Partially integrable almost complex structures
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analysis and mathematical physics, Tome 311 (2020), pp. 228-240
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

In this survey we review some old and new results on the problem for local existence of holomorphic functions on almost complex manifolds.
Keywords: almost complex structures, holomorphic functions, partially integrable almost complex structures, nearly Kähler manifolds, homogeneous almost complex spaces, twistor spaces.
@article{TM_2020_311_a12,
     author = {Oleg Mushkarov},
     title = {Partially integrable almost complex structures},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {228--240},
     year = {2020},
     volume = {311},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2020_311_a12/}
}
TY  - JOUR
AU  - Oleg Mushkarov
TI  - Partially integrable almost complex structures
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2020
SP  - 228
EP  - 240
VL  - 311
UR  - http://geodesic.mathdoc.fr/item/TM_2020_311_a12/
LA  - ru
ID  - TM_2020_311_a12
ER  - 
%0 Journal Article
%A Oleg Mushkarov
%T Partially integrable almost complex structures
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2020
%P 228-240
%V 311
%U http://geodesic.mathdoc.fr/item/TM_2020_311_a12/
%G ru
%F TM_2020_311_a12
Oleg Mushkarov. Partially integrable almost complex structures. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analysis and mathematical physics, Tome 311 (2020), pp. 228-240. http://geodesic.mathdoc.fr/item/TM_2020_311_a12/

[1] Abbena E., “An example of an almost Kähler manifold which is not Kählerian”, Boll. Unione Mat. Ital. Ser. 4 A, 3 (1984), 383–392 | MR | Zbl

[2] Atiyah M.F., Hitchin N.J., Singer I.M., “Self-duality in four-dimensional Riemannian geometry”, Proc. R. Soc. London A, 362 (1978), 425–461 | DOI | MR | Zbl

[3] Bär C., “Real Killing spinors and holonomy”, Commun. Math. Phys., 154:3 (1993), 509–521 | DOI | MR | Zbl

[4] Benson C., Gordon C.S., “Kähler and symplectic structures on nilmanifolds”, Topology, 27:4 (1988), 513–518 | DOI | MR | Zbl

[5] A. L. Besse, Einstein Manifolds, Springer, Berlin, 1987 | MR | Zbl

[6] Blair D.E., Davidov J., Muškarov O., “Hyperbolic twistor spaces”, Rocky Mt. J. Math., 35:5 (2005), 1437–1465 | DOI | MR | Zbl

[7] Butruille J.-B., “Classification des variété approximativement kähleriennes homogénes”, Ann. Global Anal. Geom., 27:3 (2005), 201–225 | DOI | MR | Zbl

[8] Calabi E., “Construction and properties of some 6-dimensional almost complex manifolds”, Trans. Amer. Math. Soc., 87:2 (1958), 407–438 | MR | Zbl

[9] Chern S.-s., Complex manifolds without potential theory, D. van Nostrand Co., Princeton, NJ, 1967 | MR | Zbl

[10] Cordero L.A., Fernández M., Gray A., “Symplectic manifolds with no Kähler structure”, Topology, 25 (1986), 375–380 | DOI | MR | Zbl

[11] Cordero L.A., Fernandez M., de Leon M., “Examples of compact non-Kähler almost Kähler manifolds”, Proc. Amer. Math. Soc., 95:2 (1985), 280–286 | MR | Zbl

[12] Davidov J., Díaz-Ramos J.C., García-Río E., Matsushita Y., Muškarov O., Vázquez-Lorenzo R., “Hermitian–Walker 4-manifolds”, J. Geom. Phys., 58:3 (2008), 307–323 | DOI | MR | Zbl

[13] Davidov J., Muškarov O., “Existence of holomorphic functions on twistor spaces”, Bull. Soc. math. Belg. B, 40 (1989), 131–151 | MR

[14] Davidov J., Muškarov O., Grantcharov G., “Almost complex structures on twistor spaces”, Almost complex structures, Proc. Int. Workshop (Sofia, 1992), World Scientific, Singapore, 1994, 113–149 | MR | Zbl

[15] Dimiev S., “Propriétés locales des fonctions presque-holomorphes”, Analytic functions, Proc. Conf. (Błażejewko, 1982), Lect. Notes Math., 1093, Springer, Berlin, 1983, 102–117 | DOI | MR

[16] Eells J., Salamon S., “Twistorial construction of harmonic maps of surfaces into four-manifolds”, Ann. Sc. Norm. Super. Pisa. Cl. Sci. Ser. 4, 12 (1985), 589–640 | MR | Zbl

[17] Ehresmann C., “Sur la théorie des espaces fibres”, Topologie algébrique (Paris, 1947), Colloq. Int. CNRS, 12, CNRS, Paris, 1949, 3–15 | MR

[18] Foscolo L., Haskins M., “New $\mathrm G_2$-holonomy cones and exotic nearly Kähler structures on $S^6$ and $S^3\times S^3$”, Ann. Math. Ser. 2, 185:1 (2017), 59–130 | DOI | MR | Zbl

[19] Friedrich T., Grunewald R., “On the first eigenvalue of the Dirac operator on 6-dimensional manifolds”, Ann. Global Anal. Geom., 3:3 (1985), 265–273 | DOI | MR | Zbl

[20] Fukami T., Ishihara S., “Almost Hermitian structure on $S^6$”, Tohoku Math J. Ser. 2, 7 (1955), 151–156 | DOI | MR | Zbl

[21] Gardner R.B., “Invariants of Pfaffian systems”, Trans. Amer. Math. Soc., 126 (1967), 514–533 | DOI | MR | Zbl

[22] De Graaf W.A., “Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2”, J. Algebra, 309:2 (2007), 640–653 | DOI | MR | Zbl

[23] Grantcharov G., Muškarov O., “On a class of Riemannian $k$-symmetric spaces”, Rend. Semin. Mat. Torino, 48:1 (1990), 61–80 | MR | Zbl

[24] Gray A., “Nearly Kähler manifolds”, J. Diff. Geom., 4 (1970), 283–309 | DOI | MR | Zbl

[25] Gray A., “The structure of nearly Kähler manifolds”, Math. Ann., 223 (1976), 233–248 | DOI | MR | Zbl

[26] Gromov M., “Pseudo holomorphic curves in symplectic manifolds”, Invent. math., 82 (1985), 307–347 | DOI | MR | Zbl

[27] Gualtieri M., Generalized complex geometry, PhD thesis, St John's College, Univ. Oxford, Oxford, 2003;, arXiv: math.DG/0401221 [math.DG]

[28] Han C.-K., Kim H., “Holomorphic functions on almost complex manifolds”, J. Korean Math. Soc., 49:2 (2012), 379–394 | DOI | MR | Zbl

[29] Hermann R., “Compact homogeneous almost complex spaces of positive characteristic”, Trans. Amer. Math. Soc., 83 (1956), 471–481 | DOI | MR | Zbl

[30] Hitchin N., “Compact four-dimensional Einstein manifolds”, J. Diff. Geom., 9 (1974), 435–441 | DOI | MR | Zbl

[31] Hitchin N.J., “Kählerian twistor spaces”, Proc. London Math. Soc. Ser. 3, 43 (1981), 133–150 | DOI | MR | Zbl

[32] Hitchin N., “Instantons, Poisson structures and generalized Kähler geometry”, Commun. Math. Phys., 265:1 (2006), 131–164 | DOI | MR | Zbl

[33] Hopf H., “Zur Topologie der komplexen Mannigfaltigkeiten”, Studies and essays presented to R. Courant on his 60th birthday, January 8, 1948, Interscience, New York, 1948, 167–185 | MR

[34] Kim H., “Partial integrability on Thurston manifolds”, Ann. Pol. math., 109:3 (2013), 261–269 | DOI | MR | Zbl

[35] V. F. Kirichenko, “$K$-spaces of maximal rank”, Math. Notes, 22:4 (1977), 751–757 | DOI | MR | MR | Zbl

[36] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, v. 2, Interscience, New York, 1969 | MR | Zbl

[37] Kodaira K., Spencer D.C., “On the variation of almost-complex structure”, Algebraic geometry and topology: A symposium in honor of S. Lefschetz, Princeton Math. Ser., 12, Princeton Univ. Press, Princeton, NJ, 1957, 139–150 | MR | Zbl

[38] Kruglikov B.S., “Non-existence of higher-dimensional pseudoholomorphic submanifolds”, Manuscr. math., 111:1 (2003), 51–69 | DOI | MR | Zbl

[39] Ledger A.J., Obata M., “Affine and Riemannian $s$-manifolds”, J. Diff. Geom., 2:4 (1968), 451–459 | DOI | MR | Zbl

[40] Matsushita Y., “Fields of 2-planes and two kinds of almost complex structures on compact 4-dimensional manifolds”, Math. Z., 207:2 (1991), 281–291 | DOI | MR | Zbl

[41] McDuff D., Salamon D., $J$-holomorphic curves and quantum cohomology, Univ. Lect. Ser., 6, Amer. Math. Soc., Providence, RI, 1994 | DOI | MR | Zbl

[42] Milnor J., “Curvatures of left invariant metrics on Lie groups”, Adv. Math., 21:3 (1976), 293–329 | DOI | MR | Zbl

[43] Momchilov H.A., Mushkarov O.K., “Some examples of Calabi's manifolds of type 1”, C. r. Acad. Bulg. sci., 39:1 (1986), 27–29 | MR | Zbl

[44] Mushkarov O.K., “Familles normales de fonctions presque-holomorphes”, PLISKA. Stud. Math. Bulg., 4 (1981), 58–61 | MR | Zbl

[45] Mushkarov O.K., “Almost complex manifolds without almost holomorphic functions”, C. r. Acad. Bulg. sci., 34:9 (1981), 1225–1228 | MR | Zbl

[46] Muškarov O., “Existence of holomorphic functions on almost complex manifolds”, Math. Z., 192 (1986), 283–295 | DOI | MR

[47] Muškarov O., “Two remarks on Thurston's example”, Complex analysis and applications, Proc. Conf. (Varna, 1985), Bulg. Acad. Sci., Sofia, 1986, 461–468 | MR

[48] Muškarov O., “On a class of almost complex group manifolds”, Simon Stevin, 61:3–4 (1987), 197–216 | MR

[49] Muškarov O., “Some remarks on generalized Grassmann manifolds”, Acta math. Hung., 54:1–2 (1989), 69–78 | MR

[50] Mushkarov O., Yankov C., “Existence of holomorphic functions on nilpotent Lie groups”, C. r. Acad. Bulg. sci., 67:9 (2014), 1187–1192 | MR | Zbl

[51] Mushkarov O., Yankov C., “Partial integrability of almost complex structures on Thurston manifolds”, Ann. Pol. math., 118:2–3 (2016), 141–148 | DOI | MR | Zbl

[52] Nagy P.-A., “Nearly Kähler geometry and Riemannian foliations”, Asian J. Math., 6:3 (2002), 481–504 | DOI | MR | Zbl

[53] Newlander A., Nirenberg L., “Complex analytic coordinates in almost complex manifolds”, Ann. Math. Ser. 2, 65 (1957), 391–404 | DOI | MR | Zbl

[54] Penrose R., “Twistor theory, its aims and achievements”, Quantum gravity: An Oxford symposium, Clarendon Press, Oxford, 1975, 268–407 | MR

[55] Petean J., “Indefinite Kähler–Einstein metrics on compact complex surfaces”, Commun. Math. Phys., 189:1 (1997), 227–235 | DOI | MR | Zbl

[56] Salamon S.M., “Complex structures on nilpotent Lie algebras”, J. Pure Appl. Algebra, 157:2–3 (2001), 311–333 | DOI | MR | Zbl

[57] Spencer D.C., “Potential theory and almost-complex manifolds”, Lectures on functions of a complex variable, Univ. Mich. Press, Ann Arbor, 1955, 15–43 | MR

[58] Steenrod N., The topology of fibre bundles, Princeton Math. Ser., 14, Princeton Univ. Press, Princeton, NJ, 1951 | MR | Zbl

[59] Tachibana S.-I., “On almost-analytic vectors in certain almost-hermitian manifolds”, Tohoku Math. J. Ser. 2, 11:3 (1959), 351–363 | DOI | MR | Zbl

[60] Thurston W.P., “Some simple examples of symplectic manifolds”, Proc. Amer. Math. Soc., 55:2 (1976), 467–468 | MR | Zbl

[61] Wolf J.A., “The geometry and structure of isotropy irreducible homogeneous spaces”, Acta math., 120 (1968), 59–148 | DOI | MR | Zbl

[62] Wolf J.A., Gray A., “Homogeneous spaces defined by Lie group automorphisms. I, II”, J. Diff. Geom., 2 (1968), 77–114, 115–159 | DOI | MR | Zbl