Details

Weiyi Zhang ^[a]

Almost complex Hodge theory

Pages: 481-504
Received: 2 December 2021
Accepted: 15 February 2022
Mathematics Subject Classification: 32Q60, 53C15, 58A14.
Keywords: Almost complex manifolds, Hodge theory, Harmonic analysis, Stokes phenomenon.
Author address:
[a]: University of Warwick, Mathematics Institute, England

Abstract: We review the recent development of Hodge theory for almost complex manifolds. This includes the determination of whether the Hodge numbers defined by \(\bar\partial\)-Laplacian are almost complex, almost Kähler, or birational invariants in dimension four.

References

[1]: D. Angella and V. Tosatti, Leafwise flat forms on Inoue-Bombieri surfaces, arXiv:2106.16141 [math.DG], preprint, 2021. DOI
[2]: L. Auslander, Lecture notes on nil-theta functions, Regional Conference Series in Mathematics, No. 34, AMS, Providence, R.I.,1977. MR0466409
[3]: L. Auslander and R. Tolimieri, Abelian harmonic analysis, theta functions and function algebras on a nilmanifold, Lecture Notes in Mathematics, 436, Springer-Verlag, Berlin-New York, 1975. MR0414785
[4]: L. Bonthrone and W. Zhang, \(J\)-holomorphic curves from closed \(J\)-anti-invariant forms, Comm. Anal. Geom., to appear.
[5]: N. Buchdahl, On compact Kähler surfaces, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 1, 287-302. MR1688136
[6]: H. Chen and W. Zhang, Kodaira Dimensions of almost complex manifolds I, Amer. J. Math., to appear.
[7]: H. Chen and W. Zhang, Kodaira dimensions of almost complex manifolds II, Comm. Anal. Geom., to appear.
[8]: J. Cirici and S. Wilson, Topological and geometric aspects of almost Kähler manifolds via harmonic theory, Selecta Math. (N.S.) 26 (2020), no. 3, Paper No. 35, 27 pp. MR4110721
[9]: C. Deninger and W. Singhof, The \(e\)-invariant and the spectrum of the Laplacian for compact nilmanifolds covered by Heisenberg groups, Invent. Math. 78 (1984), no. 1, 101-112. MR0762355
[10]: S. K. Donaldson, Yang-Mills invariants of four-manifolds, Geometry of low-dimensional manifolds, 1 (Durham, 1989), London Math. Soc. Lecture Note Ser., 150, Cambridge Univ. Press, Cambridge, 1990, 5-40. MR1171888
[11]: T. Draghici, T.-J. Li and W. Zhang, Symplectic forms and cohomology decomposition of almost complex four-manifolds, Int. Math. Res. Not. IMRN 2010, no. 1, 1-17. MR2576281
[12]: G. B. Folland, Harmonic analysis in phase space, Annals of Mathematics Studies, 122, Princeton University Press, Princeton, NJ, 1989. MR0983366
[13]: F. Hirzebruch, Some problems on differentiable and complex manifolds, Ann. of Math. (2) 60 (1954), 213-236. MR0066013
[14]: T. Holt, Bott-Chern and \(\bar\partial\) harmonic forms on almost Hermitian \(4\)-manifolds, Math. Z. 302 (2022), no. 1, 47-72. MR4462670
[15]: T. Holt, Solving the Kodaira-Spencer problem using harmonic analysis on torus bundles over \(S^1\), Warwick thesis, 2022.
[16]: T. Holt and W. Zhang, Harmonic forms on the Kodaira-Thurston manifold, Adv. Math. 400 (2022), Paper No. 108277, 30 pp. MR4385143
[17]: T. Holt and W. Zhang, Almost Kähler Kodaira-Spencer problem, Math. Res. Lett., to appear.
[18]: A. Lamari, Courants kählériens et surfaces compactes, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 1, 263-285. MR1688140
[19]: T.-J. Li and W. Zhang, Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Comm. Anal. Geom. 17 (2009), no. 4, 651-683. MR2601348
[20]: Y. Miyaoka, Kähler metrics on elliptic surfaces, Proc. Japan Acad. 50 (1974), 533-536. MR0460730
[21]: R. Piovani and A. Tomassini, Bott-Chern Laplacian on almost Hermitian manifolds, Math. Z. 301 (2022), no. 3, 2685-2707. MR4437335
[22]: R. Piovani and A. Tomassini, On the dimension of Dolbeault harmonic \((1,1)\)-forms on almost Hermitian \(4\)-manifolds, Pure Appl. Math. Q. 18 (2022), no. 3, 1187-1201.
[23]: R. Piovani, Dolbeault harmonic \((1,1)\)-forms on \(4\)-dimensional compact quotients of Lie groups with a left invariant almost Hermitian structure, J. Geom. Phys. 180 (2022), Paper No. 104639, 18 pp. MR4470102
[24]: L. F. Richardson, Global solvability on compact Heisenberg manifolds, Trans. Amer. Math. Soc. 273 (1982), no. 1, 309-317. MR0664044
[25]: D. Ruberman and N. Saveliev, On the spectral sets of Inoue surfaces, in ''Gauge theory and low-dimensional topology: progress and interaction'', The Open Book Series, 5, MSP, Berkeley, CA, 2022, 285-297, DOI: 10.2140/obs.2022.5.285.
[26]: Y. T. Siu, Analyticity of sets associated to Lelong numbers and the extension of closed positive currents, Invent. Math. 27 (1974), 53-156. MR0352516
[27]: E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, With the assistance of T. S. Murphy, Princeton Mathematical Series, 43, Monographs in Harmonic Analysis, III, Princeton University Press, Princeton, NJ, 1993. MR1232192
[28]: N. Tardini and A. Tomassini, \(\bar{\partial}\)-Harmonic forms on \(4\)-dimensional almost-Hermitian manifolds, Math. Res. Lett., to appear.
[29]: C. H. Taubes, \({\rm SW}\Rightarrow{\rm Gr}\): from the Seiberg-Witten equations to pseudo-holomorphic curves, J. Amer. Math. Soc. 9 (1996), no. 3, 845-918. MR1362874
[30]: M. Verbitsky, Hodge theory on nearly Kähler manifolds, Geom. Topol. 15 (2011), no. 4, 2111-2133. MR2860989
[31]: C. Voisin, Hodge theory and complex algebraic geometry, I, Translated from the French original by L. Schneps, Cambridge Studies in Advanced Mathematics, 76, Cambridge University Press, Cambridge, 2002. MR1967689
[32]: W. Zhang, Intersection of almost complex submanifolds, Camb. J. Math. 6 (2018), no. 4, 451-496. MR3870361