Classification of non-Kähler surfaces and locally conformally Kähler geometry

M. S. Verbitsky; V. Vuletescu; L. Ornea

M. S. Verbitsky ; V. Vuletescu ; L. Ornea

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 76 (2021) no. 2, pp. 261-289 Cet article a éte moissonné depuis la source Math-Net.Ru

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Résumé

The Enriques–Kodaira classification treats non-Kähler surfaces as a special case within the Kodaira framework. We prove the classification results for non-Kähler complex surfaces without relying on the machinery of the Enriques–Kodaira classification, and deduce the classification theorem for non-Kähler surfaces from the Buchdahl–Lamari theorem. We also prove that all non-Kähler surfaces which are not of class VII are locally conformally Kähler. Bibliography: 64 titles.

Keywords: locally conformally Kähler surface, Kato surface, elliptic fibration.

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M. S. Verbitsky; V. Vuletescu; L. Ornea. Classification of non-Kähler surfaces and locally conformally Kähler geometry. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 76 (2021) no. 2, pp. 261-289. http://geodesic.mathdoc.fr/item/RM_2021_76_2_a1/

Bibliographie
Cité par

[1] D. Angella, A. Tomassini, M. Verbitsky, “On non-Kähler degrees of complex manifolds”, Adv. Geom., 19:1 (2019), 65–69 ; (2017 (v1 – 2016)), 5 pp., arXiv: 1605.03368 | DOI | MR | Zbl

[2] V. Apostolov, G. Dloussky, “On the Lee classes of locally conformally symplectic complex surfaces”, J. Symplectic Geom., 16:4 (2018), 931–958 ; (2016), 18 pp., arXiv: 1611.00074 | DOI | MR | Zbl

[3] D. Barlet, Gauduchon's form and compactness of the space of divisors, 2017, 12 pp., arXiv: 1705.01743

[4] D. Barlet, J. Magnússon, Cycles analytiques complexes, v. I, Cours Spéc., 22, Théorèmes de préparation des cycles, Soc. Math. France, Paris, 2014, 525 pp. | MR | Zbl

[5] W. P. Barth, K. Hulek, C. A. M. Peters, A. Van de Ven, Compact complex surfaces, Ergeb. Math. Grenzgeb. (3), 4, 2nd ed., Springer-Verlag, Berlin, 2004, xii+436 pp. | DOI | MR | Zbl

[6] F. A. Belgun, “On the metric structure of non-Kähler complex surfaces”, Math. Ann., 317:1 (2000), 1–40 | DOI | MR | Zbl

[7] A. L. Besse, Einstein manifolds, Ergeb. Math. Grenzgeb. (3), 10, Springer-Verlag, Berlin, 1987, xii+510 pp. | DOI | MR | MR | Zbl | Zbl

[8] E. Bishop, “Conditions for the analyticity of certain sets”, Michigan Math. J., 11:4 (1964), 289–304 | DOI | MR | Zbl

[9] A. Blanchard, “Sur les variétés analytiques complexes”, Ann. Sci. Ecole Norm. Sup. (3), 73:2 (1956), 157–202 | DOI | MR | Zbl

[10] F. A. Bogomolov, “Classification of surfaces of class $\mathrm{VII}_0$ with $b_2=0$”, Math. USSR-Izv., 10:2 (1976), 255–269 | DOI | MR | Zbl

[11] F. A. Bogomolov, “Surfaces of class $\mathrm{VII}_0$ and affine geometry”, Math. USSR-Izv., 21:1 (1983), 31–73 | DOI | MR | Zbl

[12] V. Brînzănescu, “Neron-Severi group for nonalgebraic elliptic surfaces. II. Non-kählerian case”, Manuscripta Math., 84:3-4 (1994), 415–420 | DOI | MR | Zbl

[13] V. Brînzănescu, Holomorphic vector bundles over compact complex surfaces, Lecture Notes in Math., 1624, Springer-Verlag, Berlin, 1996, x+170 pp. | DOI | MR | Zbl

[14] V. Brînzănescu, P. Flondor, “Holomorphic 2-vector bundles on nonalgebraic 2-tori”, J. Reine Angew. Math., 1985:363 (1985), 47–58 | DOI | MR | Zbl

[15] V. Brînzănescu, P. Flondor, “Quadratic intersection form and 2-vector bundles on nonalgebraic surfaces”, Proceedings of the conference on algebraic geometry (Berlin, 1985), Teubner-Texte Math., 92, Teubner, Leipzig, 1986, 53–64 | MR | Zbl

[16] M. Brunella, “Locally conformally Kähler metrics on certain non-Kählerian surfaces”, Math. Ann., 346:3 (2010), 629–639 | DOI | MR | Zbl

[17] M. Brunella, “Locally conformally Kähler metrics on Kato surfaces”, Nagoya Math. J., 202 (2011), 77–81 ; (2010), 4 pp., arXiv: 1001.0530 | DOI | MR | Zbl

[18] N. Buchdahl, “On compact Kähler surfaces”, Ann. Inst. Fourier (Grenoble), 49:1 (1999), 287–302 | DOI | MR | Zbl

[19] C. H. Clemens, “Degeneration of Kähler manifolds”, Duke Math. J., 44:2 (1977), 215–290 | DOI | MR | Zbl

[20] J.-P. Demailly, “Estimations $L^2$ pour l'opérateur $\bar\partial$ d'un fibré vectoriel holomorphe semi-positif au-dessus d'une variété kählérienne complète”, Ann. Sci. École Norm. Sup. (4), 15:3 (1982), 457–511 | DOI | MR | Zbl

[21] J.-P. Demailly, Complex analytic and differential geometry, 2012, 455 pp.\par https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf

[22] G. Dloussky, Structure des surfaces de Kato, Mém. Soc. Math. France (N. S.), 14, Soc. Math. France, Paris, 1984, ii+120 pp. | DOI | MR | Zbl

[23] G. Dloussky, K. Oeljeklaus, M. Toma, “Class $\mathrm{VII}_0$ surfaces with {$b_2$} curves”, Tohoku Math. J. (2), 55:2 (2003), 283–309 | DOI | MR | Zbl

[24] S. Dragomir, L. Ornea, Locally conformal Kähler manifolds, Progr. Math., 155, Birkhäuser Boston, Inc., Boston, MA, 1998, xiv+327 pp. | DOI | MR | Zbl

[25] W. Fischer, H. Grauert, “Lokal-triviale Familien kompakter komplexer Mannigfaltigkeiten”, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1965 (1965), 89–94 | MR | Zbl

[26] A. Fujiki, M. Pontecorvo, “Anti-self-dual bihermitian structures on Inoue surfaces”, J. Differential Geom., 85:1 (2010), 15–72 | DOI | MR | Zbl

[27] A. Fujiki, M. Pontecorvo, “Twistors and bi-Hermitian surfaces of non-Kähler type”, SIGMA, 10 (2014), 042, 13 pp. | DOI | MR | Zbl

[28] A. Fujiki, M. Pontecorvo, “Bi-Hermitian metrics on Kato surfaces”, J. Geom. Phys., 138 (2019), 33–43 ; (2018 (v1 – 2016)), 19 pp., arXiv: 1607.00192v2 | DOI | MR | Zbl

[29] P. Gauduchon, “La 1-forme de torsion d'une variété hermitienne compacte”, Math. Ann., 267:4 (1984), 495–518 | DOI | MR | Zbl

[30] P. Gauduchon, L. Ornea, “Locally conformally Kähler metrics on Hopf surfaces”, Ann. Inst. Fourier (Grenoble), 48:4 (1998), 1107–1127 | DOI | MR | Zbl

[31] H. Grauert, “On Levi's problem and the imbedding of real-analytic manifolds”, Ann. of Math. (2), 68:2 (1958), 460–472 | DOI | MR | Zbl

[32] P. A. Griffiths (ed.), Topics in transcendental algebraic geometry, Ann. of Math. Stud., 106, Princeton Univ. Press, Princeton, NJ, 1984, viii+316 pp. | DOI | MR | Zbl

[33] T. Höfer, “Remarks on torus principal bundles”, J. Math. Kyoto Univ., 33:1 (1993), 227–259 | DOI | MR | Zbl

[34] N. Istrati, A. Otiman, M. Pontecorvo, “On a class of Kato manifolds”, Int. Math. Res. Not. IMRN, 2020, rnz354 ; 2019, 29 pp., arXiv: 1905.03224 | DOI

[35] S. Ivashkovich, “Extension properties of meromorphic mappings with values in non-Kähler manifolds”, Ann. of Math. (2), 160:3 (2004), 795–837 ; (2004 (v1 – 1997)), 37 pp., arXiv: math/9704219 | DOI | MR | Zbl

[36] D. Kaledin, M. Verbitsky, “Non-Hermitian Yang–Mills connections”, Selecta Math. (N. S.), 4:2 (1998), 279–320 | DOI | MR | Zbl

[37] Y. Kamishima, L. Ornea, “Geometric flow on compact locally conformally Kähler manifolds”, Tohoku Math. J. (2), 57:2 (2005), 201–221 ; (2002 (v1 – 2001)), 21 pp., arXiv: math/0105040 | DOI | MR | Zbl

[38] Ma. Kato, “Compact complex manifolds containing “global” spherical shells”, Proc. Japan Acad., 53:1 (1977), 15–16 | DOI | MR | Zbl

[39] Ma. Kato, “Compact complex manifolds containing “global” spherical shells. I”, Proceedings of the international symposium on algebraic geometry (Kyoto Univ., Kyoto, 1977), Kinokuniya Book Store, Tokyo, 1978, 45–84 | MR | Zbl

[40] Ma. Kato, “On a certain class of non-algebraic non-Kähler compact complex manifolds”, Recent progress of algebraic geometry in Japan, North-Holland Math. Stud., 73, North-Holland, Amsterdam, 1983, 28–50 | DOI | MR | Zbl

[41] K. Kodaira, D. C. Spencer, “On deformations of complex analytic structures. III. Stability theorems for complex structures”, Ann. of Math. (2), 71 (1960), 43–76 | DOI | MR | Zbl

[42] A. Lamari, “Courants kählériens et surfaces compactes”, Ann. Inst. Fourier (Grenoble), 49:1 (1999), 263–285 | DOI | MR | Zbl

[43] J. Li, S.-T. Yau, F. Zheng, “On projectively flat Hermitian manifolds”, Comm. Anal. Geom., 2:1 (1994), 103–109 | DOI | MR | Zbl

[44] A. C. López-Martín, “Relative Jacobians of elliptic fibrations with reducible fibers”, J. Geom. Phys., 56:3 (2006), 375–385 ; (2004), 12 pp., arXiv: math/0410394 | DOI | MR | Zbl

[45] D. R. Morrison, “The Clemens–Schmid exact sequence and applications”, Topics in transcendental algebraic geometry (Princeton, NJ, 1981/1982), Ann. of Math. Stud., 106, Princeton Univ. Press, Princeton, NJ, 1984, 101–119 | DOI | MR | Zbl

[46] I. Nakamura, “Classification of non-Kählerian surfaces”, Sugaku Expo., 2:2 (1989), 209–229 | MR | Zbl

[47] I. Nakamura, “On surfaces of class $\mathrm{VII}_0$ with curves. II”, Tohoku Math. J. (2), 42:4 (1990), 475–516 | DOI | MR | Zbl

[48] L. Ornea, M. Verbitsky, “Locally conformally Kähler metrics obtained from pseudoconvex shells”, Proc. Amer. Math. Soc., 144:1 (2016), 325–335 ; (2014 (v1 – 2012)), 13 pp., arXiv: 1210.2080 | DOI | MR | Zbl

[49] U. Persson, On degenerations of algebraic surfaces, Mem. Amer. Math. Soc., 11, No 189, Amer. Math. Soc., Providence, RI, 1977, xv+144 pp. | DOI | MR | Zbl

[50] R. Remmert, “Sur les espaces analytiques holomorphiquement séparables et holomorphiquement convexes”, C. R. Acad. Sci. Paris, 243 (1956), 118–121 | MR | Zbl

[51] V. Rogov, Kähler submanifolds in Iwasawa manifolds, 2018 (v1 – 2017), 18 pp., arXiv: 1710.02180

[52] W. Schmid, “Variation of Hodge structure: the singularities of the period mapping”, Invent. Math., 22 (1973), 211–319 | DOI | MR | Zbl

[53] A.-D. Teleman, “Projectively flat surfaces and Bogomolov's theorem on class $\mathrm{VII}_0$ surfaces”, Internat. J. Math., 5:2 (1994), 253–264 | DOI | MR | Zbl

[54] A. Teleman, “Donaldson theory on non-Kählerian surfaces and class VII surfaces with $b_2=1$”, Invent. Math., 162:3 (2005), 493–521 ; (2007), 29 pp., arXiv: 0704.2638 | DOI | MR | Zbl

[55] A. Teleman, “The pseudo-effective cone of a non-Kählerian surface and applications”, Math. Ann., 335:4 (2006), 965–989 ; (2007), 25 pp., arXiv: 0704.2948 | DOI | MR | Zbl

[56] A. Teleman, “Instantons and curves on class VII surfaces”, Ann. of Math. (2), 172:3 (2010), 1749–1804 ; (2009 (v1 – 2007)), 48 pp., arXiv: 0704.2634 | DOI | MR | Zbl

[57] F. Tricceri, “Some examples of locally conformal Kähler manifolds”, Rend. Sem. Mat. Univ. Politec. Torino, 40:1 (1982), 81–92 | MR | Zbl

[58] I. Vaisman, “A geometric condition for an l.c.K. manifold to be Kähler”, Geom. Dedicata, 10:1-4 (1981), 129–134 | DOI | MR | Zbl

[59] I. Vaisman, “Generalized Hopf manifolds”, Geom. Dedicata, 13:3 (1982), 231–255 | DOI | MR | Zbl

[60] M. Verbitsky, “Pseudoholomorphic curves on nearly Kähler manifolds”, Comm. Math. Phys., 324:1 (2013), 173–177 ; (2012), 6 pp., arXiv: 1208.6321 | DOI | MR | Zbl

[61] M. Verbitsky, “Rational curves and special metrics on twistor spaces”, Geom. Topol., 18:2 (2014), 897–909 ; (2012), 12 pp., arXiv: 1210.6725 | DOI | MR | Zbl

[62] C. Voisin, Hodge theory and complex algebraic geometry, v. I, Cambridge Stud. Adv. Math., 76, Cambridge Univ. Press, Cambridge, 2002, x+322 pp. ; v. II, Cambridge Stud. Adv. Math., 77, 2003, x+351 pp. | DOI | MR | Zbl | DOI | MR | Zbl

[63] V. Vuletescu, “Blowing-up points on l.c.K. manifolds”, Bull. Math. Soc. Sci. Math. Roumanie (N. S.), 52(100):3 (2009), 387–390 | MR | Zbl

[64] V. Vuletescu, LCK metrics on elliptic principal bundles, 2010, 5 pp., arXiv: 1001.0936

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