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Deformed Aeppli cohomology: canonical deformations and jumping formulas

Part of: Homology and cohomology theories Holomorphic functions of several complex variables Deformations of analytic structures

Published online by Cambridge University Press:  21 July 2025

Yan Hu  and
Wei Xia
Yan Hu
Affiliation:
Mathematical Science Research Center, Chongqing University of Technology, Chongqing, China e-mail: huyan1128@126.com xiawei@cqut.edu.cn
Wei Xia*
Affiliation:
Mathematical Science Research Center, Chongqing University of Technology, Chongqing, China e-mail: huyan1128@126.com xiawei@cqut.edu.cn
*
e-mail: xiaweiwei3@126.com
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Abstract

Given a complex analytic family of complex manifolds, we consider canonical Aeppli deformations of $(p,q)$-forms and study its relations to the varying of dimension of the deformed Aeppli cohomology $\dim H^{\bullet ,\bullet }_{A\phi (t)}(X)$. In particular, we prove the jumping formula for the deformed Aeppli cohomology $H^{\bullet ,\bullet }_{A\phi (t)}(X)$. As a direct consequence, $\dim H^{p,q}_{A\phi (t)}(X)$ remains constant iff the Bott–Chern deformations of $(n-p,n-q)$-forms and the Aeppli deformations of $(n-p-1,n-q-1)$-forms are canonically unobstructed. Furthermore, the Bott–Chern/Aeppli deformations are shown to be unobstructed if some weak forms of ${ \partial }{ \bar {\partial } }$-lemma is satisfied.

Keywords

Deformations of complex structuresAeppli cohomologyBott-Chern cohomologyjumping formulas

MSC classification

Primary: 32G05: Deformations of complex structures
Secondary: 32A05: Power series, series of functions 55N99: None of the above, but in this section

Information

Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 34
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

This work is supported by the National Natural Science Foundation of China (Grant No. 11901590), the Natural Science Foundation of Chongqing (China) (Grant No. CSTB2022NSCQ-MSX0876). This work is also partially supported by the Natural Science Foundation of Chongqing (China) (Grant Nos. CSTB2024NSCQ-LZX0040 and CSTB2023NSCQ-LZX0042).

References

Angella, D., Dloussky, G., and Tomassini, A., On Bott-Chern cohomology of compact complex surfaces . Ann. Mat. Pura Appl. (4) 195(2016), no. 1, 199–217.10.1007/s10231-014-0458-7CrossRefGoogle Scholar
Angella, D. and Kasuya, H., Bott-Chern cohomology of solvmanifolds . Ann. Global Anal. Geom. 52(2017), no. 4, 363–411.10.1007/s10455-017-9560-6CrossRefGoogle Scholar
Angella, D. and Kasuya, H., Cohomologies of deformations of solvmanifolds and closedness of some properties . North-West. Eur. J. Math. 3(2017), 75–105.Google Scholar
Alessandrini, L., Weak forms of $\partial \overline{\partial}$ -lemma on compact complex manifolds. Preprint. arXiv:1909.07037.Google Scholar
Angella, D., The cohomologies of the Iwasawa manifold and of its small deformations . J. Geom. Anal. 23(2013), no. 3, 1355–1378.10.1007/s12220-011-9291-zCrossRefGoogle Scholar
Angella, D. and Tomassini, A., On the $\partial \overline{\partial}$ -lemma and Bott-Chern cohomology . Invent. Math. 192(2013), no. 1, 71–81.10.1007/s00222-012-0406-3CrossRefGoogle Scholar
Angella, D. and Tomassini, A., Inequalities á la Frölicher and cohomological decompositions . J. Noncommut. Geom. 9(2015), no. 2, 505–542.10.4171/jncg/199CrossRefGoogle Scholar
Angella, D. and Ugarte, L., Locally conformal Hermitian metrics on complex non-Kähler manifolds . Mediterr. J. Math. 13(2016), no. 4, 2105–2145.10.1007/s00009-015-0586-3CrossRefGoogle Scholar
Angella, D. and Ugarte, L., On small deformations of balanced manifolds . Differ. Geom. Appl. 54(2017), 464–474. (part B).10.1016/j.difgeo.2017.07.010CrossRefGoogle Scholar
Bertin, J., Demailly, J.-P., Illusie, L., and Peters, C., Introduction to Hodge theory, volume 8 of SMF/AMS Texts and Monographs, American Mathematical Society, Providence, RI, 2002. Société Mathématique de France, Paris, Translated from the 1996 French original by James Lewis and Chris Peters.Google Scholar
Bandiera, R. and Manetti, M., Algebraic models of local period maps and Yukawa algebras . Lett. Math. Phys. 108(2018), no. 9, 2055–2097.CrossRefGoogle Scholar
Clemens, H., Geometry of formal Kuranishi theory . Adv. Math. 198(2005), no. 1, 311–365.10.1016/j.aim.2005.04.015CrossRefGoogle Scholar
De Bartolomeis, P. and Tomassini, A., On solvable generalized Calabi-Yau manifolds . Ann. Inst. Fourier (Grenoble) 56(2006), no. 5, 1281–1296.10.5802/aif.2213CrossRefGoogle Scholar
Flenner, H., Ein Kriterium für die Offenheit der Versalität . Math. Z. 178(1981), no. 4, 449–473.CrossRefGoogle Scholar
Fiorenza, D. and Manetti, M., L-infinity algebras, Cartan homotopies and period maps. Preprint, 2006. arXiv: math/0605297.Google Scholar
Fiorenza, D. and Manetti, M., A period map for generalized deformations . J. Noncommut. Geom. 3(2009), no. 4, 579–597.10.4171/jncg/47CrossRefGoogle Scholar
Fu, J. and Yau, S.-T., A note on small deformations of balanced manifolds . C. R. Math. Acad. Sci. Paris 349(2011), nos. 13–14, 793–796.10.1016/j.crma.2011.06.023CrossRefGoogle Scholar
Hasegawa, K., Small deformations and non-left-invariant complex structures on six-dimensional compact solvmanifolds . Differ. Geom. Appl. 28(2010), no. 2, 220–227.10.1016/j.difgeo.2009.10.003CrossRefGoogle Scholar
Liu, K., Rao, S., and Yang, X., Quasi-isometry and deformations of Calabi-Yau manifolds . Invent. Math. 199(2015), no. 2, 423–453.10.1007/s00222-014-0516-1CrossRefGoogle Scholar
Liu, K., Sun, X., and Yau, S.-T., Recent development on the geometry of the Teichmüller and moduli spaces of Riemann surfaces . In: L. Ji, S. A. Wolpert and S.-T. Yau. (eds.), Geometry of Riemann surfaces and their moduli spaces, volume XIV of Surveys in differential geometry, 2009, pp. 221–259.10.4310/SDG.2009.v14.n1.a9CrossRefGoogle Scholar
Lin, J. and Ye, X., Dimension jumps in Bott-Chern and Aeppli cohomology groups . Pacific J. Math. 273(2015), no. 2, 461–482.10.2140/pjm.2015.273.461CrossRefGoogle Scholar
Nakamura, I., Complex parallelisable manifolds and their small deformations . J. Differ. Geom. 10(1975), no. 1, 85–112.10.4310/jdg/1214432677CrossRefGoogle Scholar
Popovici, D., Aeppli cohomology classes associated with Gauduchon metrics on compact complex manifolds . Bull. Soc. Math. France 143(2015), no. 4, 763–800.10.24033/bsmf.2704CrossRefGoogle Scholar
Popovici, D., Holomorphic deformations of balanced Calabi-Yau $\partial \overline{\partial}$ -manifolds . Ann. Inst. Fourier (Grenoble) 69(2019), no. 2, 673–728.10.5802/aif.3254CrossRefGoogle Scholar
Popovici, D., Stelzig, J., and Ugarte, L.. Higher-page Bott–Chern and Aeppli cohomologies and applications . J. Reine Angew. Math. 777(2021), 157–194.10.1515/crelle-2021-0014CrossRefGoogle Scholar
Piovani, R. and Tomassini, A., Aeppli cohomology and Gauduchon metrics . Complex Anal. Oper. Theory 14(2020), Article number 22.10.1007/s11785-020-00984-6CrossRefGoogle Scholar
Rao, S., Wan, X., and Zhao, Q., On local stabilities of $p$ -Kähler structures . Compos. Math. 155(2019), no. 3, 455–483.10.1112/S0010437X19007085CrossRefGoogle Scholar
Rao, S., Wan, X., and Zhao, Q., Power series proofs for local stabilities of Kähler and balanced structures with mild $\partial \overline{\partial}$ -lemma . Nagoya Math. J. 246(2022), 305–354. https://doi.org/10.1017/nmj.2021.4.CrossRefGoogle Scholar
Rao, S. and Zhao, Q., Several special complex structures and their deformation properties . J. Geom. Anal. 28(2018), no. 4, 2984–3047.10.1007/s12220-017-9944-7CrossRefGoogle Scholar
Rao, S. and Zhang, R., On extension of closed complex (basic) differential forms: Hodge numbers and (transversely) $p$ -Kähler structures. Preprint, 2022. arXiv:2204.06870.Google Scholar
Rao, S. and Zou, Y., $\partial \overline{\partial}$ -lemma and double complex . to appear in Commun. Math. Stat. (2024). https://doi.org/10.1007/s40304-024-00400-x CrossRefGoogle Scholar
Sakane, Y., On compact complex parallelisable solvmanifolds . Osaka Math. J. 13(1976), no. 1, 187–212.Google Scholar
Schweitzer, M., Autour de la cohomologie de Bott-Chern. Preprint, 2007. arXiv:0709.3528v1 [math.AG]Google Scholar
Stelzig, J., On the structure of double complexes . J. Lond. Math. Soc. (2) 104(2021), no. 2, 956–988.10.1112/jlms.12453CrossRefGoogle Scholar
Wei, D. and Zhu, S., Two applications of the $\partial \overline{\partial}$ -Hodge theory . Chinese Ann. Math. Ser. B 45(2024), no. 1, 137–150.10.1007/s11401-024-0007-7CrossRefGoogle Scholar
Xia, W., Derivations on almost complex manifolds . Proc. Amer. Math. Soc. 147(2019), 559–566. Errata in arXiv:1809.07443v3.10.1090/proc/14281CrossRefGoogle Scholar
Xia, W., On the deformed Bott-Chern cohomology . J. Geom. Phys. 166(2021), Paper No. 104250. 19.10.1016/j.geomphys.2021.104250CrossRefGoogle Scholar
Xia, W., Deformations of Dolbeault cohomology classes . Math. Z. 300(2022), 2931–2973.10.1007/s00209-021-02900-wCrossRefGoogle Scholar
Yang, S. and Yang, X., Bott-Chern blow-up formulae and the bimeromorphic invariance of the $\partial \overline{\partial}$ -lemma for threefolds . Trans. Amer. Math. Soc. 373(2020), no. 12, 8885–8909.10.1090/tran/8213CrossRefGoogle Scholar
Zhao, Q. and Rao, S., Extension formulas and deformation invariance of Hodge numbers . C. R. Math. Acad. Sci. Paris 353(2015), no. 11, 979–984.10.1016/j.crma.2015.09.004CrossRefGoogle Scholar