On Gauduchon K\"ahler-like manifolds

In a paper by Angella, Otal, Ugarte, and Villacampa, the authors conjectured that on a compact Hermitian manifold, if a Gauduchon connection other than Chern or Strominger is Kähler-like, then the Hermitian metric must be Kähler. They also conjectured that if two Gauduchon connections are both Kähler-like, then the metric must be Kähler. In this paper, we discuss some partial answers to the first conjecture, and give a proof to the second conjecture. In the process, we discovered an interesting ‘duality’ phenomenon amongst Gauduchon connections, which seems to be intimately tied to the question, though we do not know if there is any underlying reason for that from physics.

[1]  Alfred Gray,et al.  Curvature identities for Hermitian and almost Hermitian manifolds , 1976 .

[2]  S. Yau,et al.  Constructing balanced metrics on some families of non-Kahler Calabi-Yau threefolds , 2008 .

[3]  Jixiang Fu.  On non-Kahler Calabi-Yau Threefolds with Balanced Metrics , 2011 .

[4]  Vanishing theorems and string backgrounds , 2000, math/0010038.

[5]  Nicola Enrietti,et al.  A correction to “Tamed symplectic forms and strong Kähler with torsion metrics” , 2012, Journal of Symplectic Geometry.

[6]  Li Jun.  THE EXISTENCE OF SUPERSYMMETRIC STRING THEORY WITH TORSION , 2004 .

[7]  Kefeng Liu,et al.  Hermitian Harmonic maps and non-degenerate curvatures , 2014, 1402.3726.

[8]  D. Popovici.  Limits of Projective Manifolds Under Holomorphic Deformations , 2009, 0910.2032.

[9]  F. Zheng.  Some recent progress in non-Kähler geometry , 2019, Science China Mathematics.

[10]  Complex nilmanifolds and Kähler-like connections , 2019, 1904.09707.

[11]  S. Yau,et al.  GEOMETRIC ANALYSIS , 2005 .

[12]  F. Zheng,et al.  On Bismut flat manifolds , 2016, Transactions of the American Mathematical Society.

[13]  Luigi Vezzoni,et al.  Lie groups with flat Gauduchon connections , 2018, Mathematische Zeitschrift.

[14]  X. Zhou,et al.  Scalar curvatures in almost Hermitian geometry and some applications , 2019, Science China Mathematics.

[15]  Valentino Tosatti.  Non-Kähler Calabi-Yau manifolds , 2014 .

[16]  F. Belgun.  On the metric structure of non-Kähler complex surfaces , 2000 .

[17]  S. Yau,et al.  On Strominger Kähler-like manifolds with degenerate torsion , 2019, Transactions of the American Mathematical Society.

[18]  J. Bismut.  A local index theorem for non Kähler manifolds , 1989 .

[19]  Kefeng Liu,et al.  Geometry of Hermitian manifolds , 2010, 1011.0207.

[20]  A. Fino,et al.  On the existence of balanced and SKT metrics on nilmanifolds , 2015, 1506.05273.

[21]  A. Fino,et al.  A survey on strong KT structures , 2009 .

[22]  Valentino Tosatti,et al.  Gauduchon metrics with prescribed volume form , 2015, 1503.04491.

[23]  On Curvature Tensors of Hermitian Manifolds , 2016, 1602.01189.

[24]  Vanishing theorems on Hermitian manifolds , 1999, math/9901090.

[25]  M. Verbitsky,et al.  Locally conformal Kähler manifolds with potential , 2010 .

[26]  A. Strominger.  Superstrings with Torsion , 1986 .

[27]  P. Gauduchon.  La 1-forme de torsion d'une variété hermitienne compacte , 1984 .

[28]  S. Yau,et al.  The theory of superstring with flux on non-Kähler manifolds and the complex Monge-Ampère equation , 2006, hep-th/0604063.

[29]  L. Ugarte,et al.  Locally Conformal Hermitian Metrics on Complex Non-Kähler Manifolds , 2015, 1505.08007.

[30]  J. Streets.  Pluriclosed Flow and the Geometrization of Complex Surfaces , 2018, Geometric Analysis.