Hermitian ranks of compact complex manifolds

Angella, Daniele; Tomassini, Adriano

doi:10.1090/proc/13938

Hermitian ranks of compact complex manifolds
HTML articles powered by AMS MathViewer

by Daniele Angella and Adriano Tomassini PDF

Proc. Amer. Math. Soc. 146 (2018), 2195-2205 Request permission

Abstract:

We investigate degenerate special-Hermitian metrics on compact complex manifolds; in particular, degenerate Kähler and locally conformally Kähler metrics on special classes of non-Kähler manifolds.

References

Daniele Angella, Georges Dloussky, and Adriano Tomassini, On Bott-Chern cohomology of compact complex surfaces, Ann. Mat. Pura Appl. (4) 195 (2016), no. 1, 199–217. MR 3453598, DOI 10.1007/s10231-014-0458-7
Jean-Michel Bismut, A local index theorem for non-Kähler manifolds, Math. Ann. 284 (1989), no. 4, 681–699. MR 1006380, DOI 10.1007/BF01443359
Nicholas Buchdahl, On compact Kähler surfaces, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 1, vii, xi, 287–302 (English, with English and French summaries). MR 1688136
M. Ceballos, A. Otal, L. Ugarte, and R. Villacampa, Invariant complex structures on 6-nilmanifolds: classification, Frölicher spectral sequence and special Hermitian metrics, J. Geom. Anal. 26 (2016), no. 1, 252–286. MR 3441513, DOI 10.1007/s12220-014-9548-4
Ionuţ Chiose, The Kähler rank of compact complex manifolds, J. Geom. Anal. 26 (2016), no. 1, 603–615. MR 3441529, DOI 10.1007/s12220-015-9564-z
Ionuţ Chiose and Matei Toma, On compact complex surfaces of Kähler rank one, Amer. J. Math. 135 (2013), no. 3, 851–860. MR 3068405, DOI 10.1353/ajm.2013.0022
Sergio Console, Dolbeault cohomology and deformations of nilmanifolds, Rev. Un. Mat. Argentina 47 (2006), no. 1, 51–60 (2007). MR 2292941
Sorin Dragomir and Liviu Ornea, Locally conformal Kähler geometry, Progress in Mathematics, vol. 155, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1481969, DOI 10.1007/978-1-4612-2026-8
Anna Fino, Gueo Grantcharov, and Misha Verbitsky, Algebraic dimension of complex nilmanifolds, J. Math. Pures Appl., DOI 10.1016/jmatpur.2017.11.010
Akira Fujiki, On automorphism groups of compact Kähler manifolds, Invent. Math. 44 (1978), no. 3, 225–258. MR 481142, DOI 10.1007/BF01403162
Paul Gauduchon, Le théorème de l’excentricité nulle, C. R. Acad. Sci. Paris Sér. A-B 285 (1977), no. 5, A387–A390 (French, with English summary). MR 470920
Rosa Gini, Liviu Ornea, Maurizio Parton, and Paolo Piccinni, Reduction of Vaisman structures in complex and quaternionic geometry, J. Geom. Phys. 56 (2006), no. 12, 2501–2522. MR 2252875, DOI 10.1016/j.geomphys.2006.01.005
Reese Harvey and H. Blaine Lawson Jr., An intrinsic characterization of Kähler manifolds, Invent. Math. 74 (1983), no. 2, 169–198. MR 723213, DOI 10.1007/BF01394312
Akio Hattori, Spectral sequence in the de Rham cohomology of fibre bundles, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1960), 289–331 (1960). MR 124918
Jürgen Jost and Shing-Tung Yau, A nonlinear elliptic system for maps from Hermitian to Riemannian manifolds and rigidity theorems in Hermitian geometry, Acta Math. 170 (1993), no. 2, 221–254. MR 1226528, DOI 10.1007/BF02392786
Ahcène Lamari, Courants kählériens et surfaces compactes, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 1, vii, x, 263–285 (French, with English and French summaries). MR 1688140
Tian-Jun Li and Weiyi Zhang, Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Comm. Anal. Geom. 17 (2009), no. 4, 651–683. MR 2601348, DOI 10.4310/CAG.2009.v17.n4.a4
Gunnar \thornór Magnússon, Automorphisms and examples of compact non-Kähler manifolds, Math. Scand. 121 (2017), no. 1, 49–56. MR 3708963, DOI 10.7146/math.scand.a-25983
M. L. Michelsohn, On the existence of special metrics in complex geometry, Acta Math. 149 (1982), no. 3-4, 261–295. MR 688351, DOI 10.1007/BF02392356
Dan Popovici, Deformation limits of projective manifolds: Hodge numbers and strongly Gauduchon metrics, Invent. Math. 194 (2013), no. 3, 515–534. MR 3127061, DOI 10.1007/s00222-013-0449-0
Sönke Rollenske, Dolbeault cohomology of nilmanifolds with left-invariant complex structure, Complex and differential geometry, Springer Proc. Math., vol. 8, Springer, Heidelberg, 2011, pp. 369–392. MR 2964483, DOI 10.1007/978-3-642-20300-8_{1}8
Sage Mathematics Software (Version 6.10), The Sage Developers, 2015, http://www.sagemath.org.
S. M. Salamon, Complex structures on nilpotent Lie algebras, J. Pure Appl. Algebra 157 (2001), no. 2-3, 311–333. MR 1812058, DOI 10.1016/S0022-4049(00)00033-5
Hiroshi Sawai, Locally conformal Kähler structures on compact nilmanifolds with left-invariant complex structures, Geom. Dedicata 125 (2007), 93–101. MR 2322542, DOI 10.1007/s10711-007-9140-1
Jeffrey Streets and Gang Tian, A parabolic flow of pluriclosed metrics, Int. Math. Res. Not. IMRN 16 (2010), 3101–3133. MR 2673720, DOI 10.1093/imrn/rnp237
Valentino Tosatti, Non-Kähler Calabi-Yau manifolds, Analysis, complex geometry, and mathematical physics: in honor of Duong H. Phong, Contemp. Math., vol. 644, Amer. Math. Soc., Providence, RI, 2015, pp. 261–277. MR 3372471, DOI 10.1090/conm/644/12770
Luis Ugarte, Hermitian structures on six-dimensional nilmanifolds, Transform. Groups 12 (2007), no. 1, 175–202. MR 2308035, DOI 10.1007/s00031-005-1134-1
Hisao Yoshihara, Structure of complex tori with the automorphisms of maximal degree, Tsukuba J. Math. 4 (1980), no. 2, 303–311. MR 623443, DOI 10.21099/tkbjm/1496159182