Almost complex structures, transverse complex structures, and transverse Dolbeault cohomology. (English) Zbl 07928006
Summary: We define a transverse Dolbeault cohomology associated to any almost complex structure \(j\) on a smooth manifold \(M\). This we do by extending the notion of transverse complex structure and by introducing a natural \(j\)-stable involutive limit distribution with such a transverse complex structure. We relate this transverse Dolbeault cohomology to the generalized Dolbeault cohomology of \((M, j)\) introduced by Cirici and Wilson in 2001, showing that the \((p, 0)\) cohomology spaces coincide. This study of transversality leads us to suggest a notion of minimally non-integrable almost complex structure.
MSC:
| 53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |
Keywords:
almost complex structures; transverse structures; Dolbeault cohomology; derived flag of distributionsReferences:
| [1] | Bozzetti, C. and Medori, C.: Almost complex manifolds with non-degenerate torsion. Int. J. Geom. Methods Mod. Phys. 14 (2017), no. 3, article no. 1750033, 25 pp. · Zbl 1378.32017 · doi:10.1142/S0219887817500335 |
| [2] | Cahen, M., Gérard, M., Gutt, S. and Hayyani, M.: Distributions associated to almost complex structures on symplectic manifolds. J. Symplectic Geom. 19 (2021), no. 5, 1071-1094. · Zbl 1495.53088 · doi:10.4310/jsg.2021.v19.n5.a2 |
| [3] | Cahen, M., Gutt, S. and Rawnsley, J.: On twistor almost complex structures. J. Geom. Mech. 13 (2021), no. 3, 313-331. · Zbl 1489.53036 · doi:10.3934/jgm.2021006 |
| [4] | Cirici, J. and Wilson, S. O.: Dolbeault cohomology for almost complex manifolds. Adv. Math. 391 (2021), article no. 107970, 52 pp. · Zbl 1478.32085 · doi:10.1016/j.aim.2021.107970 |
| [5] | Coelho, R., Placini, G. and Stelzig, J.: Maximally non-integrable almost complex structures: an h-principle and cohomological properties. Selecta Math. (N.S.) 28 (2022), no. 5, article no. 83, 25 pp. · Zbl 1504.32074 · doi:10.1007/s00029-022-00792-0 |
| [6] | Draghici, T., Li, T.-J. and Zhang, W.: On the J -anti-invariant cohomology of almost complex 4-manifolds. Q. J. Math. 64 (2013), no. 1, 83-111. · Zbl 1271.32029 · doi:10.1093/qmath/har034 |
| [7] | Eells, J. and Salamon, S.: Twistorial construction of harmonic maps of surfaces into four-manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), no. 4, 589-640. · Zbl 0627.58019 |
| [8] | Hirzebruch, F.: Some problems on differentiable and complex manifolds. Ann. of Math. (2) 60 (1954), no. 2, 213-236. · Zbl 0056.16803 · doi:10.2307/1969629 |
| [9] | Hodge, W. V. D.: The theory and applications of harmonic integrals. Cambridge University Press, Cambridge; The Macmillan Company, New York, 1941. · Zbl 0024.39703 |
| [10] | Holt, T. and Zhang, W.: Harmonic forms on the Kodaira-Thurston manifold. Adv. Math. 400 (2022), article no. 108277, 30 pp. · Zbl 1523.32046 · doi:10.1016/j.aim.2022.108277 |
| [11] | Kobayashi, S. and Nomizu, K.: Foundations of differential geometry. Vol. II. Interscience Tracts in Pure and Applied Mathematics 15, Vol. II, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1969. · Zbl 0175.48504 |
| [12] | Li, T.-J. and Zhang, W.: Comparing tamed and compatible symplectic cones and cohomolo-gical properties of almost complex manifolds. Comm. Anal. Geom. 17 (2009), nno. 4, 651-683. · Zbl 1225.53066 · doi:10.4310/CAG.2009.v17.n4.a4 |
| [13] | Muškarov, O.: Existence of holomorphic functions on almost complex manifolds. Math. Z. 192 (1986), no. 2, 283-295. · Zbl 0579.53032 · doi:10.1007/BF01179429 |
| [14] | Sillari, L. and Tomassini, A.: Dolbeault and J -invariant cohomologies on almost complex manifolds. Complex Anal. Oper. Theory 15 (2021), no. 7, article no. 112, 28 pp. · Zbl 1484.32045 · doi:10.1007/s11785-021-01156-w |
| [15] | Sillari, L. and Tomassini, A.: On Bott-Chern and Aeppli cohomologies of almost com-plex manifolds and related spaces of harmonic forms. Expo. Math. 41 (2023), no. 4, article no. 125517, 34 pp. · Zbl 1537.32092 · doi:10.1016/j.exmath.2023.09.001 |
| [16] | Sillari, L. and Tomassini, A.: Rank of the Nijenhuis tensor on parallelizable almost complex manifolds. Internat. J. Math. 34 (2023), no. 11, article no. 2350063, 35 pp. · Zbl 1523.32048 · doi:10.1142/S0129167X23500635 |
| [17] | Tardini, N. and Tomassini, A.: N @-Harmonic forms on 4-dimensional almost-Hermitian mani-folds. Math. Res. Lett. 30 (2023), no. 5, 1617-1637. · Zbl 07849872 · doi:10.4310/MRL.2023.v30.n5.a14 |
| [18] | Verbitsky, M.: Hodge theory on nearly Kähler manifolds. Geom. Topol. 15 (2011), no. 4, 2111-2133. · Zbl 1246.58002 · doi:10.2140/gt.2011.15.2111 |
| [19] | Zhang, W.: Almost complex Hodge theory. Riv. Math. Univ. Parma (N.S.) 13 (2022), no. 2, 481-504. · Zbl 1517.32091 |
| [20] | . |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
