Geometry of logarithmic forms and deformations of complex structures
Authors:
Kefeng Liu, Sheng Rao and Xueyuan Wan
Journal:
J. Algebraic Geom. 28 (2019), 773-815
DOI:
https://doi.org/10.1090/jag/723
Published electronically:
July 19, 2019
MathSciNet review:
4093453
Full-text PDF
Abstract |
References |
Additional Information
Abstract:
We present a new method to solve certain $\bar \partial$-equations for logarithmic differential forms by using harmonic integral theory for currents on Kähler manifolds. The result can be considered as a $\partial \bar \partial$-lemma for logarithmic forms. As applications, we generalize the result of Deligne about closedness of logarithmic forms, give geometric and simpler proofs of Deligne’s degeneracy theorem for the logarithmic Hodge to de Rham spectral sequences at $E_1$-level, as well as a certain injectivity theorem on compact Kähler manifolds.
Furthermore, for a family of logarithmic deformations of complex structures on Kähler manifolds, we construct the extension for any logarithmic $(n,q)$-form on the central fiber and thus deduce the local stability of log Calabi-Yau structure by extending an iteration method to the logarithmic forms. Finally we prove the unobstructedness of the deformations of a log Calabi-Yau pair and a pair on a Calabi-Yau manifold by the differential geometric method.
References
- Florin Ambro, An injectivity theorem, Compos. Math. 150 (2014), no. 6, 999–1023. MR 3223880, DOI https://doi.org/10.1112/S0010437X13007768
- Vincenzo Ancona and Bernard Gaveau, Differential forms on singular varieties, Pure and Applied Mathematics (Boca Raton), vol. 273, Chapman & Hall/CRC, Boca Raton, FL, 2006. De Rham and Hodge theory simplified. MR 2168647
- Wolf P. Barth, Klaus Hulek, Chris A. M. Peters, and Antonius Van de Ven, Compact complex surfaces, 2nd ed., 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. 4, Springer-Verlag, Berlin, 2004. MR 2030225
- F. A. Bogomolov, Hamiltonian Kählerian manifolds, Dokl. Akad. Nauk SSSR 243 (1978), no. 5, 1101–1104 (Russian). MR 514769
- Luis A. Cordero, Marisa Fernández, Luis Ugarte, and Alfred Gray, A general description of the terms in the Frölicher spectral sequence, Differential Geom. Appl. 7 (1997), no. 1, 75–84. MR 1441920, DOI https://doi.org/10.1016/S0926-2245%2896%2900038-1
- P. Deligne, Théorème de Lefschetz et critères de dégénérescence de suites spectrales, Inst. Hautes Études Sci. Publ. Math. 35 (1968), 259–278 (French). MR 244265
- Pierre Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, Vol. 163, Springer-Verlag, Berlin-New York, 1970 (French). MR 0417174
- Pierre Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5–57 (French). MR 498551
- 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
- J.-P. Demailly, Complex analytic and differential geometry, book online, http://www-fourier.ujf-grenoble.fr/~demailly/books.html.
- Georges de Rham and Kunihiko Kodaira, Harmonic Integrals, Institute for Advanced Study, Princeton, N. J., 1950. MR 0037549
- Hélène Esnault and Eckart Viehweg, Logarithmic de Rham complexes and vanishing theorems, Invent. Math. 86 (1986), no. 1, 161–194. MR 853449, DOI https://doi.org/10.1007/BF01391499
- Hélène Esnault and Eckart Viehweg, Lectures on vanishing theorems, DMV Seminar, vol. 20, Birkhäuser Verlag, Basel, 1992. MR 1193913
- Osamu Fujino, Foundations of the minimal model program, MSJ Memoirs, vol. 35, Mathematical Society of Japan, Tokyo, 2017. MR 3643725
- Osamu Fujino, Injectivity theorems, Higher dimensional algebraic geometry—in honour of Professor Yujiro Kawamata’s sixtieth birthday, Adv. Stud. Pure Math., vol. 74, Math. Soc. Japan, Tokyo, 2017, pp. 131–157. MR 3791211, DOI https://doi.org/10.2969/aspm/07410131
- Osamu Fujino, On semipositivity, injectivity and vanishing theorems, Hodge theory and $L^2$-analysis, Adv. Lect. Math. (ALM), vol. 39, Int. Press, Somerville, MA, 2017, pp. 245–282. MR 3751293
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364
- Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. MR 507725
- A. Grothendieck, A general theory of fiber spaces with structure sheaf, Univ. of Kansas report, 1958.
- C. Huang, K. Liu, X. Wan, and X. Yang, Logarithmic vanishing theorems on compact Kähler manifolds I, arXiv:1611.07671v1, 2016.
- Donatella Iacono, Deformations and obstructions of pairs $(X,D)$, Int. Math. Res. Not. IMRN 19 (2015), 9660–9695. MR 3431606, DOI https://doi.org/10.1093/imrn/rnu242
- Kazuya Kato and Sampei Usui, Classifying spaces of degenerating polarized Hodge structures, Annals of Mathematics Studies, vol. 169, Princeton University Press, Princeton, NJ, 2009. MR 2465224
- L. Katzarkov, M. Kontsevich, and T. Pantev, Hodge theoretic aspects of mirror symmetry, From Hodge theory to integrability and TQFT tt*-geometry, Proc. Sympos. Pure Math., vol. 78, Amer. Math. Soc., Providence, RI, 2008, pp. 87–174. MR 2483750, DOI https://doi.org/10.1090/pspum/078/2483750
- Yujiro Kawamata, On deformations of compactifiable complex manifolds, Math. Ann. 235 (1978), no. 3, 247–265. MR 499279, DOI https://doi.org/10.1007/BF01420124
- Yujiro Kawamata and Yoshinori Namikawa, Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties, Invent. Math. 118 (1994), no. 3, 395–409. MR 1296351, DOI https://doi.org/10.1007/BF01231538
- Kunihiko Kodaira, The theorem of Riemann-Roch on compact analytic surfaces, Amer. J. Math. 73 (1951), 813–875. MR 48107, DOI https://doi.org/10.2307/2372120
- Kunihiko Kodaira, Complex manifolds and deformation of complex structures, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 283, Springer-Verlag, New York, 1986. Translated from the Japanese by Kazuo Akao; With an appendix by Daisuke Fujiwara. MR 815922
- Kefeng Liu and Sheng Rao, Remarks on the Cartan formula and its applications, Asian J. Math. 16 (2012), no. 1, 157–169. MR 2904916, DOI https://doi.org/10.4310/AJM.2012.v16.n1.a6
- Kefeng Liu, Sheng Rao, and Xiaokui Yang, Quasi-isometry and deformations of Calabi-Yau manifolds, Invent. Math. 199 (2015), no. 2, 423–453. MR 3302118, DOI https://doi.org/10.1007/s00222-014-0516-1
- Kefeng Liu, Xiaofeng Sun, and Shing-Tung Yau, Recent development on the geometry of the Teichmüller and moduli spaces of Riemann surfaces, Surveys in differential geometry. Vol. XIV. Geometry of Riemann surfaces and their moduli spaces, Surv. Differ. Geom., vol. 14, Int. Press, Somerville, MA, 2009, pp. 221–259. MR 2655329, DOI https://doi.org/10.4310/SDG.2009.v14.n1.a9
- Shin-ichi Matsumura, A transcendental approach to injectivity theorem for log canonical pairs, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19 (2019), no. 1, 311–334. MR 3923849
- James Morrow and Kunihiko Kodaira, Complex manifolds, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1971. MR 0302937
- Junjiro Noguchi, A short analytic proof of closedness of logarithmic forms, Kodai Math. J. 18 (1995), no. 2, 295–299. MR 1346909, DOI https://doi.org/10.2996/kmj/1138043426
- S. Rao, X. Wan, and Q. Zhao, Power series proofs for local stabilities of Kähler and balanced structures with mild $\partial \bar \partial$-lemma, arXiv:1609.05637v1, 2016.
- Sheng Rao and Quanting Zhao, Several special complex structures and their deformation properties, J. Geom. Anal. 28 (2018), no. 4, 2984–3047. MR 3881963, DOI https://doi.org/10.1007/s12220-017-9944-7
- K. Saito, On the uniformization of complements of discriminant loci, AMS Summer Institute, 1975.
- Michael Schneider, Halbstetigkeitssätze für relativ analytische Räume, Invent. Math. 16 (1972), 161–176 (German). MR 301232, DOI https://doi.org/10.1007/BF01391215
- Michael E. Taylor, Partial differential equations, Texts in Applied Mathematics, vol. 23, Springer-Verlag, New York, 1996. Basic theory. MR 1395147
- Gang Tian, Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric, Mathematical aspects of string theory (San Diego, Calif., 1986) Adv. Ser. Math. Phys., vol. 1, World Sci. Publishing, Singapore, 1987, pp. 629–646. MR 915841
- Andrey N. Todorov, The Weil-Petersson geometry of the moduli space of ${\rm SU}(n\geq 3)$ (Calabi-Yau) manifolds. I, Comm. Math. Phys. 126 (1989), no. 2, 325–346. MR 1027500
- J. Tu, The iteration method in analytic deformation theory, preprint, 2016.
- Claire Voisin, Hodge theory and complex algebraic geometry. I, Cambridge Studies in Advanced Mathematics, vol. 76, Cambridge University Press, Cambridge, 2002. Translated from the French original by Leila Schneps. MR 1967689
- Quanting Zhao and Sheng Rao, Applications of the deformation formula of holomorphic one-forms, Pacific J. Math. 266 (2013), no. 1, 221–255. MR 3105782, DOI https://doi.org/10.2140/pjm.2013.266.221
- Quanting Zhao and Sheng Rao, Extension formulas and deformation invariance of Hodge numbers, C. R. Math. Acad. Sci. Paris 353 (2015), no. 11, 979–984 (English, with English and French summaries). MR 3419846, DOI https://doi.org/10.1016/j.crma.2015.09.004
References
- Florin Ambro, An injectivity theorem, Compos. Math. 150 (2014), no. 6, 999–1023. MR 3223880, DOI https://doi.org/10.1112/S0010437X13007768
- Vincenzo Ancona and Bernard Gaveau, Differential forms on singular varieties, Pure and Applied Mathematics (Boca Raton), vol. 273, Chapman & Hall/CRC, Boca Raton, FL, 2006. De Rham and Hodge theory simplified. MR 2168647
- Wolf P. Barth, Klaus Hulek, Chris A. M. Peters, and Antonius Van de Ven, Compact complex surfaces, 2nd ed., 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. 4, Springer-Verlag, Berlin, 2004. MR 2030225
- F. A. Bogomolov, Hamiltonian Kählerian manifolds, Dokl. Akad. Nauk SSSR 243 (1978), no. 5, 1101–1104 (Russian). MR 514769
- Luis A. Cordero, Marisa Fernández, Luis Ugarte, and Alfred Gray, A general description of the terms in the Frölicher spectral sequence, Differential Geom. Appl. 7 (1997), no. 1, 75–84. MR 1441920, DOI https://doi.org/10.1016/S0926-2245%2896%2900038-1
- P. Deligne, Théorème de Lefschetz et critères de dégénérescence de suites spectrales, Inst. Hautes Études Sci. Publ. Math. 35 (1968), 259–278 (French). MR 0244265
- Pierre Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, Vol. 163, Springer-Verlag, Berlin-New York, 1970 (French). MR 0417174
- Pierre Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5–57 (French). MR 0498551
- 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
- J.-P. Demailly, Complex analytic and differential geometry, book online, http://www-fourier.ujf-grenoble.fr/~demailly/books.html.
- Georges de Rham and Kunihiko Kodaira, Harmonic Integrals, Institute for Advanced Study, Princeton, N. J., 1950. MR 0037549
- Hélène Esnault and Eckart Viehweg, Logarithmic de Rham complexes and vanishing theorems, Invent. Math. 86 (1986), no. 1, 161–194. MR 853449, DOI https://doi.org/10.1007/BF01391499
- Hélène Esnault and Eckart Viehweg, Lectures on vanishing theorems, DMV Seminar, vol. 20, Birkhäuser Verlag, Basel, 1992. MR 1193913
- Osamu Fujino, Foundations of the minimal model program, MSJ Memoirs, vol. 35, Mathematical Society of Japan, Tokyo, 2017. MR 3643725
- Osamu Fujino, Injectivity theorems, Higher dimensional algebraic geometry—in honour of Professor Yujiro Kawamata’s sixtieth birthday, Adv. Stud. Pure Math., vol. 74, Math. Soc. Japan, Tokyo, 2017, pp. 131–157. MR 3791211, DOI https://doi.org/10.2969/aspm/07410131
- Osamu Fujino, On semipositivity, injectivity and vanishing theorems, Hodge theory and $L^2$-analysis, Adv. Lect. Math. (ALM), vol. 39, Int. Press, Somerville, MA, 2017, pp. 245–282. MR 3751293
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364
- Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. MR 507725
- A. Grothendieck, A general theory of fiber spaces with structure sheaf, Univ. of Kansas report, 1958.
- C. Huang, K. Liu, X. Wan, and X. Yang, Logarithmic vanishing theorems on compact Kähler manifolds I, arXiv:1611.07671v1, 2016.
- Donatella Iacono, Deformations and obstructions of pairs $(X,D)$, Int. Math. Res. Not. IMRN 19 (2015), 9660–9695. MR 3431606, DOI https://doi.org/10.1093/imrn/rnu242
- Kazuya Kato and Sampei Usui, Classifying spaces of degenerating polarized Hodge structures, Annals of Mathematics Studies, vol. 169, Princeton University Press, Princeton, NJ, 2009. MR 2465224
- L. Katzarkov, M. Kontsevich, and T. Pantev, Hodge theoretic aspects of mirror symmetry, From Hodge theory to integrability and TQFT tt*-geometry, Proc. Sympos. Pure Math., vol. 78, Amer. Math. Soc., Providence, RI, 2008, pp. 87–174. MR 2483750, DOI https://doi.org/10.1090/pspum/078/2483750
- Yujiro Kawamata, On deformations of compactifiable complex manifolds, Math. Ann. 235 (1978), no. 3, 247–265. MR 499279, DOI https://doi.org/10.1007/BF01420124
- Yujiro Kawamata and Yoshinori Namikawa, Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties, Invent. Math. 118 (1994), no. 3, 395–409. MR 1296351, DOI https://doi.org/10.1007/BF01231538
- Kunihiko Kodaira, The theorem of Riemann-Roch on compact analytic surfaces, Amer. J. Math. 73 (1951), 813–875. MR 0048107, DOI https://doi.org/10.2307/2372120
- Kunihiko Kodaira, Complex manifolds and deformation of complex structures, translated from the Japanese by Kazuo Akao, with an appendix by Daisuke Fujiwara, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 283, Springer-Verlag, New York, 1986. MR 815922
- Kefeng Liu and Sheng Rao, Remarks on the Cartan formula and its applications, Asian J. Math. 16 (2012), no. 1, 157–169. MR 2904916, DOI https://doi.org/10.4310/AJM.2012.v16.n1.a6
- Kefeng Liu, Sheng Rao, and Xiaokui Yang, Quasi-isometry and deformations of Calabi-Yau manifolds, Invent. Math. 199 (2015), no. 2, 423–453. MR 3302118, DOI https://doi.org/10.1007/s00222-014-0516-1
- Kefeng Liu, Xiaofeng Sun, and Shing-Tung Yau, Recent development on the geometry of the Teichmüller and moduli spaces of Riemann surfaces, Surveys in differential geometry. Vol. XIV. Geometry of Riemann surfaces and their moduli spaces, Surv. Differ. Geom., vol. 14, Int. Press, Somerville, MA, 2009, pp. 221–259. MR 2655329, DOI https://doi.org/10.4310/SDG.2009.v14.n1.a9
- Shin-ichi Matsumura, A transcendental approach to injectivity theorem for log canonical pairs (English summary), Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19 (2019), no. 1, 311–334. MR 3923849
- James Morrow and Kunihiko Kodaira, Complex manifolds, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1971. MR 0302937
- Junjiro Noguchi, A short analytic proof of closedness of logarithmic forms, Kodai Math. J. 18 (1995), no. 2, 295–299. MR 1346909, DOI https://doi.org/10.2996/kmj/1138043426
- S. Rao, X. Wan, and Q. Zhao, Power series proofs for local stabilities of Kähler and balanced structures with mild $\partial \bar \partial$-lemma, arXiv:1609.05637v1, 2016.
- Sheng Rao and Quanting Zhao, Several special complex structures and their deformation properties, J. Geom. Anal. 28 (2018), no. 4, 2984–3047. MR 3881963, DOI https://doi.org/10.1007/s12220-017-9944-7
- K. Saito, On the uniformization of complements of discriminant loci, AMS Summer Institute, 1975.
- Michael Schneider, Halbstetigkeitssätze für relativ analytische Räume, Invent. Math. 16 (1972), 161–176 (German). MR 0301232, DOI https://doi.org/10.1007/BF01391215
- Michael E. Taylor, Partial differential equations, Texts in Applied Mathematics, vol. 23, Springer-Verlag, New York, 1996. Basic theory. MR 1395147
- Gang Tian, Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric, Mathematical aspects of string theory (San Diego, Calif., 1986) Adv. Ser. Math. Phys., vol. 1, World Sci. Publishing, Singapore, 1987, pp. 629–646. MR 915841
- Andrey N. Todorov, The Weil-Petersson geometry of the moduli space of $\textrm {SU}(n\geq 3)$ (Calabi-Yau) manifolds. I, Comm. Math. Phys. 126 (1989), no. 2, 325–346. MR 1027500
- J. Tu, The iteration method in analytic deformation theory, preprint, 2016.
- Claire Voisin, Hodge theory and complex algebraic geometry. I, translated from the French original by Leila Schneps, Cambridge Studies in Advanced Mathematics, vol. 76, Cambridge University Press, Cambridge, 2002. MR 1967689
- Quanting Zhao and Sheng Rao, Applications of the deformation formula of holomorphic one-forms, Pacific J. Math. 266 (2013), no. 1, 221–255. MR 3105782, DOI https://doi.org/10.2140/pjm.2013.266.221
- Quanting Zhao and Sheng Rao, Extension formulas and deformation invariance of Hodge numbers, C. R. Math. Acad. Sci. Paris 353 (2015), no. 11, 979–984 (English, with English and French summaries). MR 3419846, DOI https://doi.org/10.1016/j.crma.2015.09.004
Additional Information
Kefeng Liu
Affiliation:
Mathematical Sciences Research Center, Chongqing University of Technology, Chongqing 400054, China — and — Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90095
Email:
liu@math.ucla.edu
Sheng Rao
Affiliation:
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
Email:
raoshengmath@gmail.com, likeanyone@whu.edu.cn
Xueyuan Wan
Affiliation:
Department of Mathematical Sciences, Chalmers University of Technology, University of Gothenburg, 412 96 Gothenburg, Sweden
MR Author ID:
1158681
Email:
xwan@chalmers.se
Received by editor(s):
November 20, 2017
Received by editor(s) in revised form:
April 3, 2018
Published electronically:
July 19, 2019
Additional Notes:
The first author was partially supported by NSF (Grant No. 1510216). The second author was partially supported by NSFC (Grants No. 11671305, 11771339).
Article copyright:
© Copyright 2019
University Press, Inc.