An injectivity theorem with multiplier ideal sheaves of singular metrics with transcendental singularities
Author:
Shin-ichi Matsumura
Journal:
J. Algebraic Geom. 27 (2018), 305-337
DOI:
https://doi.org/10.1090/jag/687
Published electronically:
August 17, 2017
MathSciNet review:
3764278
Full-text PDF
Abstract |
References |
Additional Information
Abstract: The purpose of this paper is to establish an injectivity theorem generalized to pseudo-effective line bundles with transcendental (non-algebraic) singular hermitian metrics and multiplier ideal sheaves. As an application, we obtain a Nadel type vanishing theorem. For the proof, we study the asymptotic behavior of the harmonic forms with respect to a family of regularized metrics, and give a method to obtain $L^{2}$-estimates of solutions of the $\overline {\partial }$-equation by using the de Rham-Weil isomorphism between the $\overline {\partial }$-cohomology and the $\rm {\check {C}}$ech cohomology.
References
- Junyan Cao, Numerical dimension and a Kawamata-Viehweg-Nadel-type vanishing theorem on compact Kähler manifolds, Compos. Math. 150 (2014), no. 11, 1869–1902. MR 3279260, DOI https://doi.org/10.1112/S0010437X14007398
- Jean-Pierre Demailly, Lawrence Ein, and Robert Lazarsfeld, A subadditivity property of multiplier ideals, Michigan Math. J. 48 (2000), 137–156. Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786484, DOI https://doi.org/10.1307/mmj/1030132712
- Jean-Pierre Demailly, Analytic methods in algebraic geometry, Surveys of Modern Mathematics, vol. 1, International Press, Somerville, MA; Higher Education Press, Beijing, 2012. MR 2978333
- J.-P. Demailly, Complex analytic and differential geometry, lecture notes on the webpage of the author.
- Jean-Pierre Demailly, Estimations $L^{2}$ pour l’opérateur $\bar \partial $ d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 3, 457–511 (French). MR 690650
- Jean-Pierre Demailly, Christopher D. Hacon, and Mihai Păun, Extension theorems, non-vanishing and the existence of good minimal models, Acta Math. 210 (2013), no. 2, 203–259. MR 3070567, DOI https://doi.org/10.1007/s11511-013-0094-x
- Jean-Pierre Demailly, Thomas Peternell, and Michael Schneider, Pseudo-effective line bundles on compact Kähler manifolds, Internat. J. Math. 12 (2001), no. 6, 689–741. MR 1875649, DOI https://doi.org/10.1142/S0129167X01000861
- Ichiro Enoki, Kawamata-Viehweg vanishing theorem for compact Kähler manifolds, Einstein metrics and Yang-Mills connections (Sanda, 1990) Lecture Notes in Pure and Appl. Math., vol. 145, Dekker, New York, 1993, pp. 59–68. MR 1215279
- Lawrence Ein and Mihnea Popa, Global division of cohomology classes via injectivity, Michigan Math. J. 57 (2008), 249–259. Special volume in honor of Melvin Hochster. MR 2492451, DOI https://doi.org/10.1307/mmj/1220879407
- Hélène Esnault and Eckart Viehweg, Lectures on vanishing theorems, DMV Seminar, vol. 20, Birkhäuser Verlag, Basel, 1992. MR 1193913
- Osamu Fujino, A transcendental approach to Kollár’s injectivity theorem, Osaka J. Math. 49 (2012), no. 3, 833–852. MR 2993068
- Osamu Fujino, A transcendental approach to Kollár’s injectivity theorem II, J. Reine Angew. Math. 681 (2013), 149–174. MR 3181493, DOI https://doi.org/10.1515/crelle-2012-0036
- Yoshinori Gongyo and Shin-ichi Matsumura, Versions of injectivity and extension theorems, Ann. Sci. Éc. Norm. Supér. (4) 50 (2017), no. 2, 479–502 (English, with English and French summaries). MR 3621435, DOI https://doi.org/10.24033/asens.2325
- Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR 0180696
- Qi’an Guan and Xiangyu Zhou, Effectiveness of Demailly’s strong openness conjecture and related problems, Invent. Math. 202 (2015), no. 2, 635–676. MR 3418242, DOI https://doi.org/10.1007/s00222-014-0575-3
- Pham Hoang Hiep, The weighted log canonical threshold, C. R. Math. Acad. Sci. Paris 352 (2014), no. 4, 283–288 (English, with English and French summaries). MR 3186914, DOI https://doi.org/10.1016/j.crma.2014.02.010
- Lars Hörmander, $L^{2}$ estimates and existence theorems for the $\bar \partial $ operator, Acta Math. 113 (1965), 89–152. MR 179443, DOI https://doi.org/10.1007/BF02391775
- Y. Kawamata, Pluricanonical systems on minimal algebraic varieties, Invent. Math. 79 (1985), no. 3, 567–588. MR 782236, DOI https://doi.org/10.1007/BF01388524
- János Kollár, Higher direct images of dualizing sheaves. I, Ann. of Math. (2) 123 (1986), no. 1, 11–42. MR 825838, DOI https://doi.org/10.2307/1971351
- L. Lempert, Modules of square integrable holomorphic germs, preprint, arXiv:1404.0407v2.
- 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
- Shin-ichi Matsumura, A Nadel vanishing theorem via injectivity theorems, Math. Ann. 359 (2014), no. 3-4, 785–802. MR 3231016, DOI https://doi.org/10.1007/s00208-014-1018-6
- Shin-ichi Matsumura, Some applications of the theory of harmonic integrals, Complex Manifolds 2 (2015), no. 1, 16–25. MR 3370350, DOI https://doi.org/10.1515/coma-2015-0003
- Shin-ichi Matsumura, A Nadel vanishing theorem for metrics with minimal singularities on big line bundles, Adv. Math. 280 (2015), 188–207. MR 3350216, DOI https://doi.org/10.1016/j.aim.2015.03.019
- Noboru Nakayama, The lower semicontinuity of the plurigenera of complex varieties, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 551–590. MR 946250, DOI https://doi.org/10.2969/aspm/01010551
- Takeo Ohsawa, On a curvature condition that implies a cohomology injectivity theorem of Kollár-Skoda type, Publ. Res. Inst. Math. Sci. 41 (2005), no. 3, 565–577. MR 2153535
- Mihai Păun, Siu’s invariance of plurigenera: a one-tower proof, J. Differential Geom. 76 (2007), no. 3, 485–493. MR 2331528
- Francesco Russo, A characterization of nef and good divisors by asymptotic multiplier ideals, Bull. Belg. Math. Soc. Simon Stevin 16 (2009), no. 5, Linear systems and subschemes, 943–951. MR 2574371
- Yum-Tong Siu, Invariance of plurigenera, Invent. Math. 134 (1998), no. 3, 661–673. MR 1660941, DOI https://doi.org/10.1007/s002220050276
- Kensho Takegoshi, On cohomology groups of nef line bundles tensorized with multiplier ideal sheaves on compact Kähler manifolds, Osaka J. Math. 34 (1997), no. 4, 783–802. MR 1618661
- S. G. Tankeev, $n$-dimensional canonically polarized varieties, and varieties of basic type, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 31–44 (Russian). MR 0277528
References
- Junyan Cao, Numerical dimension and a Kawamata-Viehweg-Nadel-type vanishing theorem on compact Kähler manifolds, Compos. Math. 150 (2014), no. 11, 1869–1902. MR 3279260, DOI https://doi.org/10.1112/S0010437X14007398
- Jean-Pierre Demailly, Lawrence Ein, and Robert Lazarsfeld, A subadditivity property of multiplier ideals, Michigan Math. J. 48 (2000), 137–156. Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786484, DOI https://doi.org/10.1307/mmj/1030132712
- Jean-Pierre Demailly, Analytic methods in algebraic geometry, Surveys of Modern Mathematics, vol. 1, International Press, Somerville, MA; Higher Education Press, Beijing, 2012. MR 2978333
- J.-P. Demailly, Complex analytic and differential geometry, lecture notes on the webpage of the author.
- Jean-Pierre Demailly, Estimations $L^{2}$ pour l’opérateur $\bar \partial$ d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 3, 457–511 (French). MR 690650
- Jean-Pierre Demailly, Christopher D. Hacon, and Mihai Păun, Extension theorems, non-vanishing and the existence of good minimal models, Acta Math. 210 (2013), no. 2, 203–259. MR 3070567, DOI https://doi.org/10.1007/s11511-013-0094-x
- Jean-Pierre Demailly, Thomas Peternell, and Michael Schneider, Pseudo-effective line bundles on compact Kähler manifolds, Internat. J. Math. 12 (2001), no. 6, 689–741. MR 1875649, DOI https://doi.org/10.1142/S0129167X01000861
- Ichiro Enoki, Kawamata-Viehweg vanishing theorem for compact Kähler manifolds, Einstein metrics and Yang-Mills connections (Sanda, 1990) Lecture Notes in Pure and Appl. Math., vol. 145, Dekker, New York, 1993, pp. 59–68. MR 1215279
- Lawrence Ein and Mihnea Popa, Global division of cohomology classes via injectivity, Michigan Math. J. 57 (2008), 249–259. Special volume in honor of Melvin Hochster. MR 2492451, DOI https://doi.org/10.1307/mmj/1220879407
- Hélène Esnault and Eckart Viehweg, Lectures on vanishing theorems, DMV Seminar, vol. 20, Birkhäuser Verlag, Basel, 1992. MR 1193913
- Osamu Fujino, A transcendental approach to Kollár’s injectivity theorem, Osaka J. Math. 49 (2012), no. 3, 833–852. MR 2993068
- Osamu Fujino, A transcendental approach to Kollár’s injectivity theorem II, J. Reine Angew. Math. 681 (2013), 149–174. MR 3181493
- Yoshinori Gongyo and Shin-ichi Matsumura, Versions of injectivity and extension theorems, Ann. Sci. Éc. Norm. Supér. (4) 50 (2017), no. 2, 479–502 (English, with English and French summaries). MR 3621435
- Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR 0180696
- Qi’an Guan and Xiangyu Zhou, Effectiveness of Demailly’s strong openness conjecture and related problems, Invent. Math. 202 (2015), no. 2, 635–676. MR 3418242
- Pham Hoang Hiep, The weighted log canonical threshold, C. R. Math. Acad. Sci. Paris 352 (2014), no. 4, 283–288 (English, with English and French summaries). MR 3186914, DOI https://doi.org/10.1016/j.crma.2014.02.010
- Lars Hörmander, $L^{2}$ estimates and existence theorems for the $\bar \partial$ operator, Acta Math. 113 (1965), 89–152. MR 0179443
- Y. Kawamata, Pluricanonical systems on minimal algebraic varieties, Invent. Math. 79 (1985), no. 3, 567–588. MR 782236, DOI https://doi.org/10.1007/BF01388524
- János Kollár, Higher direct images of dualizing sheaves. I, Ann. of Math. (2) 123 (1986), no. 1, 11–42. MR 825838, DOI https://doi.org/10.2307/1971351
- L. Lempert, Modules of square integrable holomorphic germs, preprint, arXiv:1404.0407v2.
- Robert Lazarsfeld, Positivity in algebraic geometry. I, Classical setting: line bundles and linear series, 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. MR 2095471
- Shin-ichi Matsumura, A Nadel vanishing theorem via injectivity theorems, Math. Ann. 359 (2014), no. 3-4, 785–802. MR 3231016, DOI https://doi.org/10.1007/s00208-014-1018-6
- Shin-ichi Matsumura, Some applications of the theory of harmonic integrals, Complex Manifolds 2 (2015), 16–25. MR 3370350, DOI https://doi.org/10.1515/coma-2015-0003
- Shin-ichi Matsumura, A Nadel vanishing theorem for metrics with minimal singularities on big line bundles, Adv. Math. 280 (2015), 188–207. MR 3350216, DOI https://doi.org/10.1016/j.aim.2015.03.019
- Noboru Nakayama, The lower semicontinuity of the plurigenera of complex varieties, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 551–590. MR 946250
- Takeo Ohsawa, On a curvature condition that implies a cohomology injectivity theorem of Kollár-Skoda type, Publ. Res. Inst. Math. Sci. 41 (2005), no. 3, 565–577. MR 2153535
- Mihai Păun, Siu’s invariance of plurigenera: a one-tower proof, J. Differential Geom. 76 (2007), no. 3, 485–493. MR 2331528
- Francesco Russo, A characterization of nef and good divisors by asymptotic multiplier ideals, Bull. Belg. Math. Soc. Simon Stevin 16 (2009), no. 5, Linear systems and subschemes, 943–951. MR 2574371
- Yum-Tong Siu, Invariance of plurigenera, Invent. Math. 134 (1998), no. 3, 661–673. MR 1660941, DOI https://doi.org/10.1007/s002220050276
- Kensho Takegoshi, On cohomology groups of nef line bundles tensorized with multiplier ideal sheaves on compact Kähler manifolds, Osaka J. Math. 34 (1997), no. 4, 783–802. MR 1618661
- S. G. Tankeev, $n$-dimensional canonically polarized varieties, and varieties of basic type, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 31–44 (Russian). MR 0277528
Additional Information
Shin-ichi Matsumura
Affiliation:
Mathematical Institute, Tohoku University, 6-3, Aramaki Aza-Aoba, Aoba-ku, Sendai 980-8578, Japan
MR Author ID:
1006398
Email:
mshinichi@m.tohoku.ac.jp, mshinichi0@gmail.com
Received by editor(s):
December 21, 2015
Received by editor(s) in revised form:
February 4, 2016
Published electronically:
August 17, 2017
Additional Notes:
The author was partially supported by Grant-in-Aid for Young Scientists (B) #25800051, (A) #17H04821 from JSPS, and the JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers
Article copyright:
© Copyright 2017
University Press, Inc.
