On metric and cohomological properties of Oeljeklaus-Toma manifolds

• Autores: Danielle Angella, Arturas Dubickas, Alexandra Otiman, Jonas Stelzig
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 68, Nº. 1, 2024, págs. 219-239
• Idioma: inglés
• DOI: 10.5565/PUBLMAT6812409
• Enlaces
  • Texto completo
• Resumen

  • We study metric and cohomological properties of Oeljeklaus–Toma manifolds. In particular, we describe the structure of the double complex of differential forms and its Bott–Chern cohomology and we characterize the existence of pluriclosed (aka SKT) metrics in number-theoretic and cohomological terms. Moreover, we prove that they do not admit any Hermitian metric ω such that ∂∂ω¯ k = 0, for 2 ≤ k ≤ n − 2, and we give explicit formulas for the Dolbeault cohomology of Oeljeklaus–Toma manifolds admitting pluriclosed metrics.

• Referencias bibliográficas
  • D. Angella, M. Parton, and V. Vuletescu, Rigidity of Oeljeklaus–Toma manifolds, Ann. Inst. Fourier (Grenoble) 70(6) (2020), 2409–2423. DOI:...
  • J.-M. Bismut, A local index theorem for non K¨ahler manifolds, Math. Ann. 284(4) (1989), 681–699. DOI: 10.1007/BF01443359
  • E. Bombieri, Letter to Kodaira (1973).
  • O. Braunling, Oeljeklaus–Toma manifolds and arithmetic invariants, Math. Z. 286(1-2) (2017), 291–323. DOI: 10.1007/s00209-016-1763-1
  • G. R. Cavalcanti, Hodge theory of SKT manifolds, Adv. Math. 374 (2020), 107270, 42 pp. DOI: 10.1016/j.aim.2020.107270
  • S¸. Deaconu and V. Vuletescu, On locally conformally K¨ahler metrics on Oeljeklaus–Toma manifolds, Manuscripta Math. 171(3-4) (2023), 643–647....
  • S. Dinew, Pluripotential theory on compact Hermitian manifolds, Ann. Fac. Sci. Toulouse Math. (6) 25(1) (2016), 91–139. DOI: 10.5802/afst.1488
  • A. Dubickas, Nonreciprocal units in a number field with an application to Oeljeklaus–Toma manifolds, New York J. Math. 20 (2014), 257–274
  • A. Dubickas, Units in number fields satisfying a multiplicative relation with application to Oeljeklaus–Toma manifolds, Results Math. 76(2)...
  • A. Fino, H. Kasuya, and L. Vezzoni, SKT and tamed symplectic structures on solvmanifolds. Tohoku Math. J. (2) 67(1) (2015), 19–37. DOI: 10.2748/tmj/1429549577
  • A. Fino and A. Tomassini, A survey on strong KT structures, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 52(100), no. 2 (2009), 99–116.
  • A. Fino and A. Tomassini, On astheno-K¨ahler metrics, J. Lond. Math. Soc. (2) 83(2) (2011), 290–308. DOI: 10.1112/jlms/jdq066
  • J. Fu, Z. Wang, and D. Wu, Semilinear equations, the γk function, and generalized Gauduchon metrics, J. Eur. Math. Soc. (JEMS) 15(2) (2013),...
  • P. Gauduchon, Le theoreme de l’excentricit´e nulle, C. R. Acad. Sci. Paris S´er. A-B 285(5) (1977), A387–A390.
  • D. Grantcharov, G. Grantcharov, and Y. S. Poon, Calabi–Yau connections with torsion on toric bundles, J. Differential Geom. 78(1) (2008),...
  • A. Gray and L. M. Hervella, The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl. (4) 123 (1980),...
  • M. Inoue, On surfaces of Class VII0, Invent. Math. 24 (1974), 269–310. DOI: 10.1007/ BF01425563
  • N. Istrati and A. Otiman, De Rham and twisted cohomology of Oeljeklaus–Toma manifolds, Ann. Inst. Fourier (Grenoble) 69(5) (2019), 2037–2066....
  • S. Ivanov and G. Papadopoulos, Vanishing theorems and string backgrounds, Classical Quantum Gravity 18(6) (2001), 1089–1110. DOI: 10.1088/0264-9381/18/6/309
  • J. Jost and S.-T. Yau, Correction to: “A nonlinear elliptic system for maps from Hermitian to Riemannian manifolds and rigidity theorems in...
  • H. Kasuya, Vaisman metrics on solvmanifolds and Oeljeklaus–Toma manifolds, Bull. Lond. Math. Soc. 45(1) (2013), 15–26. DOI: 10.1112/blms/bds057
  • H. Kasuya, Remarks on Dolbeault cohomology of Oeljeklaus–Toma manifolds and Hodge theory, Proc. Amer. Math. Soc. 149(7) (2021), 3129–3137....
  • M. Khovanov and Y. Qi, A faithful braid group action on the stable category of tricomplexes, SIGMA Symmetry Integrability Geom. Methods Appl....
  • K. Oeljeklaus and M. Toma, Non-K¨ahler compact complex manifolds associated to number fields, Ann. Inst. Fourier (Grenoble) 55(1) (2005),...
  • A. Otiman, Special Hermitian metrics on Oeljeklaus–Toma manifolds, Bull. Lond. Math. Soc. 54(2) (2022), 655–667. DOI: 10.1112/blms.12590
  • A. Otiman and M. Toma, Hodge decomposition for Cousin groups and for Oeljeklaus–Toma manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 22(2)...
  • D. Popovici, Deformation limits of projective manifolds: Hodge numbers and strongly Gauduchon metrics, Invent. Math. 194(3) (2013), 515–534....
  • V. V. Prasolov, Polynomials, Translated from the 2001 Russian second edition by Dimitry Leites, Algorithms Comput. Math. 11, Springer-Verlag,...
  • J. Stelzig, On the structure of double complexes, J. Lond. Math. Soc. (2) 104(2) (2021), 956–988. DOI: 10.1112/jlms.12453
  • J. Stelzig, On linear combinations of cohomological invariants of compact complex manifolds, Adv. Math. 407 (2022), Paper no. 108560. DOI:...
  • J. Streets and G. Tian, A parabolic flow of pluriclosed metrics, Int. Math. Res. Not. IMRN 2010(16) (2010), 3101–3133. DOI: 10.1093/imrn/rnp237
  • A. Tomassini and S. Torelli, On the cohomology of Oeljeklaus–Toma manifolds, Preprint (2015).
  • F. Tricerri, Some examples of locally conformal K¨ahler manifolds, Rend. Sem. Mat. Univ. Politec. Torino 40(1) (1982), 81–92.
  • S. M. Verbitskaya, Curves on Oeljeklaus–Toma manifolds (Russian), Funktsional. Anal. i Prilozhen. 48(3) (2014), 84–88; translation in: Funct....
  • S. Verbitsky, Surfaces on Oeljeklaus–Toma manifolds, Preprint (2013). arXiv:1306.2456v1.
  • V. Vuletescu, LCK metrics on Oeljeklaus–Toma manifolds versus Kronecker’s theorem, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 57(105), no....