Algebraic approximations of fibrations in abelian varieties over a curve
Author:
Hsueh-Yung Lin
Journal:
J. Algebraic Geom. 31 (2022), 75-103
DOI:
https://doi.org/10.1090/jag/791
Published electronically:
October 25, 2021
MathSciNet review:
4372407
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Abstract |
References |
Additional Information
Abstract: For every fibration $f : X \to B$ with $X$ a compact Kähler manifold, $B$ a smooth projective curve, and a general fiber of $f$ an abelian variety, we prove that $f$ has an algebraic approximation.
References
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- F. Campana, Coréduction algébrique d’un espace analytique faiblement kählérien compact, Invent. Math. 63 (1981), no. 2, 187–223 (French). MR 610537, DOI 10.1007/BF01393876
- F. Campana, Réduction algébrique d’un morphisme faiblement Kählérien propre et applications, Math. Ann. 256 (1981), no. 2, 157–189 (French). MR 620706, DOI 10.1007/BF01450796
- Frédéric Campana and Thomas Peternell, Complex threefolds with non-trivial holomorphic $2$-forms, J. Algebraic Geom. 9 (2000), no. 2, 223–264. MR 1735771
- Junyan Cao, On the approximation of Kähler manifolds by algebraic varieties, Math. Ann. 363 (2015), no. 1-2, 393–422. MR 3394384, DOI 10.1007/s00208-014-1097-4
- Benoît Claudon, Smooth families of tori and linear Kähler groups [addendum to MR3322791], Ann. Fac. Sci. Toulouse Math. (6) 27 (2018), no. 3, 477–496 (English, with English and French summaries). MR 3869072, DOI 10.5802/afst.1576
- Benoît Claudon, Andreas Höring, and Hsueh-Yung Lin, The fundamental group of compact Kähler threefolds, Geom. Topol. 23 (2019), no. 7, 3233–3271. MR 4059086, DOI 10.2140/gt.2019.23.3233
- Fouad El Zein and Steven Zucker, Extendability of normal functions associated to algebraic cycles, Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982) Ann. of Math. Stud., vol. 106, Princeton Univ. Press, Princeton, NJ, 1984, pp. 269–288. MR 756857
- Akira Fujiki, Closedness of the Douady spaces of compact Kähler spaces, Publ. Res. Inst. Math. Sci. 14 (1978/79), no. 1, 1–52. MR 0486648, DOI 10.2977/prims/1195189279
- Akira Fujiki, On the structure of compact complex manifolds in ${\cal C}$, Algebraic varieties and analytic varieties (Tokyo, 1981) Adv. Stud. Pure Math., vol. 1, North-Holland, Amsterdam, 1983, pp. 231–302. MR 715653, DOI 10.2969/aspm/00110231
- Patrick Graf, Algebraic approximation of Kähler threefolds of Kodaira dimension zero, Math. Ann. 371 (2018), no. 1-2, 487–516. MR 3788856, DOI 10.1007/s00208-017-1577-4
- K. Kodaira, On compact analytic surfaces. II, III, Ann. of Math. (2) 77 (1963), 563–626; ibid. 78 (1963), 1–40. MR 0184257, DOI 10.2307/1970500
- János Kollár, Lectures on resolution of singularities, Annals of Mathematics Studies, vol. 166, Princeton University Press, Princeton, NJ, 2007. MR 2289519
- Hsueh-Yung Lin. Algebraic approximations of compact Kähler threefolds, ArXiv:1710.01083, 2018.
- Hsueh-Yung Lin, Algebraic approximations of compact Kähler manifolds of algebraic codimension 1, Duke Math. J. 169 (2020), no. 14, 2767–2826. MR 4149508, DOI 10.1215/00127094-2020-0018
- Hsueh-Yung Lin, Algebraic approximations of compact Kähler manifolds of algebraic codimension 1, Duke Math. J. 169 (2020), no. 14, 2767–2826. MR 4149508, DOI 10.1215/00127094-2020-0018
- Noboru Nakayama, Compact Kähler manifolds whose universal covering spaces are biholomorphic to $\mathbf {C}^n$, Preprint RIMS-1230, Res. Inst. Math. Sci. Kyoto Univ., 1999.
- Chris A. M. Peters and Joseph H. M. Steenbrink, Mixed Hodge structures, 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. 52, Springer-Verlag, Berlin, 2008. MR 2393625
- Morihiko Saito, Admissible normal functions, J. Algebraic Geom. 5 (1996), no. 2, 235–276. MR 1374710
- Florian Schrack, Algebraic approximation of Kähler threefolds, Math. Nachr. 285 (2012), no. 11-12, 1486–1499. MR 2959971, DOI 10.1002/mana.201100228
- Jean Varouchas, Kähler spaces and proper open morphisms, Math. Ann. 283 (1989), no. 1, 13–52. MR 973802, DOI 10.1007/BF01457500
- Claire Voisin, Hodge theory and complex algebraic geometry. II, Cambridge Studies in Advanced Mathematics, vol. 77, Cambridge University Press, Cambridge, 2003. Translated from the French by Leila Schneps. MR 1997577, DOI 10.1017/CBO9780511615177
- Claire Voisin, On the homotopy types of compact Kähler and complex projective manifolds, Invent. Math. 157 (2004), no. 2, 329–343. MR 2076925, DOI 10.1007/s00222-003-0352-1
- Jarosław Włodarczyk, Resolution of singularities of analytic spaces, Proceedings of Gökova Geometry-Topology Conference 2008, Gökova Geometry/Topology Conference (GGT), Gökova, 2009, pp. 31–63. MR 2500573
- Steven Zucker, Generalized intermediate Jacobians and the theorem on normal functions, Invent. Math. 33 (1976), no. 3, 185–222. MR 412186, DOI 10.1007/BF01404203
- Steven Zucker, Hodge theory with degenerating coefficients. $L_{2}$ cohomology in the Poincaré metric, Ann. of Math. (2) 109 (1979), no. 3, 415–476. MR 534758, DOI 10.2307/1971221
- Steven Zucker, Degeneration of Hodge bundles (after Steenbrink), Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982) Ann. of Math. Stud., vol. 106, Princeton Univ. Press, Princeton, NJ, 1984, pp. 121–141. MR 756849
References
- Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York, 1994. Corrected reprint of the 1982 original. MR 1324339
- F. Campana, Coréduction algébrique d’un espace analytique faiblement kählérien compact, Invent. Math. 63 (1981), no. 2, 187–223 (French). MR 610537, DOI 10.1007/BF01393876
- F. Campana, Réduction algébrique d’un morphisme faiblement Kählérien propre et applications, Math. Ann. 256 (1981), no. 2, 157–189 (French). MR 620706, DOI 10.1007/BF01450796
- Frédéric Campana and Thomas Peternell, Complex threefolds with non-trivial holomorphic $2$-forms, J. Algebraic Geom. 9 (2000), no. 2, 223–264. MR 1735771
- Junyan Cao, On the approximation of Kähler manifolds by algebraic varieties, Math. Ann. 363 (2015), no. 1-2, 393–422. MR 3394384, DOI 10.1007/s00208-014-1097-4
- Benoît Claudon, Smooth families of tori and linear Kähler groups [addendum to MR3322791], Ann. Fac. Sci. Toulouse Math. (6) 27 (2018), no. 3, 477–496 (English, with English and French summaries). MR 3869072, DOI 10.5802/afst.1576
- Benoît Claudon, Andreas Höring, and Hsueh-Yung Lin, The fundamental group of compact Kähler threefolds, Geom. Topol. 23 (2019), no. 7, 3233–3271. MR 4059086, DOI 10.2140/gt.2019.23.3233
- Fouad El Zein and Steven Zucker, Extendability of normal functions associated to algebraic cycles, Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982) Ann. of Math. Stud., vol. 106, Princeton Univ. Press, Princeton, NJ, 1984, pp. 269–288. MR 756857
- Akira Fujiki, Closedness of the Douady spaces of compact Kähler spaces, Publ. Res. Inst. Math. Sci. 14 (1978/79), no. 1, 1–52. MR 0486648, DOI 10.2977/prims/1195189279
- Akira Fujiki, On the structure of compact complex manifolds in ${\mathcal {C}}$, Algebraic varieties and analytic varieties (Tokyo, 1981) Adv. Stud. Pure Math., vol. 1, North-Holland, Amsterdam, 1983, pp. 231–302. MR 715653, DOI 10.2969/aspm/00110231
- Patrick Graf, Algebraic approximation of Kähler threefolds of Kodaira dimension zero, Math. Ann. 371 (2018), no. 1-2, 487–516. MR 3788856, DOI 10.1007/s00208-017-1577-4
- K. Kodaira, On compact analytic surfaces. II, III, Ann. of Math. (2) 77 (1963), 563–626; ibid. 78 (1963), 1–40. MR 0184257, DOI 10.2307/1970500
- János Kollár, Lectures on resolution of singularities, Annals of Mathematics Studies, vol. 166, Princeton University Press, Princeton, NJ, 2007. MR 2289519
- Hsueh-Yung Lin. Algebraic approximations of compact Kähler threefolds, ArXiv:1710.01083, 2018.
- Hsueh-Yung Lin, Algebraic approximations of compact Kähler manifolds of algebraic codimension 1, Duke Math. J. 169 (2020), no. 14, 2767–2826. MR 4149508, DOI 10.1215/00127094-2020-0018
- Hsueh-Yung Lin, Algebraic approximations of compact Kähler manifolds of algebraic codimension 1, Duke Math. J. 169 (2020), no. 14, 2767–2826. MR 4149508, DOI 10.1215/00127094-2020-0018
- Noboru Nakayama, Compact Kähler manifolds whose universal covering spaces are biholomorphic to $\mathbf {C}^n$, Preprint RIMS-1230, Res. Inst. Math. Sci. Kyoto Univ., 1999.
- Chris A. M. Peters and Joseph H. M. Steenbrink, Mixed Hodge structures, 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. 52, Springer-Verlag, Berlin, 2008. MR 2393625
- Morihiko Saito, Admissible normal functions, J. Algebraic Geom. 5 (1996), no. 2, 235–276. MR 1374710
- Florian Schrack, Algebraic approximation of Kähler threefolds, Math. Nachr. 285 (2012), no. 11-12, 1486–1499. MR 2959971, DOI 10.1002/mana.201100228
- Jean Varouchas, Kähler spaces and proper open morphisms, Math. Ann. 283 (1989), no. 1, 13–52. MR 973802, DOI 10.1007/BF01457500
- Claire Voisin, Hodge theory and complex algebraic geometry. II, Cambridge Studies in Advanced Mathematics, vol. 77, Cambridge University Press, Cambridge, 2003. Translated from the French by Leila Schneps. MR 1997577, DOI 10.1017/CBO9780511615177
- Claire Voisin, On the homotopy types of compact Kähler and complex projective manifolds, Invent. Math. 157 (2004), no. 2, 329–343. MR 2076925, DOI 10.1007/s00222-003-0352-1
- Jarosław Włodarczyk, Resolution of singularities of analytic spaces, Proceedings of Gökova Geometry-Topology Conference 2008, Gökova Geometry/Topology Conference (GGT), Gökova, 2009, pp. 31–63. MR 2500573
- Steven Zucker, Generalized intermediate Jacobians and the theorem on normal functions, Invent. Math. 33 (1976), no. 3, 185–222. MR 412186, DOI 10.1007/BF01404203
- Steven Zucker, Hodge theory with degenerating coefficients. $L_{2}$ cohomology in the Poincaré metric, Ann. of Math. (2) 109 (1979), no. 3, 415–476. MR 534758, DOI 10.2307/1971221
- Steven Zucker, Degeneration of Hodge bundles (after Steenbrink), Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982) Ann. of Math. Stud., vol. 106, Princeton Univ. Press, Princeton, NJ, 1984, pp. 121–141. MR 756849
Additional Information
Hsueh-Yung Lin
Affiliation:
Department of Mathematics, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd., Taipei 10617, Taiwan
MR Author ID:
986899
Email:
hsuehyunglin@ntu.edu.tw
Received by editor(s):
February 4, 2019
Received by editor(s) in revised form:
June 30, 2021
Published electronically:
October 25, 2021
Article copyright:
© Copyright 2021
University Press, Inc.
