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Register a new userOn Gauduchon Kahler-like manifolds
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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 Kahler-like, then the Hermitian metric must be Kahler. They also conjectured that if two Gauduchon connections are both Kahler-like, then the metric must be Kahler. 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.
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A special Kahler-Ricci potential on a Kahler manifold is any nonconstant $C^infty$ function $tau$ such that $J( ablatau)$ is a Killing vector field and, at every point with $dtau e 0$, all nonzero tangent vectors orthogonal to $ ablatau$ and $J( ablatau)$ are eigenvectors of both $ abla dtau$ and the Ricci tensor. For instance, this is always the case if $tau$ is a nonconstant $C^infty$ function on a Kahler manifold $(M,g)$ of complex dimension $m>2$ and the metric $tilde g=g/tau^2$, defined wherever $tau e 0$, is Einstein. (When such $tau$ exists, $(M,g)$ may be called {it almost-everywhere conformally Einstein}.) We provide a complete classification of compact Kahler manifolds with special Kahler-Ricci potentials and use it to prove a structure theorem for compact Kahler manifolds of any complex dimension $m>2$ which are almost-everywhere conformally Einstein.
The moduli space NK of infinitesimal deformations of a nearly Kahler structure on a compact 6-dimensional manifold is described by a certain eigenspace of the Laplace operator acting on co-closed primitive (1,1) forms. Using the Hermitian Laplace operator and some representation theory, we compute the space NK on all 6-dimensional homogeneous nearly Kahler manifolds. It turns out that the nearly Kahler structure is rigid except for the flag manifold F(1,2)=SU_3/T^2, which carries an 8-dimensional moduli space of infinitesimal nearly Kahler deformations, modeled on the Lie algebra su_3 of the isometry group.
In this paper, we prove a Liouville theorem for holomorphic functions on a class of complete Gauduchon manifolds. This generalizes a result of Yau for complete Kahler manifolds to the complete non-Kahler case.
In this paper, we consider Hessian equations with its structure as a combination of elementary symmetric functions on closed Kahler manifolds. We provide a sufficient and necessary condition for the solvability of these equations, which generalize the results of Hessian equations and Hessian quotient equations.
We characterise the actions, by holomorphic isometries on a Kahler manifold with zero first Betti number, of an abelian Lie group of dimgeq 2, for which the moment map is horizontally weakly conformal (with respect to some Euclidean structure on the Lie algebra of the group). Furthermore, we study the hyper-Kahler moment map $phi$ induced by an abelian Lie group T acting by triholomorphic isometries on a hyper-Kahler manifold M, with zero first Betti number, thus obtaining the following: If dim T=1 then $phi$ is a harmonic morphism. Moreover, we illustrate this on the tangent bundle of the complex projective space equipped with the Calabi hyper-Kahler structure, and we obtain an explicit global formula for the map. If dim Tgeq 2 and either $phi$ has critical points, or M is nonflat and dim M=4 dim T then $phi$ cannot be horizontally weakly conformal.
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