 3082873
- W. L. Baily Jr. and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2) 84 (1966), 442–528. MR 0216035 (35 \#6870)
- Mauro C. Beltrametti and Paltin Ionescu, A view on extending morphisms from ample divisors, Interactions of classical and numerical algebraic geometry, Contemp. Math., vol. 496, Amer. Math. Soc., Providence, RI, 2009, pp. 71–110. MR 2555950, DOI https://doi.org/10.1090/conm/496/09719
- Mauro C. Beltrametti and Andrew J. Sommese, The adjunction theory of complex projective varieties, de Gruyter Expositions in Mathematics, vol. 16, Walter de Gruyter & Co., Berlin, 1995. MR 1318687 (96f:14004)
- Bhargav Bhatt and Aise Johan de Jong, Lefschetz for local Picard groups, Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), no. 4, 833–849. MR 3250065
- Richard E. Borcherds, Ludmil Katzarkov, Tony Pantev, and N. I. Shepherd-Barron, Families of $K3$ surfaces, J. Algebraic Geom. 7 (1998), no. 1, 183–193. MR 1620702 (99d:14029)
- Damian Brotbek, Projective varieties with ample cotangent bundle, Université Rennes 1, 2001.
- Roger W. Carter and George Lusztig, On the modular representations of the general linear and symmetric groups, Math. Z. 136 (1974), 193–242. MR 0354887 (50 \#7364)
- François Charles, Birational boundedness for holomorphic symplectic varieties, Zarhin’s trick for $K3$ surfaces, and the Tate conjecture, Ann. of Math. (2) 184 (2016), no. 2, 487–526. MR 3548531
- Brian Conrad, Deligne’s notes on Nagata compactifications, J. Ramanujan Math. Soc. 22 (2007), no. 3, 205–257. MR 2356346 (2009d:14002)
- Olivier Debarre, Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, 2001. MR 1841091 (2002g:14001)
- Olivier Debarre, Varieties with ample cotangent bundle, Compos. Math. 141 (2005), no. 6, 1445–1459. MR 2188444 (2006j:14063)
- Pierre Deligne and Luc Illusie, Relèvements modulo $p^2$ et décomposition du complexe de de Rham, Invent. Math. 89 (1987), no. 2, 247–270. MR 894379 (88j:14029)
- Jean Dieudonné and Alexander Grothendieck, Éléments de géométrie algébrique, Inst. Hautes Études Sci. Publ. Math., Vol. 4, I. Le langage des schémas, 1960, 228 pp. MR 0217083; Vol. 8, I. Étude globale élémentaire de quelques classes de morphismes, 1961, 222 pp. MR 0217084; Vol. 11, III. Étude cohomologique des faisceaux cohérents. I, 1961, 167 pp. MR 0217085; Vol. 17, III. Étude cohomologique des faisceaux cohérents. II, 1963, 91 pp. MR 0163911; Vol. 20, IV. Étude locale des schémas et des morphismes de schémas. I (French), 1964, 259 pp. MR 0173675; Vol. 24, IV. Étude locale des schémas et des morphismes de schémas. II (French), 1965, 231 pp. MR 0199181; Vol. 28, IV. Étude locale des schémas et des morphismes de schémas. III, 1966, 255 pp. MR 0217086; Vol. 32, IV. Étude locale des schémas et des morphismes de schémas IV (French), 1967, 361 pp. MR 0238860.
- N. Fakhruddin, Restriction of sections of abelian schemes, arXiv e-prints (2003-10), available at math/0310405.
- Phillip A. Griffiths, Periods of integrals on algebraic manifolds. III. Some global differential-geometric properties of the period mapping, Inst. Hautes Études Sci. Publ. Math. 38 (1970), 125–180. MR 0282990 (44 \#224)
- Alexander Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux $(SGA$ $2)$ (French), augmenté d’un exposé par Michèle Raynaud, Séminaire de Géométrie Algébrique du Bois-Marie, 1962, Advanced Studies in Pure Mathematics, Vol. 2, North-Holland Publishing Co., Amsterdam; Masson & Cie, Éditeur, Paris, 1968. MR 0476737
- Andreas Höring, Manifolds with nef cotangent bundle, Asian J. Math. 17 (2013), no. 3, 561–568. MR 3119799
- Daniel Huybrechts, Moduli spaces of hyperkähler manifolds and mirror symmetry, Intersection theory and moduli, ICTP Lect. Notes, XIX, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004, pp. 185–247 (electronic). MR 2172498
- Kelly Jabbusch, Positivity of cotangent bundles, Michigan Math. J. 58 (2009), no. 3, 723–744. MR 2595561 (2011c:14122)
- Seán Keel, Basepoint freeness for nef and big line bundles in positive characteristic, Ann. of Math. (2) 149 (1999), no. 1, 253–286. MR 1680559 (2000j:14011)
- Seán Keel and Shigefumi Mori, Quotients by groupoids, Ann. of Math. (2) 145 (1997), no. 1, 193–213. MR 1432041 (97m:14014)
- Dennis S. Keeler, Ample filters and Frobenius amplitude, J. Algebra 323 (2010), no. 10, 3039–3053. MR 2609191 (2011f:14029)
- János Kollár, Projectivity of complete moduli, J. Differential Geom. 32 (1990), no. 1, 235–268. MR 1064874 (92e:14008)
- János Kollár, Rational curves on algebraic varieties, 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. 32, Springer-Verlag, Berlin, 1996. MR 1440180
- Sándor J. Kovács, Families over a base with a birationally nef tangent bundle, Math. Ann. 308 (1997), no. 2, 347–359. MR 1464907 (98h:14039)
- Henrik Kratz, Compact complex manifolds with numerically effective cotangent bundles, Doc. Math. 2 (1997), 183–193 (electronic). MR 1464070 (98j:32033)
- Robert Lazarsfeld, Positivity in algebraic geometry. II. Positivity for vector bundles, and multiplier ideals, 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. 49, Springer-Verlag, Berlin, 2004. MR 2095472
- Daniel Litt, Non-abelian Lefschetz hyperplane theorems II,\nopunct. in preparation.
- Martin Möller, Eckart Viehweg, and Kang Zuo, Special families of curves, of abelian varieties, and of certain minimal manifolds over curves, Global aspects of complex geometry, Springer, Berlin, 2006, pp. 417–450. MR 2264111 (2007k:14054)
- Laurent Moret-Bailly, Familles de courbes et de variétés abeliennes sur $\mathbb {P}^1.$ I. Descente des polarisations (1981) (French).
- Martin C. Olsson, On proper coverings of Artin stacks, Adv. Math. 198 (2005), no. 1, 93–106. MR 2183251
- Michèle Raynaud, Théoremes de Lefschetz en cohomologie des faisceaux cohérents et en cohomologie étale. Application au groupe fondamental, Collection of articles dedicated to Henri Cartan on the occasion of his 70th birthday, I, Ann. Sci. École Norm. Sup. (4) 7 (1974), 29–52. MR 0379503 (52 \#408)
- Alessandro Silva, Relative vanishing theorems. I. Applications to ample divisors, Comment. Math. Helv. 52 (1977), no. 4, 483–489. MR 0460733
- Carlos T. Simpson, Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. Amer. Math. Soc. 1 (1988), no. 4, 867–918. MR 944577 (90e:58026)
- Carlos T. Simpson, Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 5–95. MR 1179076 (94d:32027)
- Andrew John Sommese, On manifolds that cannot be ample divisors, Math. Ann. 221 (1976), no. 1, 55–72. MR 0404703
- Michael J. Spurr, Nef cotangent bundles of branched coverings, Proc. Amer. Math. Soc. 118 (1993), no. 1, 57–66. MR 1124150 (93f:14011)
- The Stacks Project Authors, Stacks Project.
- Andrey Todorov, Local and global theory of the moduli of polarized Calabi-Yau manifolds, Proceedings of the International Conference on Algebraic Geometry and Singularities (Spanish) (Sevilla, 2001), 2003, pp. 687–730. MR 2023203, DOI https://doi.org/10.4171/RMI/365
- Misha Verbitsky, Mapping class group and a global Torelli theorem for hyperkähler manifolds, Appendix A by Eyal Markman, Duke Math. J. 162 (2013), no. 15, 2929–2986. MR 3161308
- Eckart Viehweg, Quasi-projective moduli for polarized manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 30, Springer-Verlag, Berlin, 1995. MR 1368632 (97j:14001)
- Eckart Viehweg and Kang Zuo, On the isotriviality of families of projective manifolds over curves, J. Algebraic Geom. 10 (2001), no. 4, 781–799. MR 1838979 (2002g:14012)
- Eckart Viehweg and Kang Zuo, On the Brody hyperbolicity of moduli spaces for canonically polarized manifolds, Duke Math. J. 118 (2003), no. 1, 103–150. MR 1978884 (2004h:14042)
Additional Information
Daniel Litt
Affiliation:
Department of Mathematics, Columbia University, New York, New York 10027
MR Author ID:
916147
ORCID:
0000-0003-2273-4630
Email:
dlitt@math.columbia.edu
Received by editor(s):
April 6, 2016
Received by editor(s) in revised form:
November 28, 2016
Published electronically:
May 17, 2018
Article copyright:
© Copyright 2018
University Press, Inc.